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Schrodinger equation for dummies

"Anyone who is not shocked by quantum mechanics has not probably understood it"

Quantum mechanics for chemists

Matrix mechanics   |   Dirac Equation   |   Quantum electrodynamics    

Theory of relativity

Special theory of relativity   |   General theory of relativity

quantum mechanics for dummies, this page is about quantum mechanics for dummies

Mathematical priliminaries

Kinetic energy
Kinetic energy of a moving body is its mass times velocity squared. So mathematicall it is :
E(ke) = 1/2mv^2 = p^2/2m where p = mv = momentum of the body
When dealing with wave function ψ(r,t) we use Δ operator to find the kinetic energy of the moving particle. We use momentum operator ih(d/dx) to find momentum. There are various other operators which are used in Quantum mechanics. Operators do the same thing as variables (x, p) do in classical physics. One thing that is different is that in Qunatum mechanics operators have expectation values.
Potential energy of a system is another kind of energy which varies due to a change in the configuration of the system. It is a function of position in Newtonian gravity. In electrostatics it is a function of distance R from charged body.
potential energy

The remarkable property of potential energy is that it is conservative. That is to say it is path indepenent. in Classical mechanics mass of charge create potential and its associated force. When you study quantum field theory you will see quantum fields create such kind of potential and forces. In quantum field theory which is somewhat complicated theory due to crazy integrations. Quantum field is like matraces which can be perturbed to create energy in them.

Philosophical thoughts

The atomicity of atom is a hypotheses, as old as the Greeks and no way distasteful to our mental habits. The theory that atom is composed of electron and proton is beautiful throughout its success, but it is not difficult to imagine. It is otherwise, with the form introduced by the theory of quanta. This might not possibly have surprised Pythagoras, but it would definitely have astonished every latter man of science as it have astonished every man of our own day. It is necessary to understand the general principles of the theory before attempting a modern philosophical account of it. But there are still unsolved physical problems connected with the theory , which makes it improbable that any satisfactory philosophical account can yet be constructed. But we must do what we can.
Everybody knows that the revolution began when Plank gave his hypotheses on the observation of black-body radiation. Plank showed that when we consider the vibration of a body , the frequencies of the vibration do not arrange themselves in all possible ways accordinng to usual law of frequency or frequency distribution. But they do so in a certain way. Let e be the energy of any radiation and v be the frequency. Then e/v will always be equal to certain amount, i.e plank constant h, or 2h or 3h and so on. We can not have the ratio (e/v) to be a fraction of plank constant. This happens to be so in all occurences involving quanta. Nobody know why this is the case but it seems to be the fact. There is no reason known for its non-occurrance , which remains , so far, the nature of brute fact. At first it was an isolated fact . But now Plank's constant has been found to be involved in various other kinds of phenomena ; in fact, wherever observation is sufficiently minute to make it possible to discover whether it is involved or not .

black body radiation
A second field for the quantum mechanics was found in the photo-electric effect. This can be explained in the following way:
If high frequency sunlight falls on a metal surface, the electrons bound in the atom of the surface seems to produce a potential at the two far ends of the surface. This phenomena has been shown quite exclusively with photo-electric effect. It is the emission of electron by the incident of the light. The minimum energy of the light which set the electron free from the surface is called the threshold energy of the surface. Below that frequency no electron gets emitted. Einstein discovered this unique phenomena relating to metal surface. He was awarded novel prize for this discovery.
The explanation of the above phenomena is that four-fifth of the energy hv is absoved by the electron , which makes it possible to liberate the electron from the atom. Other one-fifth of hv is absorbed by the atom. So total amount of energy absorbed by the electron and atom is one quanta hv.
The most interesting application of quantum mechanics is the explanation of Bohr's atomic model. Electron jumps from one orbit to another while emitting or absorbing radiation in the form of quanta. When an electron jumps from higher orbit to lower orbit it emits light. When it jump from lower to higher orbit it absorbs radiation. The amount of energy is exactly the same as predicted by Planck's law. It had been found that lines in the hydrozen spectrum which were known had frequencies obtained from the difference of two terms according to the formula :

bohr's radius

Where R is Rayleigh constant and n, k are small integers , usually corresponding to the orbit. It is apparent that the equation above does not say that the line of hydrozen spectrum is not connected to a single orbit where electron involves in periodic vibration. But it correspond to the transition connected with a change from a state defined by an integer to a state defined by another. This also suggests that orbits of electron is not a simple one as described by Newtonian mechanics but orbit designated by integral "quantum number" with a multiple of h.
At this point we do not know what causes electron to jump from one orbit to another. Our knowledge of the atom is only statistical. But we know of course , when an electron jump from lower to higher orbit , electron absorbs energy from incident light. We do not know , in a large number of atoms , of the electrons which are not in minimum orbits , some jump at one time and some other time just as we do not know some radioactive elements disintegrates and some do not. Nature seems to be full of revolutionary occurences as to which we can say that , if they take place, they will be one the many possible kinds, but we can not say they will take place at all or if they will , at what time. This way we enter into the physics which must be modeled with probability.
The probability of an event is the ratio of the number of ways the event can happen to the all possible events that can occur during an experiment. So if x is the number of ways an event A can occur and n is the total possible events then the probability of A confusing is P(A) = x/n .
Previously De Broglie developed his wave-particle duality hypotheses. According to his hypotheses the electron wave wrapped around the nucleus has integer number of wavelengths.
de broglie
All the quantum phenomena that happen are the special case of a more general law known as generalized quantum principle. There are generalized coordinates and momentum associated with them. First thing is to observe that energy is not fundamental entity but action is. Action is energy multiplied by time. Now suppose we have a system that depends upon several coordinates , periodic with respect to each. The coordinates are conditionally periodic. The momentum p(k) associated with the coordinates q(k) will be d(E)/d(q'(k) where q'(k) is the derivative with respect to time. E is the kinetic energy of the system. Now we have the quantum principle
Integral[p(k)dq(k) = nh , where h is planck constant. The motion of the earth around the sun is a system which depends on several coordinates i.e (earth spins on its own axis as well as orbits around the sun). When applied to quantum system we divide the motion into smaller parts and take associated momenta and coordinates. Then we multiply them together and sum each individual bits. In the limit the sum will give total amount which is exactly equal to plank constant h or multiple of h.


bohr's radius

Integral is to be taken over one complete period. There is quantum law which can be derived using this general principle mentioned above. Classical physics thus differs from Quantum physics in a very profound way. What causes the transition from Quantum world to classical world is very hard to find out but a large number of quantum phenomena can average out to give classical behaviour. In the atomic realm space and time , no longer, have usual properties like smoothness and continuity. Space and time appear to have discreteness and discontinuity, at least, inside the atom.
"the man who did the waking buys the man who was sleeping a drink and the man who was sleeping drinks it while listening to a proposition from the man who did the waking.. Savvy"

Quantum mechanics

Mysterious talks of quarks, spin, and cats in boxes have given quantum physics a reputation as the scientific theory that no one really understands. The practical applications of quantum physics are all around us in daily life. Were it not for quantum physics, computers would not function, kettles would not boil, and power stations would not heat our homes.
To many people mathematics represents a significant barrier to understanding of science. Certainly, mathematics has been the language for physics for the last four hundred years and it has become impossible to progress in physics without mathematics. Why is that? it seems like nature is governed by cause and effect of changes. Mathematics is able to describe this relationship of cause and effect very precisely. One example is the differntial law where the law is necessary to be written in differntial equation. For example a man is walking across a road and his dog is in the field beside it. When the man whistles the dog it follows his master. The dog follows a curve which describes the dog's velocity exactly towards his master at each moment of time. This is a kind of curve which can be described by a differential equation. Similarly many laws of nature are necessary to be written in the language of calculus.
"Don not fool yourself, you are the easiest person to fool.."
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the microscopic scales of energy levels of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, describes nature at ordinary (macroscopic) scale. Most formulationas in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics deviates from classical physics in that energy, momentum, angular momentum and other quantities of a bound system are constrained to discrete values (quantization);

Objects have characteristics of both particles and waves (wave-particle duality); and there are limits to the precision with which quantities can be measured (uncertainty principle).
Quantum mechanics gradually arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation catastrophy, and from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was pdeeply re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.


"The best thing about me is that there is so many me's"

Introduction

In the beginning there was continuous flow, and then Max Planck came along and proposed discretization. Quantization basically just implies, that instead of being continuous, things such as EM radiation, can only take multiples of certain values. It’s a little bit like having a tube of smarties. The whole tube represents a beam of light. Inside it you have collection of smarties. You can divide the tube, so you can have less number of smarties in there, or you can get another tube and have smarties, but you have to have a whole number of smarties, because they can’t be split (if anyone e-mails me suggesting I squash/crush/split a smartie, I will hunt them down and make them pay!).
Planck came to this conclusion when solving on the “Ultraviolet Catastrophe”. According to principles of classical electromagnetism, the number of ways an electromagnetic wave can undulates a in a 3-dimensional cavity, per unit frequency, is proportional to the square of the frequency. This means that the power you would get out of per unit frequency should follow the Rayleigh-Jeans law, which again , means that the power would be commensurate to the frequency squared. So if you put the frequency up higher and higher the power would be limitless. Planck said that electromagnetic energy did not follow the classical description. He said that it could only be emitted in discrete packets or chunks of energy proportional to the frequency
E=hω
where \hbar (pronounced “h bar”) is h/(2\pi). These equations mean the the radiation eventually goes to zero at infinite frequencies, and the total power is finite. Planck called these packets of energy “Quanta”. The value of h is 6.626\times 10^{-34} J·s and the value of \hbar is 1.06\times 10^{-34} J·s.

"Science is the manipulation of nature"

Little Packets of Light

If you shine a light onto a metal body for long enough the surface of the will heat up. This must mean that the light is transferring energy to the metal, so in theory it is possible that if you shone a light on a surface for long enough, sufficient energy would be transferred to liberate an electron from an orbit. Even with a weak light you should be able to wait long enough for the energy to build up and an electron to be emitted from atom. So physicists tried the experiment. It failed horribly. For some metals specific light would cause electron separation, for other metals the same light source wouldn’t, no matter how long it was left. And it was found that the electrons came out with higher energies depending on the colour of the light, not the intensity. The problem of the photoelectric effect was resolved in 1905 by Einstein, and was what he won the Nobel Prize for in 1921. Einstein applied Plancks theory of Quantization to light and said that the light is not a continuous stream of energy but rather bundle of little packets of a certain energy value that depended on its wavelength. This explained why no matter how long you left the light on the surface there would be no emission unless the individual photons had enough energy. This also explained why different colours gave the emitted electrons different energy values. The energy was shown to be related to wavelength by Plancks equation. Einstein also showed that the energy of the emitted electrons would be equal to [E = hf - ψ] where φ is the energy needed to get the electron from inside the metal to just outside the surface, and is called the “Work Function”.

Schrodinger equation

Schrodinger equation describes the behavior of subatomic particles which are associated with a wave function Ψ. Wave function Ψ(r,t) is basically function of space and time, which solve Schrodinger equation when some potential energy that influences the particle is given. Schrodinger equation is the quantum mechanical version of newton's force (F=ma) equation. If we know the position of a particle and force acting on it, we can find the position of the particle at another time. This simply involves solving an equation describing the time evolution of the particle with mass m. Similar thing happens in case of Schrodinger's equation. In this scenerio, the solution gives the probability of finding the particle. It also says that energy of the particle associated with the wave function is conserved. The derivation is simple :
First let us derive time-independent Schrodinger equation
Schrodinger equation

We can similarly derive time-dependent Schrodinger equation as follows:

Schrodinger equation
The equation described below is time independent equation where the parameter t is absent. h is the plank constant. Schrodinger equation can be interpreted as an operator known as Hamiltonian (H) that acts on some wave function Ψ(r,t) and produces an corresponding eigenvalue (E) of eigenvector |Ψ(r,t)> . Hamiltonian contains the dynamics of a quantum system. Schrodinger equation is second order linear differential equation. It is not a covariant equaton , i.e it does not hold for relativistic partcle. Space and time are not given equal footing in the equation. So theory of relativity is not consistent with the equation of Schrodinger.


quantum mechanics for dummies

Schrodinger equation
Operator Hamiltonian (H) acts on the wave function and gives the value of an observable E for wave function Ψ(r,t) . Another operator is the momentum operator P which can be defined like :

Schrodinger wave function

Momentum operator thus acts on wavefunction to give observable p(momentum). In quantum mechanics we are always interested in quantities that can be observed. Wave function itself has no physical properties. So it needs to be acted upon by some operator which can give us observable associated with the wave function. This, consequently becomes an Eigenvalue equation where the observable serves as the eigenvalue

Schrodinger equation

It is one of the most successful foundation of modern quantum mechanics that Schrodinger discovered. It is tantamount(equivalent) to Heigenberg's matrix mechanics. The symbol |> enclosing the wavefunction is called ket vector which is a column vector. Bra vector <| is a row vector. This two vector notation is known as braket notation which Dirac introduced. The bra <| or ket |> vector is the representation of state vector i.e |ψ>. A state vector is a state of an isolated quantum system. Mathematical interpretation of state vector is related to the probability of outcome of certain measurement. More specifically state vector assigns probability distribution for the value of each observable( position, momentum). Quantum state is a set of integer, that evolves toward another set of integers denoting another state. That is why states are usually represented by bra or ket vectors. In schrodinger picture the state vector evolves with time.


bra ket notation


quantum mechanics for dummies


"we are middle children of history man, no purpose and no place. We have got no war and no great depression, our great war is the spiritual war. Our great depression is our lives"

The wave function describes the probability of finding a particle at a certain position, which Schrodinger was not aware at the time when he baked his revolutionary equation. Max born interpreted this wave function as the probability that an electron can be found at certain place at a certain time. The squares of modulus of wavefunction represent the apllitude of such probability. Given specific physical condition there is always a finite , though very meager , probability that an electron can be found anywhere in the entire universe. That is only slight craziness that the quantum particle can show. If you are not still shocked , there are many more absurdities to come to make you that. When we perform a measurement we can find electron at specific place. The wave function collapses at the moment of measurement. We force the electron to be in some definite state by doing the measurement.


Schrodinger equation

A typical wave function that solves the Schrodinger wave equation is interpreted above. Each individual state has a probability amplitude |a{j}| (squared) associated with it. Total amplitude is unity.


Schrodinger equation

Schrodinger equation is a linear differential equation. So a linear combination of multiple solutions is also a solution to the wave equation. This creates a superposition of states. A quantum system can exist simultaneously in multiple states. But this seems a bizarre idea which is interpreted in many ways. One of these is the copenhagen Interpretation. According to Copenhagan interpretation quantum system does not possess definite state prior to measurement. It is the measurement that causes the system to have one of many possible states. Another interpretation is the many-worlds interpretation. In this case the quantum system exist simultaneously in different states each of which resides in a unique universe known as parallel universe. A collection of many such parallel universes is called multiverse.


Schrodinger equation

Pauli's exclusion principle

Pauli's exclusion principle asserts that no fermions can be in the same or identical quantum state. Mathematically it can be expressed with wave function ψ

Schrodinger equation explained

"uncertainty is what make quantum physics very interesting.."

Schrodinger Cat


Schrodinger Cat

Thus a wave function representing a particular ket vector can be viewed as a column vector of complex quantity. The wave function is thus a complex quantity.
Schrodinger equation

The wave function cab be thought as an element of Hilbert space of N dimension. There are N number of components in vector representation of wave function. Each has an individual probabilty measure according to quantum mechanical interpretation. And each component corresponds to particular state of the system. Momentum space representation of wave function can be found by taking Fourier transform of wave function over all space. This integrates wave function taken inside three dimensional space. That is why volume integration is used in the following equation. The inverse transform will give the original wave function



Schrodinger wave function
The time evolution operator describes the evolution of quantum state and and is thus a function of time.
Evolution operator

Schrodinger equation
When Schrodinger equation is applied to electron inside atom , the wave function perfectly describes atomic orbitals of Hydrozen.
Schrodinger equation
Schrodinger equation solution This is a separable, partial differential equation which can be solved in terms of special functions. The normalized position wavefunctions, given in spherical coordinates are:

quantum mechanics for dummies

Schrodinger equation

When everything is derived using Schrodinger equation, various eigenstate of hydrozen atom can be found accordingly.
Schrodinger equation
"The only things that matters is what a man can do and what he cannot.."

Some useful quantum mechanics equations

Quantum mechanics equations normally involve bra and ket vectors. This is because this vectors express the states of quantum system. Some equations involving bra and kets are described here:

quantum mechanics for dummies
The first equation describes the scattering of elementary particles. This is a topic of quantum electrodynamics. Second equation is the state tranformation rule involving the operator ( a†) .
Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamics. For generalized coordinates q(i) and two functions f and g , the poisson bracket takes the form :

quantum mechanics poisson bracket

Double slit experiment


"all the mystery of quantum mechanics is rooted in the young's double slit experiment"

Schrodinger equation


In modern physics, the double-slit experiment is a demonstration that light and matter can show characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanical phenomena. The experiment was first executed with light by Thomas Young in 1801. In 1927, Davisson and Germer demonstrated that electrons show the same behavior, which was later extended to atoms and molecules.
Thomas Young's experiment with light was part of classical physics well before quantum mechanics, and the concept of wave-particle duality. He believed it illustrated that the wave theory of light was right, and his experiment is sometimes cited to as Young's experiment or Young's slits. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere, creating bright and dark bands on the screen — a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at separate points, as individual particles (not waves), the interference pattern appearing via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. These results demonstrate the principle of wave–particle duality.

Physics and philosophy

Physics has been a supreme branch of science and engineering. This is probably because a lot of mathematics are needed in physics. Electricity and other forces certainly make physics very interesting. Quantum mechanics raises some profound questions about reality. There was a matter of great controversy around the question "Does the moon exist if no one else is looking at it? ". So is the question "Does a tree in a lonely forest make a sound when no one else is hearing?". Electron and elementary particles do not have definite location. These only have tendency to exist. We can not say with certainty that an electron is located at a particular point in space. They behave like wave. What happens inside an electron? This raises certain problems when we consider Rutherford type electron. Heigenberg's electron removes the difficulties. According to new theory, electron is not in a definite place. An electron is essentially a collection of raditions emitting from the place in question, which are observable at other places than that where the electron would formally have been. These radiations become weak as the distance increases. This reduces electron to a law as to certain occurrances in a region. We can not say electron is a point or a region but having certain properties of different logical type.
Philosophy is good at answering these kinds of puzzling questions about reality. Einstein was a philosopher too. Some other scientists became philosophers at later parts of their lives. Philosophy is the study of world and life at large. It falls in the intermediate way between religion and science. If you want to understand you have to be some kind of a philosopher. Philosophy has the freedom of speculation. Philosophy is what we do not know contrary to science which means what we know.

Hartree-fock equation

Hartree-fock equation is the equation involving nucleus-nucleus repulsion along with contribution from electron to electron repulsion energy.
Harfree-fock equation
The equation can be broken down to interpret various terms entering into it.
Harfe - fock equation
It is basically Schrodinger equation for multi-particle system.
Quantum theory of atom

An atom is composed of electrons and protons, the latter being inside the nucleus, the former partly in the nucleus (except in hydrogen) , partly planetary. The number of the protons in nucleus gives the atomic number. When the atom is unelectrified the number of planetary electron is the same as the atomic number. If the quantum theory is correct , an atom has ceratin number of characters, each measured by integers called quantum numbers, which are allways small. It has also a property called energy, which is a function of the quantum numbers; and in connection with each of the quantum numbers there is periodic process which is subject to quantum rules. Each quantum number is capable of changing suddenly from one integer to another integer. When the atom is left to itself, the change will be such as to diminish energy. But when it recieves energy from outside the change may increase the energy. All this , however, is more or less hypothetical. What we really know is the interchange of energy between the atom and the surrounding medium; here there are simple laws as to form the radiant energy will take (classical laws). There are at present no definite laws determining when quantum change will take place in the atom.
The theory of atom had made it possible to develop periodic table where all the elements are arranged in specific order. The position of the elements depends on the number of electron they contain. These elements take part in chemical reactions and form various kinds of other compounds and mixtures. The chemical properties of the atoms can be deduced using quantum mechanical principles. A relatively new branch called "molecular quantum mechanics" has been developed. All the elements and compounds are basically replica of simple structure like hydrogen or helium. Combining multiple numbers of hydrogens and heliums we can create other atoms. The only difference lie in the number of protons, neutrons and electrons. What are these electron, proton and neutrons made of? These , in turns, are composed of quarks , gluons and other tiny particles which particle physics deal with.
How did atom come into existence in the first place? How did nature make a well-organized, symmetric and miniscule structure like atom? This is the question which should probably be addressed in TOE(theory of everything). All the three forces are in effect in the atom. If we analyze quantum characteristics inside the atom we find these fundamental forces there. In nucleus strong and weak nuclear forces exist and between proton and planetary electron electromagnetic force works. But in principle gravitational force also exist inside the atom. Proton, electron and neutron have certain mass. All the forces exist inside the atom. Atom is a tiny universe so to speak.

quantum computing

Quantum computing is based on the superposition principle of quantum mechanics. In classical computing bits are used. A bit is either 0 or 1. So in a state of classical computation these bits can take only one value (either 0 or one 1). But in quantum computation these bits can be both 1 and 0 simultaneously. These are called qubits. So qubits are the quantum version of bits. It still remains the digits 0 and 1 but the computation power increases as exponentially as the number of qubits increases. For 2 qubits the number of available states is 2^2 = 4 and for 3 it is 2^3 = 8 and so on. This follows from the principle of superposition. As the qubits can be represented with complex numbers They live in complex space of a number of dimension. The number of dimensions increases as the number of qubits increases. This kind of space is hard to visialize. A typical qubit can be represented with a Bloch Sphere
bloch sphere
Wave function can be resolved into superposition o f two states as
bloch sphere
You can put any value for θ = 90 and φ = 90 to check unique combination of the states that make up the original wave function |ψ>. In tandem with the development of quantum computing the concept of quantum computer and quantum internet have sprung. In quantum internet everything happens in quantum level. Although there are some disadvantages with quantum computing , a lot of money and research have been invested to create first quantum computer. Everybody is now on a race to win quantum supremacy.

Youmg's double slit experiment

All the mystery of quantum mechanics can be found in Young's double slit experiment as follows.
young double slit experiment

Quantum tunnelling

Another bizzare consequences of quantum mechanics is quatum tunnelling. A classical object can not pass through a rigid barrier but an electron has always some chance to be found on the other side of the a potential barrier.


quantum tunnelling
"time is the most precious commodity in the whole universe"

quantum entanglement

Quantum entanglement is a physical phenomena which occurs when pairs or groups of particles are produced or interact or share spatial proximity in a way such that the measurement of states of each particle can not determined independently of the state of others. In such a situation the particles are said to be entangled. quantum pure states are rays or elements of Hilbert space (H). Hilbert space is a generalized Euclidean space where inner product between two elements are well defined. A point in Hilbert space can be seen as a state of vibrating string , which can be projected on to coordinate axes representing different overtones or harmonics. In general Hilbert space is the infinite dimensional analog of Euclidean space. Pure states or rays are usually represented by ket vector. Particles in these pure state can be separable. But there are states where such separation is not possible. These states are entangled states which can be modelled mathematically.
Quantum entanglement should better be classified as an example of non-locality. Classical physics is always concerned with local measurement or it only enables us to compare local quantities. Non-locality violates causality . This is weird but non-locality seems to be inherent in quantum mechanics.
quantum entanglement
The states of the composite system can not be factored into prime states corresponding to individual Hilbert space. This is shown as the non-equality between density matrix of the mixed state with the tensor product of the density matrices of individual states.
This quantum entanglement permits faster than light communication which is also known as spooky action at a distance. Suppose two electrons are entangled so that when we measure one electron's spin then the spin of other is determined with certainty. Spin states can be either up or down. So when we fix one of this , the other is also fixed. This will happen no matter how far apart the two electrons are. This is quite mysterious yet experimentally verified.


Bell's inequality

Bell's inequality was developed using the assumption than reality is local. By locality it is to mean that nothing can affect other thing very far apart instantaneously. EPR paradox devised by Einstein , Podolsky and Rogen is the indication of the violation of local reality. Quantum entanglement shows that something travels faster than light to convey information. John Stewart Bell tried to explain the EPR paradox and distinguish between classical realism and this spooky action at a distance. He developed a mathematical theorem which he thought could explain this dilemma. Formally his theorem states
"no physical thoery based on hidden varibles can explain all the predictions of quantum mechanics". Bell's theorem is an inequality. Here is the actual interpretation :

Bell's inequality

The result of measurement of three test A , B and C having outcome zero(0) or one(1) must follow an inequality when expressed in terms of probability. But in context of quantum mechanics this inequality is violated. This is an indication of the fact that either local realism is false or quantum mechanics is incomplete. This is the significance of Bell's inequality or Bell's theorem. This further proves that something travels faster than light to maintain a consistent communication. Bell's inequality is the hidden variable theory..

Continuity equation and flow of probability

This is similar to continuity equation in which mass or current is conserved.
Continuity equation

Quantunglement

I must make it very clear that I am not trying to give support to the idea that ordinary information can be propagated backwards in time (nor can EPR effects be used to send classical information faster than light; see later). That kind of thing would lead to all sorts of paradoxes that we should have absolutely no truck with . Information, in the ordinary sense, cannot travel backwards in time. I am talking about something quite different that is sometimes referred to as quantum information. Now there is a difficulty about this term, namely the appearance of the word ‘information’. In my view, the prefix ‘quantum’ does not do enough to soften the association with ordinary information, so I am proposing that we adopt a new term for it:
QUANGLEMENT
At least in this book, I shall refer to what is commonly called ‘quantum information’ as quanglement. The term suggests ‘quantum mechanics’ and it suggests ‘entanglement’. This is very appropriate. This is what quanglement is all about. Quanglement also does have something very much to do with information, but it is not information. There is no way to send an ordinary signal by means of quanglement alone. This much is made clear from the fact that past-directed channels of quanglement can be used just as well as future-directed channels. If quanglement were transmittable information, then it would be possible to send messages into the past, which it isn’t. But quanglement can be used in conjunction with ordinary information channels, to enable these to achieve things that ordinary signalling alone cannot achieve. It is a very subtle thing. In a sense, quantum computing and quantum cryptography, and certainly quantum teleportation, depend crucially on the properties of quanglement and its interrelation with ordinary information.
As far as I can make out, quanglement links are always constrained by the light cones, just as are ordinary information links, but quanglement links have the novel feature that they can zig-zag backwards and forwards in time, so as to achieve an effective ‘spacelike propagation’. Since quanglement is not information, this does not allow actual signals to be sent faster than light.
quantunglement

Parametric down-conversion. A photon, emerging from a laser, impinging upon a suitable 'non-linear crystal', produces a pair of entangled photons. This entanglement manifests itself in the EPR nature of the correlated polarization states of the secondary photons, but also in the fact that their 3-momentum states must sum to that of the incident photon.
There is also an association between quanglement and ordinary spatial geometry being spatially reXected at a reversal of time direction, with interesting implications. It would take us too far aWeld to explore these in detail One of the most direct uses of the idea of quanglement is in certain experiments where a pair of entangled photons is produced according to the process referred to as parametric down-conversion. This occurs when a photon, produced by a laser, enters a particular type of (‘non-linear’) crystal which converts it into a pair of photons. These emitted photons are entangled in various ways. Their momenta must add up to the momentum of the incident photon, and their polarizations are also related to one another in an EPR way, like the examples given earlier, above.
In one particularly striking experiment, one of the photons (photon A) passes through hole of a particular shape as it speeds towards its detector DA. The other photon (photon B) passes through a lens that is positioned so as to focus it, appropriately, at its detector DB. The position of detector DB is moved around slightly as each photon pair is emitted. The situation is illustrated schematically in Figure. Whenever DA registers reception of photon A and DB also registers reception of B, the position of DB is noted. This is repeated many times, and gradually an image is built up by the detector DB, where only the positions of B are counted when simultaneously DA registers. The shape of the hole that A encounters is gradually built up at DB, even though photon B never directly encounters the hole at all! It is as though DB 'sees' the shape of the hole by looking backwards in time to the emission point C at the crystal, and then forwards in time in the guise of photon A. It can do this because the 'seeing' process in this situation is achieved by quanglement. This flitting back and forth in time is precisely the kind of thing that quanglement is allowed to do. Even the strength and positioning of the lens can be understood in terms of quanglement. To obtain the lens location, think of a mirror placed at the emission point C. The lens (a positive lens) is placed so that the image of the hole, as reXected in this mirror at C, is focused at the detectorDB. Of course there is no actual mirror at C, but the quanglement links act as though reflected at a mirror, but they are reflected in time as well as space


quantunglement

Transmission of an image via quantum effects. (a) Entangled photons A, Bare produced by parametric down-conversion at C. Photon A has to pass through a hole of some special shape to reach detector DA, while B passes through a lens, positioned so as to focus it at detector DB. Detector positions are gradually moved, appropriately in conjunction, and when they both register, the position of DB is noted. Repeated many times, an image of the hole shape is gradually built up by DB, where only those positions of B are counted whenDA also registers. (This is schematically illustrated here by having, instead, DB as a fixed photographic plate that is only activated when DA registers.) Quanglementis illustratedbythe lens positioning being determined as though C were a ‘mirror’ that reflects the photon backwards in time as well as in direction. (b) An alternative scheme using an adaption of the Elitzur– Vaidman bomb test (which is to be reflected in a horizontal line). The photographic plate at B receives the photon only when the photon ‘would have been stopped’ by the template at C, but actually took the lower route!
In case the reader finds this experiment far-fetched, I should make clear that this is a real effect. It has been successfully conWrmed in experiments18 performed at the University of Baltimore, Maryland. Various other related experiments involving parametric down-conversion, which can be best understood in terms of quanglement, have also been carried out. On the other hand, the general type of situation illustrated in Figure above might be regarded as not being 'essentially quantum mechanical'. For one could envisage a device at C which simply ejects classical particles pairwise in the appropriate directions and, apart from the lensing, similar results could be obtained. We can remedy this by using a modification of the Elitzur– Vaidman set-up. Now there is only one photon at a time. It can register at the photographic plate B only if the interference is destroyed when the alternative route for the photon would miss the hole at C.
Now,let us look again at an ordinary EPR effect, like the Stapp and Hardy examples considered earlier. In the ordinary application of the quantum R process, one imagines a particular reference frame in which there is a time coordinate t providing parallel time slices, each corresponding to a constant t value, through the spacetime. The normal procedure is to adopt the (nonrelativistic) viewpoint that, when one member of an EPR pair is measured, the state of the other is simultaneously reduced, so that a later measurement looks at a reduced (unentangled) state rather than an entangled state. This kind of description can be used, for example, in my specific EPR examples. Let us suppose that, from the point of view of a reference frame stationary with respect to the Sun, it is my colleague on Titan whose measurement takes place Wrst, some 15 minutes before my own measurement here on Earth. So, in this picture of things, it is my colleague’s measurement that reduces the state, and I subsequently perform a measurement on a particle with an unentangled state. But we might imagine that, instead, the whole situation is described from the perspective of some observer O passing by at great speed (say 2/3c) in the general direction from my colleague on Titan to me. From O's viewpoint, I was the one who First made the measurement on the EPR pair, thereby reducing the state, and it was my colleague who measured the reduced unentangled state). The joint probabilities come out the same either way, but O has a different picture of 'reality' from the one that I and mycolleague had before. If we think of R as a real process, then we seem to be in conflict with the principle of special relativity, because there are two incompatible views as to which of us


quantunglement

effected the reduction of the state and which of us observed the reduced state after reduction. We may deduce from this that EPR effects, despite their seemingly acausal nature, cannot be directly used to transmit ordinary information acausally, which one might imagine could influence the behaviour of a receiver at spacelike separation from the transmitter. A reference frame can always be chosen in which it is the ‘reception event’ which occurs first, and the ‘transmitter’ then has only the reduced state to examine. It is ‘too late’, by then, for the entanglement to be used for a signal because it has already been destroyed by the state reduction. What is the quanglement perspective on these matters?. On this picture, it is not correct to think of either measurement (mine or my colleague’s) as effecting the reduction and the other (my colleague’s or mine) as measuring the reduced state. The two measurement events are on an equal footing with one another, and we think of the quanglement as providing a connection between these events which correlates the two. It makes no difference which event is viewed as being to the past of the other, for quanglement can equally be thought of as propagating into the past as propagating into the future. Not being capable directly of carrying information, quanglement does not respect the normal restrictions of relativistic causality. It merely effects constraints on the joint probabilities of the results of different measurements.

quantum mystery


Schrodinger equation

Sometimes, if you want to understand how nature truly works, you need to break things down to the simplest levels imaginable. The macroscopic world is composed of particles that are — if you divide them until they can be divided no more — fundamental. They experience forces that are determined by the exchange of additional particles (or the curvature of spacetime, for gravity), and react to the presence of objects around them.
At least, that's how it seems. The closer two objects are, the greater the forces they exert on one another. If they're too far away, the forces reduces to zero, just like your intuition tells you they should. This is called the principle of locality, and it holds true in almost every instance. But in quantum mechanics, it's violated all the time. Locality may be nothing but a persistent illusion, and seeing through that facade may be just what physics needs.

Quantum gravity

Imagine that you had two objects located in close proximity to one another. They would attract or repel one another based on their charges and the distance between them. You might picture this as one object generating a field that affects the other, or as two objects exchanging particles that impart either a push or a pull to one or both of them. You'd expect, of course, that there would be a speed limit to this interaction: the speed of light. Relativity gives you no other way out, since the speed at which the particles responsible for forces propagate is constrained by the speed they can travel, which can never exceed the speed of light for any particle in the Universe. It seems so straightforward, and yet the Universe is full of surprises.

light cone

We have this notion of cause-and-effect that's been hard-wired into us by our experience with reality. Physicists call this causality, and it's one of the rare physics ideas that actually conforms to our intuition. Every observer in the Universe, from its own point of view, has a set of events that exist in its past and in its future. In relativity, these are events contained within either your past light-cone (for events that can causally affect you) or your future light-cone (for events that you can causally effect). Events that can be seen, perceived, or can otherwise have an effect on an observer are known as causally-connected or causal. Signals and physical effects, both from the past and into the future, can propagate at the speed of light, but no faster. At least, that's what your intuitive notions about reality tell you.

schrodinger cat

But in the quantum mechanical Universe, this notion of relativistic causality isn't as straightforward or universal as it would seem. There are many properties that a particle can have — such as its spin or polarization — that are fundamentally indeterminate until you make a definite measurement. Prior to observing the particle, or interacting with it in such a way that it's forced to be in either one state or the other, it's actually in a superposition of all possible outcomes.
Well, you can also take two quantum particles and entangle them, so that these very same quantum properties are linked between the two entangled particles. Whenever you interfere with one member of the entangled pair, you not only gain information about which particular state it's in, but also information about its entangled partner.

quantum entanglement

This would not be so bad, except for the fact that you can set up an experiment as follows. 1.You can create your pair of entangled particles at a particular location in space and time.
2.You can transport them an arbitrarily large distance apart from one another, all while maintaining that quantum entanglement.
3. Finally, you can make those measurements (or force those interactions) as close to simultaneously as possible.
In every instance where you do this, you'll find the member you measure in a particular state, and instantly "know" some information about the other entangled member.
In every instance where you do this, you'll find the member you measure in a particular state, and instantly "know" some information about the other entangled member.
What's puzzling is that you cannot check whether this information is true or not until much later, because it takes a finite amount of time for a light signal to reach from the other member. When the signal does arrive, it always confirms what you'd known just by measuring your member of the entangled pair: your expectation for the state of the far-removed particle agreed 100% with what its measurement indicated.

circular polarization
Only, there seems to be a problem. You "knew" information about a measurement that was taking place non-locally, which is to say that the measurement that happened is outside of your light cone. Yet somehow, you weren't entirely unknowledgeable about what was going on over there. Even though no information was transmitted faster than the speed of light, this measurement describes a troubling truth about quantum physics: it is fundamentally a non-local theory.
circular polarization
There are limits to this, of course.
It isn't as clean as you want: measuring the state of your particle doesn't tell us the exact state of its entangled pair, just probabilistic information about its partner.
There is still no way to send a signal faster than light; you can only use this non-locality to predict a statistical average of entangled particle properties.
And even though it has been the dream of many, from Einstein to Schrödinger to de Broglie, no one has ever come up with an improved version of quantum mechanics that tells you anything more than its original formulation. But there are many who still dream that dream.
quantum entanglement

One of them is Lee Smolin, who cowrote a paper way back in 2003 that showed an fachinating link between general ideas in quantum gravity and the fundamental non-locality of quantum physics. Although we don't have a successful quantum theory of gravity, we have established a number of important properties concerning how a quantum theory of gravity will behave and still be consistent with the known Universe.
When you attempt to quantize gravity, by replacing the concept of curved spacetime with an exchange of particles that mediate the gravitational force, huge violations of locality will arise. If you look at the consequences of those violations — which Smolin and his coauthor, Fotini Markopoulou, did — you find that they are capable of explaining the non-local behaviors of quantum mechanics via new, non-local, non-observable variables.


double slit experiment
Imagine that you pass an electron through a double slit. If you don't measure which slit it goes through, you wind up concluding that it must pass through both slits simultaneously, interfering with itself as it does. That's how you get the interference pattern on the screen behind it. But then, you ask, what happens when you try and measure its gravitational field?
gravitational field

Does the gravitational field display an interference pattern? Or does it follow a single particle-like trajectory, passing through one slit alone?

Stern-Gerlach experiments

"When we measure something we are forcing an undetermined, undefined world to assume an experimental value. We are not measuring the world, we are creating it."
You have to realize that this is something very subtle, but that is indisputable. There are experiments you can perform that show you the world behaves differently if you do or don't measure it.
For example, there's an experiment you can do called the Stern-Gerlach experiment, where you put an electron through a magnetic field oriented in a particular direction. This could be along, say, the x-axis. Electrons that spin in one direction will split in the positive direction, electrons that spin in the other direction deflect in the negative direction.
The act of determining this result along the x-axis destroys any information along the y-axis or z-axis. If you set up another Stern-Gerlach experiment in the x-axis, the particles that deflected positively will still deflect positively; those that deflected negatively will still deflect negatively.
But if you throw in another experiment in the y-direction, for instance, you'll not only see a split in that new direction, you'll destroy any information about the x-direction. It's messed up, but it's experimentally real. An illustration between the inherent uncertainty between position and momentum at the quantum level. There is a limit to how well you can measure these two quantities simultaneously, and uncertainty appear in places where people often least expect it.

uncertainty principle

this is another aspect of quantum physics that is very real: fundamental quantum uncertainty. There are certain combinations of properties that cannot be simultaneously known to better than a certain accuracy, combined. Position and momentum, energy and time, or even (as we just illustrated) spin in two mutually perpendicular directions, cannot be known to arbitrary accuracy. "Why is it this way?" We don't know! That's the problem: there's no governing principle that underlies it. This is the principle.

How do measurements collapse quantum wave functions?

In the strange realm of electrons, photons and the other fundamental particles, quantum mechanics is law. Particles don't behave like tiny balls, but rather like waves that are spread over a large area. Each particle is described by a "wavefunction," or probability distribution, which tells what its location, velocity, and other properties are more likely to be, but not what those properties are. The particle actually has a range of values for all the properties, until you experimentally measure one of them — its location, for example — at which point the particle's wavefunction "collapses" and it adopts just one location. [Newborn Babies Understand Quantum Mechanics]
But how and why does measuring a particle make its wavefunction collapse, producing the concrete reality that we perceive to exist? The issue, known as the measurement problem, may seem esoteric, but our understanding of what reality is, or if it exists at all, hinges upon the answer.

Superposition happens when the resultant wave function which is a linear combination of all different solutions of Schrodinger equation spreads across the space. It has a mathematical expression given by
superposition principle

Schrodinger equation is a linear differential equation meaning that a linear combination of different solutions will result in a new solution of the same equation.
This is why quantum mechanics and gravity do not make themselves friendly. Schrodinger equation is not consistent with theory of general relativity.
quantum gravity consistency

Berry phase and geometric phase

Geometric phase is the phase that is acquired over a course of a cycle when a system undergoes an adiabatic process. This phase results from geometric properties of the parameter space of Hamiltonian. Now adiabatic theorem in quantum mechanics states that
When a given pertubation acts on a quantum system slowly and there is a gap between the eigenvalue and rest of the spectrum of Hamiltonian, the system remains in its simultaneous eigenstate.
Now proof of adiabatic theorem and definition of Berry phase can be given
A time-dependent Schrodiner equation
time dependent schrodinger equation
can be converted into time-independent Schrodinger equation as
time independent schrodinger equation
The general solution of the Schrodinger equation ,as separatio of variables , can be written as
general solution of Schrodiner equation
This indicates the the particle in nth eigenstate remain in nth eigenstate by picking up a phase factor E(n)t/h.
In adiabatic process hamiltonian is time dependent . Hamiltonian changes with time t. This is not the same as perturbation as in pertubation the hamiltonian changes with small amount. As the hamiltonian changes with time the eigenstate and eigenvalue is time dependent too.
eigenstate and eigenvalue time dependent --- (a)
But at a particular instant of time the eigenstate still forms a complete orthogonal system
eigenstate and eigenvalue time dependent
The postion dependence is tacitly assumed as the dependence on time will be more concerned. The ψ(r,t) will be regarged as the state at time t no matter in what position the state is . The general solution of the time dependent Schrodinger equation is then
general solution time dependent Shrodinger equation
The parameter θ(t) will be called dynamic phase factor. Substituting this into the Schrodiner equation another equation for the variation of the coefficients can be obtained
variation of coefficients
Note that we have called d/dt(θ(n)) the variation of coefficients. Now d/dt(θn) gives -E(n)/h . So the third terms cancels out with the right hand side leaving us with
variation of coefficients
Now taking the inner product with an arbitrary eigenstate <ψ(t)m| , the <ψ(t);m|ψ(t)n> gives δmn , which is one for m =n and otherwise vanishes. We are left with the identity
variation of coefficients
By calcululating the expression for <ψ(t)m|ψ(t)n> by differentiating the modified time-dependent Schrodinger equation (a) we get
variation of coefficients
The second term is oscillatory and over a long period of time contribtion from this term cancels out leaving us with
variation of coefficients
Where the parameter &tao; = t1 - t0 ( difference of time holding t0 to be initial time). In the equation above the term
geometric phase
is called geometric phase and is a real number since <ψ(t`)|d/dt[ψ(t`)]> is a pure imaginary number. Putting this expression for the coefficient Cm(t) back into the equation of nth eigenstate yields
geometric phase
So for an adiabatic process, a particle starting from nth eigenstate remains in the nth eigenstate but it only picks up a few phase factors. The new pahse factor γ(t) can be cancelled out by appropriate choice of gauge for the eigenfunctions. However if the adiabatic process is cyclic the pahse factor γ(t) become an gauge invaraint physical quantity known as The Berry phase.

Perturbation theory

Perturbation is very important method in solving quantum mechanical problems. The first order perturbation theory can be given
first order perturbation theory
To find soltion to perturbation method we break down hamiltonian into two parts : one part which solution we know and other part which solution we need to find. Consider this example
first order perturbation theory

Particle in 3D box

We shall now consider the more realistic problem of a particle confined in three dimensions. If the particle has definite energy E, its wave function has the form
particle in 3D box
where c(x, y, z) satisfies the energy eigenvalue equation
particle in 3D box---1
We shall choose a potential energy function
particle in 3D box
The possible energy eigenfunctions and eigenvalues of the particle may be found by seeking solutions of Eq. (1) inside the box which are equal to zero on all six faces of the box. For example, the function
particle in 3D box
is zero on each of the faces defined by x = 0, x = a, y = 0, y = b, z = 0 and z = c, and it satisfies Eq. (4.41) inside the box where V(x, y, z) = 0, if
particle in 3D box
In general, there are an infinite set of eigenfunctions and eigenvalues labelled by three quantum numbers n_x = 1, 2, 3, . . . , n_y = 1, 2, 3, . . . , and n_z = 1, 2, 3, . . . . The eigenfunctions have the form
particle in 3D box
and the energy eigenvalues are given by
particle in 3D box
Last equation shows how the quantized energy levels of a particle in a box depend upon the dimensions of the box, a, b and c.
Electron energy as the distance from the nucleus increases.
electron's orbital energy
Possible values of quantum numbers.
possible values of quantum numbers

Scattering cross section

In aiming a beam of particles at a target which is much smaller than the beam, as in the Rutherford scattering experiment, the cross section takes on a statistical nature. The accurate definition of cross section would then be evaluated using a large number of particles which are scattered and deflected.
possible values of quantum numbers
Where σ is the cross section and &ro; is the density.

Hilbert space and wave function

A particularly important basis is the (infinite-dimensional) basis of eigenstates of the position operator. In that case, we get:
wave function eigenbasis
We can regard modulus of wave function squared as the probability distribution. In that case we get
wave function eigenbasis
The coefficients of the expansion of the quantum state ψ into the position eigenstates give the values of the quantum wave function. The probability to find the particle at some time t inside a small interval dq centered on q is given by
wave function eigenbasis
One can use any set of basis vectors to represent operators in a Hilbert space. Consider, for example, the basis of eigenstates of the position operator. How can we represent the momentum operator on this basis? It is not difficult to show that (see Fradkin for details): Schrodinger's equation again
schrodinger equation
Surprisingly, the Schrödinger equation is not restricted to nonrelativistic quantum mechanics (as it is usually thought). In quantum field theory, for example, where coordinates are replaced by field configurations and the wave function is replaced by a wave functional, the Schrödinger equation becomes, for the simple case of a scalar field ϕ(x):
schrodinger equation  in qft
where K in the integrand is a function of the spatial coordinates (see Mukhanov and Winitzki for more details):
schrodinger equation  in qft
As an aside, the only fundamental quantum scalar field that was experimentally found was the famous Higgs field.
The von Neumann equation of several pure states ψ_1, ψ_2, ψ_3.. is written in terms of the density operator as follows
schrodinger equation  in qft
A qoute of Feynman about the origin of Schrodinger equation is famous :
Where did we get [the Schrödinger’s equation] from? Nowhere. It’s not possible to derive it from anything you know. It came out of the mind of Schrödinger, invented in his struggle to find an understanding of the experimental observations of the real world.
— Richard Feynman
wave function can be written in terms of probability density &ro;
schrodinger equation  in qft
Short wave length limit
short wave length limit
In the present context, the meaning of “short-wavelength” is that the particle's quantum wavelength is much smaller than the typical distance over which the potential acting on the particle varies. Collecting terms which do not contain the Planck h we obtain:
short wave length limit
The short-wavelength limit (with h → 0) of the Schrödinger equation is the (classical) Hamilton-Jacobi equation.

Deriving Schrodinger equation via Feynman path integral

Hence, according to Feynman, the probability amplitude for a system to make the Transition from the initial position I to the final position F via a path j is
scort wave length limit
The transition amplitude is obtained by summing over all possible paths connecting the initial and final points:
scort wave length limit
The propagator K, given by the sum of exponentials of the classical action for all possible paths between I and F.
As first noted by Dirac, using this formulation of quantum mechanics one can derive all of classical mechanics (via the Principle of Least Action and the Euler-Lagrange equations) by setting h → 0.
First, divide each path into discrete infinitesimal segments as shown in the figure below
scort wave length limit
The propagator K obeys the following composition property:
scort wave length limit--1
The discrete version of Feynman’s propagator K for two infinitesimally close spacetime points is given by the expression:
scort wave length limit--2
Inserting Eq. 2 into Eq.1, making the replacements x_N -> x and t_N = t + ∇t ;
and then changing the integration variable from x to ξ one obtains:
scort wave length limit--3
Since Δt is infinitesimal, the exponential is highly oscillating except where ξ≈0. We can, therefore, expand the integrand in powers of ξ. We can also expand both sides in powers of Δt. After the expansion, the left-hand side of Eq. 3 reads:
scort wave length limit--4
The hand-hand side becomes:
scort wave length limit--5
Note that the expansion above is quadratic in ξ since the integration containing the term linear in ξ vanishes. Finally, after collecting terms linear in Δt and performing a trivial Gaussian integral over ξ one gets the following equation for the Feynman’s propagator K:
short wave length limit--6
Now, one can express the Schrödinger wave function as follows:
short wave length limit--7
where K, in this case, is the propagator in Schrödinger’s wave mechanics, which is not necessarily the same object as the K in other equation 6. However, if both Ks are the same object, we can quickly see that ψ in Eq. 7 indeed satisfies Schrödinger’s equation, as it should:
short wave length limit
Wave functions must be square-integrable since the total probability to find the particle anywhere in the Universe must be one:
short wave length limit
Time derivative of total probability to find electron anywhere in the universe.
short wave length limit
where the substitution z = i is justified by assuming this condition
short wave length limit

Stueckelberg’s Real Quantum Mechanics

Some built mathematical framework of quantum mechanics using real hilbert space. An extra, super-selection rule was necessary to obtain the correct quantum-mechanical uncertainty relations. According to this super-selection rule, all observables should commute with an operator J that obeys:
J^T = I and j^2 -I. It has two implications, as pointed out by Jauch: The operator J can always be represented as
real hilbert space--8

Reference materials:

Mathematical Quantum physics for layman (pdf)
Mathematical Quantum mechanics for student (very interesting)
Mathematical Quantum mechanics by feynman
Mathematical Grand Design by Stephen Hawking
Mathematical Higher Engineering Mathematics ( PDFDrive.com ).pdf
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Quantum theory in a nutshell

All the equations of quantum theory in one package.

quantum theory equations

Border line between quantum mechanics and classical mechanics

Some general properties can be attributed to classical and quantum mechanical world.

quantum theory equations

Quantum field theory is the combination of special relativity and quantum mechanics. It is also called relativistic quantum mechanics. The field quantization in quantum field theory like quantum electrodynamics is known as second quantization. It is done through some mathematical operations as follows:
quantum field theory equations
Everything in quantum field theory involves partial differentiation of field φ(x,t). By analogy with Heisenberg's uncertainty relation commutation relation of field and its associated momentum is derived. Euler equation is sometimes useful in quantum field theory
Euler equations
The journey began with Paul Dirac when he was the professor of mathematics at the Cambridge. The same position was later hold by Stephen Hawkin.
PaUL DIRAC EQUATION
After Paul dirac developed quantum electrodynamics, quantum field theory got an enormous pace.
Many body green's function is deined with time ordered product as
many body green function

Quantum chemistry

Quantum mechanics has described all the chemistry. Chemistry is concerned with the chemical properties of atom. The behaviour and energy of electron can be explained completely using Shrodinger wave function. The solution of Shrodinger's equation , consequently describes various orbitals where electron stay. By studying the energy associated with these orbitals their chemical properties like ionic bond, valence bonding can be explained completely.
The component functions of Schrodinger's wave equation are
Schrodinger  EQUATION
Schrodinger's equation has been decomposed into three component functions.
The momentum and position measurement will give the probalities of momentum and position respectively .
Schrodinger  EQUATION
Angular momentum eigenfunction tansforms as the axes are rotated arbitrarily
Angular momentum eigenfunction
Orbital angular momentum eigenvalues arises when separating the angular part of Schrodinger equation
Angular momentum eigenvalues
On the other hand the eigenvalue of the Schrodinger's equation can be found by solving the equation
Schrodinger equation and eigenvalue
Sometimes , orbitals contain the same energy , which are called degenerate orbitals
Degenerate orbitals

Secret of the creating lies in the hear of quantum mechanics...

These are the solutions of Schrodiner's equation of Hydrogen atom.
Solution of hydrogen atom
At the moment , it is better to know how to caclulate various kinds of Hermite polynomial
hermite polynomial
The secret of the origin of quantum theory is
matrix mechanics

Reduced mass

Reduced mass is used when two-body rotating system is replaced with one-body problem. In practice when electron moves around nucleus the nucleus also moves and this makes them a two - body problem. This can be simplified with a new system by replacing electron with reduced mass. So the nucleus can be thought as stationary. Here is the approximation
reduced mass electron

reduced mass electron
Schrodinger equation applied to two-body problem
two body problem
The hamiltonian for hydrogen atom is
hamiltoninan of hydrogen atom
Through beta decay an atom transformed into another atom of reduced atomic number.
beta decay of atom
Normal zeeman effect is associated with orbital energy.
normal zeeman effect

Miscellanous equations in mathematical physics

Momentum equation describes evolution like all other fluid dynamics equations.
momentum equation
The equation of state is
 equation of state
Where ω is the ancentic factor having an specific value. This has proven very useful in describing matter.
Classical discrete thermodynamical system is defined by
 equation of state

Let me tell you the secret, if you want to understand physics rely solely on formulas and formulas only..

Conection and diffusion equation are also very important to understand if you want to know more about heat.
convection and diffusion equation
Aerodynamics is concerned with flying aeroplane and flow of air through some medium. The lift equation is conerned with aerodynamics, although the basics can be explained by Newtonian dynamics
convection and diffusion equation
General thrust equation is given by
general thrust equation
Some important heat related formulas are worth remembering.
general heat related formula
The equation of motion of a particle in conservative field is given by
classical physics

Wave Packets

First we have a wavefunction ψ for a free particle with a definite momentum p
wave function
Where the wave number k is defined as p = hk and the angular frequency Ω satisfies E = hΩ
To create a wavepacket which is localised in space we need to add components of various wave number. In Fourier series we use function f(x) when we limit the range to -L < x < L . We do not want to limit our x in this interval so that any wave number is allowed. In the limit L -> ∞ the summation turns into an integral which is known as Fourier transform.
wave packet
With coefficients which are comparable.
wave packet
The normalizations of f(x) and A(k) are the same (symmetric) and both can represent probability amplitude.
wave packet normalization
We understant f(x) made up with definite momentum terms e^[ikx].
The square wave packet and gaussian wave packet are given by
square and Gaussian wave packets
They are both localized in p = hk_0 . The value of f(x) representing the wave packet is zero everywhere outside this range.
Checking the normalization of square wave is
square  wave renormalization
Similarly the normalization of Gaussian is
square  wave renormalization

Atom in electromagnetic radiation

The interaction of atoms with electromagnetic waves can be computed using time dependent perturbation theory. The atomic problem is solved in the absence of EM waves, then the vector potential terms in the Hamiltonian can be regarded as a perturbation.
atom in electromagnetic radiation
In a gauge where &del;.A = 0 the perturabation is
atom in electromagnetic radiation
For most atomic decays, the A^2 term can be neglected since it is much smaller than the A.p term. Both the decay of excited atomic states with the emission of radiation and the excitation of atoms with the absorption of radiation can be calculated.
An arbitrary EM field can be Fourier analyzed to give a sum of component term of definite frequency. Consider the vector potential A for one such component,
atom in electromagnetic radiation
. The energy of the vector field is then
energy of vector potential
If the field is quantized as the energy E = hμ then we can write the field strength in term of photon numbers N.
quantized photon field

Atomic world is fully interpreted as a collection of radiations and probability..

The sense or direction of the field is given by the unit polarization vector ε. The cosine term has been split into positive and negative exponentials. In time dependent perturbation theory, the positive exponential corresponds to the absorption of a photon and excitation of the atom and the negative exponential corresponds to the emission of a photon and decay of the atom to a lower energy state.
Think of the EM field as a harmonic oscillator at each frequency, the negative exponential corresponds to a raising operator for the field and the positive exponential to a lowering operator. In analogy to the quantum 1D harmonic oscillator we replace
quantized photon field
in the raising operator case.
quantized photon field
With this alteration, which will later be justified with the quantization of the field, there is a perturbation even with no applied field ( N=0)
perturbation in hamiltonian
Which can cause decays of atom. When we put N=0 in the perturbation we get the decay rate of atmic states
decay rates of atmic states
The absolute square of the time integral from perturbation theory produces the delta function of energy conservation. To get the total decay rate, we must sum over the allowed final states. We can assume that the atom remains at rest as a very good approximation, but, the final photon states must be carefully examined. Applying periodic boundary conditions in a cubic volume V, the integral over final states can be done as indicated below.
total decay rates of atmic states
With this phase space integral done supported by the delta function, the general formula for the decay rate is
total decay rates of atmic states
This decay rate still contains the integral over photon directions and a sum over final state polarization. Computation of the atomic matrix element is usually performed in the Electric Dipole approximation
electrical dipole approximation
which is valid if the wavelength of the photon is much greater than the size of the atom. With the help of some commutation relations, the decay rate formula becomes
decay rate formula
The atomic matrix element of the vector operator r is o unless certain constraints on the angular momentum of initial and final states are satisfied. The selection constraints for electric dipole (E1) transitions are:
decay rate formula
This is the result of the Wigner-Eckart theorem which states that the matrix element of a vector operator V(q) where the integer q runs from -1 to 1 , is
decay rate formula
Here α represents all the (other) quantum numbers of the state, not the angular momentum quantum numbers. In the case of a simple spatial operator like r, only the orbital angular momentum is involved.
decay rate transition
We compute a simple result for the total decay rate of a state, summed over final photon polarization and integrated over photon direction.
decay rate transition
Using this formula we can compute decay rate of hydrogen atom
decay rate transition
The total decay rate is related to the energy width of an excited state, as might be anticipated from the uncertainty principle. The Full Width at Half Maximum (FWHM) of the energy distribution of a state is hγ. The distribution in frequency follows a Breit-Wigner distribution.
decay rate transition

Matrix element

We may define the components of a state vector ψ as the projections of the state on a complete, orthonormal set of states, like the eigenfunctions of a Hermitian operator.
decay rate transition
Similarly, we may define the matrix element of an operator in terms of a pair of those orthonormal basis states
decay rate transition

The flux of probability

In analogy to the Poynting vector for EM radiation, we may want to know the probability current in some physical situation. For example, in our free particle solution, the probability density is uniform over all space, but there is a net flow along the direction of the momentum. We can derive an equation representing conservation of probability by differentiating P(x, t) = ψ*ψ and using the Schrodinger equation
Flux of probability
This is usual conservation if the j(x, t) is identified as the probability current
Flux of probability
This current can be computed from the wave function.
If we integrate if over some interval in x
Flux of probability
the equation says that the rate of change of probability in an interval is equal to the probability flux into the integral minus the flux out.
Extending this analysis to 3 dimensions,
Flux of probability
The above equation were derived using this contonuity equation
continuity equation
The equation of motion in curvilinear coordinates is not the same as the motion considered in cartesian space.
General equation of motion is
general motion equation

Classical central field problem

In classical potential theory central-force problem is to determine the motion in a single potential field. A central force is the force that points from the particle from the particle towards a single fixewd point in space and whose magnitude is determined only by the distance from the center.
central potential
A central force is always conservative force. Potential energy of such a system is
central potential
Work done is such a system governed by central force is
central potential
Equivalently, it suffices that the curl of the force field F is zero; using the formula for the curl in spherical coordinates,
central potential
because the partial derivatives are zero for a central force; the magnitude F does not depend on the angular spherical coordinates θ and φ
central potential
Since F = ma by Newton's second law of motion and since F is a central force, then only the radial component of the acceleration a can be non-zero; the angular component a_φ must be zero
central potential
Therefore
central potential
This expression in the parenthesis is usually denoted by h or angular momentum.
So the central force problem reduces to
central force problem
Schrodinger's equation in two frames
Schrodinger equation
Shcrodinger's equation can be applied to multiparticle system too.
Schrodinger equation
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