But nature makes that mean : so, over that art
Which you say adds to nature, is an art,
That nature makes."
William Shakespear, A winters tale.
"Energy makes it go and entropy tells where to go"
Quantum field theoryPath integral | S Matrix | Quantum electrodynamics
Theory of relativitySpecial theory of relativity | General theory of relativity
s matrix and scattering matrix
In physics, the s matrix or scattering matrix correlates the initial state and the final state of a physical system evolving a scattering process. It is utilized in quantum mechanics, scattering theory and quantum field theory (QFT).
More formally, in the language of QFT, the s matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of quantum physical states. A multi-particle state is assumed to be free (non-interacting) if it transforms under Lorentz transformations as a tensorial product, or direct product in physics parlance, of one-particle states as prescribed by equation (1) below. Asymptotically free then implies that the state has this appearance in either the distant past or the distant future.
While the s-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no event horizons, it has a simplistic structure in the case of the Minkowski spacetime. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group (the Poincaré group); the s matrix is the evolution operator between time equal to minus infinity (the distant past), and time equal to plus infinity (the distant future). It is defined only in the limit of zero energy density (or infinite particle separation distance).
It can be proven that if a quantum field theory in Minkowski spacetime has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.
In high-energy particle physics we are very interested in computing the probability for distinct outcomes in scattering experiments. These experiments can be broken down into three separate stages:
a)collide together a collection of incoming particles (usually two particles with high energies).
b)Allowing the incoming particles to interact with themselves. These interactions may change the types of particles present (e.g. if an electron and a positron (anto-electron) annihilate they may produce two photons).
c)Measuring the mass of resulting outgoing particles.
The process by which the incoming particles are converted (through their interaction) into the outgoing particles is called scattering process. For particle physics, a physical theory of these processes must be able to systematically compute the probability for different outgoing particles when different incoming particles collide with different energies.
The S-matrix in quantum field theory achieves exactly this goal. It is assumed that the small-energy-density approximation is valid in these cases.
An operator (a matrix) describing the process of transference of a quantum-mechanical system from one state into another under their interactions (scattering).
Under process of scattering, the system moves from one quantum state, the initial one (one may relate it to the time t=−∞), into another, the final one (related to t=+∞). If one denotes the set of quantum numbers describing the initial (final) state by i (j), then the scattering amplitude (the square of whose modulus defines the probability of a given scattering) can be written as S(ij). The collection or set of all scattering amplitudes forms a table with two inputs, and is called the scattering matrix S. Each of these scattering amplitudes or transition amplitudes represents a Feynman diagram which contributes to its corresponding scattering amplitude perturbatively. Feynman diagram tells how particles interact with each other. Thus it is the same thing as the initial assumption of s-matrix.
Determining scattering matrices is a core problem in quantum mechanics and quantum field theory. The scattering matrix contains complete information about the behaviour of a system, provided that one knows not only the numerical values, but also the analytical properties of its elements. In particular, its poles specify the bound states of the system (and thus the discrete energy levels). The most important property of a scattering matrix follows from the basic principles of quantum theory: it must be unitary.
The difference between Heisenberg picture and Schrodinger picture of atom is the time dependency of observables and operators. Fundamentally two pictures developed by Heisenberg and
Schrodinger is equivalent. They both describe same phenomena in two different ways. To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts:
A little comparison can be made with a chart.
The interaction part , as seen , is the representation of two states that interact through Hamiltonian H(0,s) which is time dependent. S-matrix is developed based on this interaction Hamiltonian H(0,s). mIf the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with H(1,S) leaving H(0,S) time-independent. S-matrix or scattering matrix is the transformation matrix between incoming and outgoing states of particles waves. This is the intermediate representation between Schrodinger picture and Heisenberg picture. In Heisenberg picture the state vector is constant or time-independent but in Schrodinger's picture this is not the case. In interaction picture both the state vector and operator carry time dependence of the observables. The interaction part of the Hamiltonian is dependent of time. Total Hamiltonian is the sum of free part H(0) and an interaction part V. So the interaction picture is :
Let us give a short description of each step taken to reach the final form of S - matrix. The first step denotes the projection of the initial state Φ(i) on the final state Φ(f). This gives the elements of S matrix and defines an operator S. In the next step the evolution operator U(,) is used and comparing the two results we see S-operator is the evolution operator U(a,a). The final S(fi) matrix is the projection of initial state onto final state Ψ(i) by the operator S which is defined above. The s - matrix is similar to transition probability amplitude in quantum mechanics. The elements of S-matrix represents scattering amplitudes. Scattering amplitudes are various ways in which wave interact with themselves or with particles.
"you can watch me, mock me and try to block me but you can not stop me.."
We are not here because we are free..we are here because we are not free...
Perturbation in calculating fine structure energy deviationFine structure describes the splitting of energy of atomic orbital due to the spin-orbital interaction and relativistic energy correction to the non-relativistic Shrodinger equation. The fine structure energy can be obtained by using perturbation theory. To perform this calculation one must add three terms to the hamiltonian. One is the leading order correction relativistic correction to kinetic energy, the second is the spin-orbital correction and the third is the Darwin term coming from quantum fluctuating motion.
The gross structure assumes that kinetic energy term of the energy takes the same as classical one.
But the relativistic energy will be
Using tylor's expansion we get
The first order correction then can be taken as
Using this as perturbation we can find first order energy correction due to the relativistic effect
a(0) is the Bohr's radius and n is the principle quantum number, l is the azimuthal quantum number, r is the distance of electron from the nucleus. Therefore the first order correction of energy of elctron of Hydrogen atom is
Spin-orbital coupling of elctron is given by
There is one last term in the non-relativistic expansion of Dirac equation. This term is called Darwin term as it was first derived by scientist Darwin.
Spin and orbital angular momentum of electron are very sophisticated concepts. Effects due to these are very minute yet very important for atomic physics. An electron is like the earth which is rotating and spinning at the same time. In case of electron the effect can only be observed applying an external magnetic field. Electron responds to the magnetic field by aligning itself properly. The avearage spin orbital coupling is measured by integrating over the entire atom .
The Bohr's radius changes as the square of the orbit quantum number n.
Atomic physics are interesting and hard at the same time. Don't be scared. You can understand the mathematics.
Schrodinger's equation created a revolution in quantum physics. Now radial Schrodinger's equation will be derived.
It is subject to normalization contraint
Variating it (subject to the normalization condition) we get:
Which gives the Schrödinger equation assuming the surface integral disappears.
Note: to apply the variation Δ correctly, one uses the definition:
The weak formulation is obtained from the above by substituting Δψ to v (the test function) so we get:
There are two ways to derive Radial Schrodinger's equation : one is from lagrangian and other is from equation itself. We will adopt the second.
The way to solve it is to decouple the equation into radial and angular parts by writing the Laplace operator in spherical coordinates as:
Substituting ψ(x)=R(ρ)Y(θ,φ) into the Schrödinger equation yields:
Using the fact that L^2Y=l(l+1)Y we can cancel Y and we get the radial Schrödinger equation:
Now we derive it from lagrangian.
We need to convert the Lagrangian to spherical coordinates. In order to easily make sure we do things covariantly, we start from the action (which is a scalar):
where we used the following properties of spherical harmonics:
We now minimize the action (subject to the normalization ∫ ρ^2 R^2dρ = 1) to obtain the radial equation:
So the radial Schrodinger equation becomes
Weak formulationThe weak formulation is obtained from the action above by substituting Δ R to v (the test function) so we get:
We can also start from the equation itself, multiply by a test function v:
We integrate it. Normally we need to be using ρ^2dρ in order to integrate covariantly, but the above equation was already multiplied by ρ^2 (i.e. strictly speaking, it is not coordinate independent anymore), so we only integrate by dρ:
After integration by parts:
Where a is the end of the domain (the origin is at 0). The boundary term is zero at the origin, so we get:
We normally want to have the boundary term 1/2ρ^2R'(a)v(a) equal to zero. This is equivalent to either letting R'(a) = 0 (we prescribe the zero derivative of the radial wave function at a) or we set v(a)=0 (which corresponds to zero Dirichlet condition for R, i.e. setting R(a)=0). We can also write all the formulas using the Dirac notation:
Then normalization is:
The operator &hat; H can be written as:
and we obtain the FE formulation by expanding |R> = ∑_j R_j |j> (note that the basis |ketj> is not orthogonal, so in particular ∑_j |j> j != 1):
This is a generalized eigenvalue problem. In the special case of an orthonormal basis, = Δ_ij (which FE is not), we reach:
Variational formulation of dirac equationThe QED lagrangian density is
We will treat the fields as classical fields, so we get the classical wave Dirac equation, after plugging this Lagrangian into the Euler-Lagrange equation of motion:
Notice that the Lagrangian appears to be zero for the solution of Dirac equation (e.g. the extremum of the action). This has nothing to do with the variational principle itself, it's just a coincindence.
In this section we are only interested in the Dirac equation, so we write the Lagrangian as:
where we introduced the potential function by V = ceA_0. We also could have done the same manipulation to the dirac equation itself and we would get the same expression:
The corresponding eigenvalue problem is
Now we will develop radial dirac equation from dirac equation itself.
The manipulations are well known, one starts by writing the Dirac spinors using the spin angular functions and radial components P and Q:
and putting this into the Dirac equation one obtains:
So one obtains the following radial equations:
You are beginning to believe....The total hamiltonian is then given by
First term is the Coulomb interaction.
The total effect is then calculated by adding up the individual components.
Reference materials:Feynman's lectures on physics (elementary)
Quantum electrodynamics and s matrix (springer)
s matrix and quantum eqlectrodynamics by Greiner W & Reinhardt J.
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf