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

quantum mechanics for dummies, Quantum mechanics for dummies

Quantum mechanics for dummies

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

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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 hypothese 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 Plank'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 .
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 plank 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 in side the atom.

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.
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.

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.

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
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


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 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

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

Some useful quantum mechanics equations


quantum mechanics for dummies

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.

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 wouldn't 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 wavefunctions?

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.


Reference materials:

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

quantum mechanics for dummies

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