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"Scattering matrix" redirects here. For the meaning in linear electrical networks, see Scattering parameters.
For the 1960s approach to particle physics, see S-matrix theory.

In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory.

More formally, the S-matrix is defined as the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels). While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no horizons[clarification needed], it has a simple form in the case of the Minkowski space. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group[clarification needed]; 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 shown that if a quantum field theory in Minkowski space has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.

History[edit]

The S-matrix was first introduced by John Archibald Wheeler in the 1937 paper "'On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure'".[1] In this paper Wheeler introduced a scattering matrix – a unitary matrix of coefficients connecting "the asymptotic behaviour of an arbitrary particular solution [of the integral equations] with that of solutions of a standard form".[2]

In the 1940s, Werner Heisenberg developed, independently, the idea of the S-matrix. Due to the problematic divergences present in quantum field theory at that time Heisenberg was motivated to isolate the essential features of the theory that would not be affected by future changes as the theory developed. In doing so he was led to introduce a unitary "characteristic" S-matrix.[2]

After World War II, the clout of Heisenberg and his attachment to the S-matrix approach may have retarded development of alternative approaches and the closer study of sub-hadronic physics for a decade or more, at least in Europe: "Pretty much like medieval Scholastic Magisters were extremely inventive in defending the Church Dogmas and blocking the way to experimental science, some great minds in the sixties developed the S-Matrix dogma with great perfection and skill before it was buried down in the seventies after discovery of quarks and asymptotic freedom" [3]

Motivation[edit]

In high-energy particle physics we are interested in computing the probability for different outcomes in scattering experiments. These experiments can be broken down into three stages:

1. Collide together a collection of incoming particles (usually two particles with high energies).

2. Allowing the incoming particles to interact. These interactions may change the types of particles present (e.g. if an electron and a positron annihilate they may produce two photons).

3. Measuring the resulting outgoing particles.

The process by which the incoming particles are transformed (through their interaction) into the outgoing particles is called scattering. For particle physics, a physical theory of these processes must be able to compute the probability for different outgoing particles when we collide different incoming particles with different energies. The S-matrix in quantum field theory is used to do exactly this. It is assumed that the small-energy-density approximation is valid in these cases.

Use of S-matrices[edit]

The S-matrix is closely related[vague] to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; the elements (individual numerical entries) in the S-matrix are known as scattering amplitudes. Poles of the S-matrix in the complex-energy plane are identified with bound states, virtual states or resonances. Branch cuts of the S-matrix in the complex-energy plane are associated to the opening of a scattering channel.

In the Hamiltonian approach to quantum field theory, the S-matrix may be calculated as a time-ordered exponential of the integrated Hamiltonian in the interaction picture; it may also be expressed using Feynman's path integrals. In both cases, the perturbative calculation of the S-matrix leads to Feynman diagrams.

In scattering theory, the S-matrix is an operator mapping free particle in-states to free particle out-states (scattering channels) in the Heisenberg picture. This is very useful because often we cannot describe the interaction (at least, not the most interesting ones) exactly.

Mathematical definition[edit]

In Dirac notation, we define |0\rangle as the vacuum quantum state. If a^{\dagger}(k) is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the vacuum as follows:

a(k)\left |0\right\rangle = 0.

Now, we define two kinds of creation/destruction operators acting on different Hilbert spaces (initial space i, final space f), a_i^\dagger (k) and a_f^\dagger (k).

So now

\mathcal H_\mathrm{IN} = \operatorname{span}\{ \left| I, k_1\ldots k_n \right\rangle = a_i^\dagger (k_1)\cdots a_i^\dagger (k_n)\left| I, 0\right\rangle\},
\mathcal H_\mathrm{OUT} = \operatorname{span}\{ \left| F, p_1\ldots p_n \right\rangle = a_f^\dagger (p_1)\cdots a_f^\dagger (p_n)\left| F, 0\right\rangle\}.

It is possible to play the trick assuming that \left| I, 0\right\rangle and \left| F, 0\right\rangle are both invariant under translation and that the states \left| I, k_1\ldots k_n \right\rangle and \left| F, p_1\ldots p_n \right\rangle are eigenstates of the momentum operator \mathcal P^\mu, by adiabatically turning on and off the interaction.

In the Heisenberg picture the states are time-independent, so we can expand initial states on a basis of final states (or vice versa) as follows:

\left| I, k_1\ldots k_n \right\rangle = C_0 \left| F, 0\right\rangle\ + \sum_{m=1}^\infty \int{d^4p_1\ldots d^4p_mC_m(p_1\ldots p_m)\left| F, p_1\ldots p_m \right\rangle}

Where \left|C_m\right|^2 is the probability that the interaction transforms \left| I, k_1\ldots k_n \right\rangle into \left| F, p_1\ldots p_m \right\rangle

According to Wigner's theorem, S must be a unitary operator such that \left \langle I,\beta \right |S\left | I,\alpha\right\rangle = S_{\alpha\beta} = \left \langle F,\beta | I,\alpha\right\rangle. Moreover, S leaves the vacuum state invariant and transforms IN-space fields in OUT-space fields:

S\left|0\right\rangle = \left|0\right\rangle
\phi_f=S\phi_i S^{-1}

If S describes an interaction correctly, these properties must be also true:

  • If the system is made up with a single particle in momentum eigenstate \left| k\right\rangle, then S\left| k\right\rangle=\left| k\right\rangle
  • The S-matrix element may be nonzero only where the output state has the same total momentum as the input state.

S-matrix and evolution operator U[edit]

Define a time-dependent creation and annihilation operator as follow

a^{\dagger}\left(k,t\right)=U^{-1}(t)a^{\dagger}_i\left(k\right)U\left( t \right)
a\left(k,t\right)=U^{-1}(t)a_i\left(k\right)U\left( t \right)

Hence

\phi_f=U^{-1}(\infty)\phi_i U(\infty)=S^{-1}\phi_i S.

where we have

S= e^{i\alpha}\, U(\infty) .

We allow a phase difference given by

e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle^{-1}

because for S:

S\left|0\right\rangle = \left|0\right\rangle  \Longrightarrow  \left\langle 0|S|0\right\rangle = \left\langle 0|0\right\rangle =1

Substituting the explicit expression for U we obtain:

S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal T e^{-i\int{d\tau H_{\rm{int}}(\tau)}}.

where  H_{\rm{int}} is the interaction part of the hamiltonian and  \mathcal T is the time ordering. By inspection it can be seen that this formula is not explicitly covariant.

Dyson series[edit]

Main article: Dyson series

The most widely used expression for the S-matrix is the Dyson series. This expresses the S-matrix operator as the series:

S = \sum_{n=0}^\infty \frac{(-i)^n}{n!} \int \cdots \int d^4x_1 d^4x_2 \ldots d^4x_n T [ H_{\rm{int}}(x_1) H_{\rm{int}}(x_2) \cdots H_{\rm{int}}(x_n)]

where:

S-matrix in one dimensional quantum mechanics[edit]

Definition[edit]

Consider a localized one dimensional potential barrier V(x), subjected to a beam of quantum particles with energy E. These particles incident on the potential barrier from left and right. The solution of Schrödinger's equation outside the potential barrier are plane waves and are given by:

\psi_L(x)= A e^{ikx} + B e^{-ikx}

for the region left to the potential barrier and

\psi_R(x)= C e^{ikx} + D e^{-ikx}

for the region right to the potential barrier, where

k=\sqrt{2m E/\hbar^{2}}

is the wave vector,while the terms with coefficients A and D represent the incoming waves,whereas terms with coefficients B and C represent the outgoing waves.The outgoing waves are connected to the incoming waves by a linear relation described by the S-matrix

\begin{pmatrix}B \\ C \end{pmatrix} = \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}\begin{pmatrix} A \\ D \end{pmatrix}\,.

Elements of this matrix completely characterize the scattering properties of the V(x).

The above relation can be written as \Psi_{out}=S \Psi_{in}
where,
\Psi_{out}=\begin{pmatrix}B \\ C \end{pmatrix},\Psi_{in}=\begin{pmatrix}A \\ D \end{pmatrix} and  S=\begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}.

Unitary property of S-matrix[edit]

The unitary property of S-matrix is directly related the to conservation of probability current in quantum mechanics.The probability current J of the wave function \psi(x)\,, is defined as

 J = \frac{\hbar}{2mi}\left(\psi^* \frac{\partial \psi }{\partial x}-  \psi \frac{\partial \psi^* }{\partial x} \right) .

The current density to the left of the barrier is J_L=\frac{\hbar k}{m}\left(|A|^2-|B|^2\right),and the current density to the right of the barrier is J_R=\frac{\hbar k}{m}\left(|C|^2-|D|^2\right). According to conservation of probability current density J_L=J_R.This implies the S-matrix is a unitary matrix.

Time-reversal symmetry[edit]

If the potential V(x) is real,then the system possesses time-reversal symmetry.Under this condition if \psi(x) is a solution of Schrödinger's equation,then \psi^*(x) is also a solution.The time-reversed solution is given by:

\psi^*_L(x)= A^* e^{-ikx} + B^* e^{ikx}

for the region left to the potential barrier and

\psi^*_R(x)= C^* e^{-ikx} + D^* e^{ikx}

for the region right to the potential barrier, where the terms with coefficient B*,C*represent incoming wave and terms with coefficient A*,D* represent outgoing wave.So they are again related by the S-matrix :

\begin{pmatrix}A^* \\ D^* \end{pmatrix} = \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}\begin{pmatrix} B^* \\ C^* \end{pmatrix}\, or \Psi^*_{in}=S \Psi^*_{out}.

Now,the relations \Psi^*_{in}=S \Psi^*_{out} and \Psi_{out}=S \Psi_{in} together yield a condition S^*S=I. This condition in conjunction with the unitary relation implies that the S-matrix is symmetric as a result of time reversal symmetry:

S^T=S.

Transmission coefficient and Reflection coefficient[edit]

Transmission coefficient from the left of the potential barrier is T_L=\frac{|C|^2}{|A|^2},when D=0.Thus T_L=|S_{21}|^2 .

Reflection Coefficient from the left of the potential barrier is R_L=\frac{|B|^2}{|A|^2},when D=0.Thus R_L=|S_{11}|^2 .

Similarly,Transmission Coefficient from the right of the potential barrier is T_R=\frac{|B|^2}{|D|^2},when A=0.Thus T_R=|S_{12}|^2 .

Reflection Coefficient from the right of the potential barrier is R_R=\frac{|C|^2}{|D|^2},when A=0.Thus R_R=|S_{22}|^2 .

The relation between transmission coefficient and reflection coefficient is: T_L+R_L=1 and T_R+R_R=1. This relation is the consequence of the unitary property of S-matrix.Thus the elements of S-matrix are basically reflection and transmission coefficients.

Optical theorem in one dimension[edit]

In the case of free particle V(x)=0.The S-matrix is then  S=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. Now whenever V(x) is different from zero,there is a departure of S-matrix from the above form.This departure is measured by two complex functions of energy,r and t,which are defined by S_{11}=2ir and S_{21}=1+2it.The relation between this two function is given by:

|r|^2+|t|^2=\operatorname{Im}(t).

The analogue of this identity in three dimension is known as optical theorem.

See also[edit]

Notes[edit]

  1. ^ John Archibald Wheeler, 'On the Mathematical Description of Light Nuclei by the Method. of Resonating Group Structure' Phys. Rev. 52, 1107–1122 (1937)
  2. ^ a b Jagdish Mehra, Helmut Rechenberg, The Historical Development of Quantum Theory (Pages 990 and 1031) Springer, 2001 ISBN 0-387-95086-9, ISBN 978-0-387-95086-0
  3. ^ Alexander Migdal, Paradise Lost, Part 1

References[edit]


Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/S-matrix — Please support Wikipedia.
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