In physics, the S-matrix or scattering matrix is a matrix that 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 (QFT). More formally, in the context 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 physical states: a multi-particle state is said to be free (or non-interacting) if it transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation (1) below. Asymptotically free then means 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 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 (the Poincaré group); the S-matrix is the evolution operator between t = − ∞ {\displaystyle t=-\infty } (the distant past), and t = + ∞ {\displaystyle t=+\infty } (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 The initial elements of S-matrix theory are found in Paul Dirac's 1927 paper "Über die Quantenmechanik der Stoßvorgänge". The S-matrix was first properly introduced by John Archibald Wheeler in the 1937 paper "On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure". 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", but did not develop it fully. In the 1940s, Werner Heisenberg independently developed and substantiated the idea of the S-matrix. Because of 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. Today, however, exact S-matrix results are important for conformal field theory, integrable systems, and several further areas of quantum field theory and string theory. S-matrices are not substitutes for a field-theoretic treatment, but rather, complement the end results of such.
Motivation In high-energy particle physics one is interested in computing the probability for different outcomes in scattering experiments. These experiments can be broken down into three stages:
Making a collection of incoming particles collide (usually two kinds of particles with high energies). 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). 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 different incoming particles collide with different energies. The S-matrix in quantum field theory achieves exactly this. It is assumed that the small-energy-density approximation is valid in these cases.
Use The S-matrix is closely related 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.
In one-dimensional quantum mechanics A simple prototype in which the S-matrix is 2-dimensional is considered first, for the purposes of illustration. In it, particles with sharp energy E scatter from a localized potential V according to the rules of 1-dimensional quantum mechanics. Already this simple model displays some features of more general cases, but is easier to handle. Each energy E yields a matrix S = S(E) that depends on V. Thus, the total S-matrix could, figuratively speaking, be visualized, in a suitable basis, as a "continuous matrix" with every element zero except for 2 × 2-blocks along the diagonal for a given V.
Definition Consider a localized one dimensional potential barrier V(x), subjected to a beam of quantum particles with energy E. These particles are incident on the potential barrier from left to right. The solutions of the Schrödinger equation outside the potential barrier are plane waves given by
ψ L ( x ) = A e i k x + B e − i k x {\displaystyle \psi _{\rm {L}}(x)=Ae^{ikx}+Be^{-ikx}}
for the region to the left of the potential barrier, and
ψ R ( x ) = C e i k x + D e − i k x {\displaystyle \psi _{\rm {R}}(x)=Ce^{ikx}+De^{-ikx}}
for the region to the right to the potential barrier, where
k = 2 m E / ℏ 2 {\displaystyle k={\sqrt {2mE/\hbar ^{2}}}}
is the wave vector. The time dependence is not needed in our overview and is hence omitted. The term with coefficient A represents the incoming wave, whereas term with coefficient C represents the outgoing wave. B stands for the reflecting wave. Since we set the incoming wave moving in the positive direction (coming from the left), D is zero and can be omitted. The "scattering amplitude", i.e., the transition overlap of the outgoing waves with the incoming waves is a linear relation defining the S-matrix,
( B C ) = ( S 11 S 12 S 21 S 22 ) ( A D ) . {\displaystyle {\begin{pmatrix}B\\C\end{pmatrix}}={\begin{pmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{pmatrix}}{\begin{pmatrix}A\\D\end{pmatrix}}.}
The above relation can be written as
Ψ o u t = S Ψ i n {\displaystyle \Psi _{\rm {out}}=S\Psi _{\rm {in}}}
where
Ψ o u t = ( B C ) , Ψ i n = ( A D ) , S = ( S 11 S 12 S 21 S 22 ) . {\displaystyle \Psi _{\rm {out}}={\begin{pmatrix}B\\C\end{pmatrix}},\quad \Psi _{\rm {in}}={\begin{pmatrix}A\\D\end{pmatrix}},\qquad S={\begin{pmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{pmatrix}}.}
The elements of S completely characterize the scattering properties of the potential barrier V(x).
Unitary property The unitary property of the S-matrix is directly related to the conservation of the probability current in quantum mechanics. The probability current density J of the wave function ψ(x) is defined as
J = ℏ 2 m i ( ψ ∗ ∂ ψ ∂ x − ψ ∂ ψ ∗ ∂ x ) . {\displaystyle J={\frac {\hbar }{2mi}}\left(\psi ^{*}{\frac {\partial \psi }{\partial x}}-\psi {\frac {\partial \psi ^{*}}{\partial x}}\right).}
The probability current density J L ( x ) {\displaystyle J_{\rm {L}}(x)} of ψ L ( x ) {\displaystyle \psi _{\rm {L}}(x)} to the left of the barrier is
J L ( x ) = ℏ k m ( | A | 2 − | B | 2 ) , {\displaystyle J_{\rm {L}}(x)={\frac {\hbar k}{m}}\left(|A|^{2}-|B|^{2}\right),}
while the probability current density J R ( x ) {\displaystyle J_{\rm {R}}(x)} of ψ R ( x ) {\displaystyle \psi _{\rm {R}}(x)} to the right of the barrier is
J R ( x ) = ℏ k m ( | C | 2 − | D | 2 ) . {\displaystyle J_{\rm {R}}(x)={\frac {\hbar k}{m}}\left(|C|^{2}-|D|^{2}\right).} For conservation of the probability current, JL = JR. When combined with the relation Ψ out = S Ψ in {\displaystyle \Psi _{\text{out}}=S\Psi _{\text{in}}} , this implies that the S-matrix is a unitary matrix. In the notation below, Ψ out † = ( B ∗ C ∗ ) {\displaystyle \Psi _{\text{out}}^{\dagger }=(\,B^{*}\quad C^{*}\,)} , and Ψ in † = ( A ∗ D ∗ ) {\displaystyle \Psi _{\text{in}}^{\dagger }=(\,A^{*}\quad D^{*}\,)} , so that Ψ out † Ψ out † {\displaystyle \Psi _{\text{out}}^{\dagger }\Psi _{\text{out}}^{\phantom {\dagger }}} , represents the inner product of a vector with its dual co-vector, and A ∗ {\displaystyle A^{*}} , etc. is the complex conjugate of A ∈ C {\displaystyle A\in \mathbb {C} } , etc., whose complex modulus is | A | {\displaystyle \vert A\vert } .
Time-reversal symmetry If the potential V(x) is real, then the system possesses time-reversal symmetry. Under this condition, if ψ(x) is a solution of the Schrödinger equation, then ψ*(x) is also a solution. The time-reversed solution is given by
ψ L ∗ ( x ) = A ∗ e − i k x + B ∗ e i k x {\displaystyle \psi _{\rm {L}}^{*}(x)=A^{*}e^{-ikx}+B^{*}e^{ikx}}
for the region to the left to the potential barrier, and
ψ R ∗ ( x ) = C ∗ e − i k x + D ∗ e i k x {\displaystyle \psi _{\rm {R}}^{*}(x)=C^{*}e^{-ikx}+D^{*}e^{ikx}}
for the region to the right to the potential barrier, where the terms with coefficient B*, C* represent incoming wave, and terms with coefficient A*, D* represent outgoing wave. They are again related by the S-matrix,
( A ∗ D ∗ ) = ( S 11 S 12 S 21 S 22 ) ( B ∗ C ∗ ) {\displaystyle {\begin{pmatrix}A^{*}\\D^{*}\end{pmatrix}}={\begin{pmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{pmatrix}}{\begin{pmatrix}B^{*}\\C^{*}\end{pmatrix}}\,}
that is,
Ψ i n ∗ = S Ψ o u t ∗ . {\displaystyle \Psi _{\rm {in}}^{*}=S\Psi _{\rm {out}}^{*}.}
Now, the relations
Ψ i n ∗ = S Ψ o u t ∗ , Ψ o u t = S Ψ i n {\displaystyle \Psi _{\rm {in}}^{*}=S\Psi _{\rm {out}}^{*},\quad \Psi _{\rm {out}}=S\Psi _{\rm {in}}}
together yield a condition
S ∗ S = I {\displaystyle S^{*}S=I}
This condition, in conjunction with the unitarity relation, implies that the S-matrix is symmetric, as a result of time reversal symmetry,
S T = S . {\displaystyle S^{T}=S.}
By combining the symmetry and the unitarity, the S-matrix can be expressed in the form:
( S 11 S 12 S 21 S 22 ) = ( e i φ e i δ ⋅ r e i φ 1 − r 2 e i φ 1 − r 2 − e i φ e − i δ ⋅ r ) = e i φ ( e i δ ⋅ r 1 − r 2 1 − r 2 − e − i δ ⋅ r ) {\displaystyle {\begin{pmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{pmatrix}}={\begin{pmatrix}e^{i\varphi }e^{i\delta }\cdot r&e^{i\varphi }{\sqrt {1-r^{2}}}\\e^{i\varphi }{\sqrt {1-r^{2}}}&-e^{i\varphi }e^{-i\delta }\cdot r\end{pmatrix}}=e^{i\varphi }{\begin{pmatrix}e^{i\delta }\cdot r&{\sqrt {1-r^{2}}}\\{\sqrt {1-r^{2}}}&-e^{-i\delta }\cdot r\end{pmatrix}}} with δ , φ ∈ [ 0 ; 2 π ] {\displaystyle \delta ,\varphi \in [0;2\pi ]} and r ∈ [ 0 ; 1 ] {\displaystyle r\in [0;1]} . So the S-matrix is determined by three real parameters.
Transfer matrix The transfer matrix M {\displaystyle M} relates the plane waves C e i k x {\displaystyle Ce^{ikx}} and D e − i k x {\displaystyle De^{-ikx}} on the right side of scattering potential to the plane waves A e i k x {\displaystyle Ae^{ikx}} and B e − i k x {\displaystyle Be^{-ikx}} on the left side:
( C D ) = ( M 11 M 12 M 21 M 22 ) ( A B ) {\displaystyle {\begin{pmatrix}C\\D\end{pmatrix}}={\begin{pmatrix}M_{11}&M_{12}\\M_{21}&M_{22}\end{pmatrix}}{\begin{pmatrix}A\\B\end{pmatrix}}} and its components can be derived from the components of the S-matrix via: M 11 = 1 / S 12 ∗ = 1 / S 21 ∗ , M 22 = M 11 ∗ {\displaystyle M_{11}=1/S_{12}^{*}=1/S_{21}^{*}{,}\ M_{22}=M_{11}^{*}} and M 12 = − S 11 ∗ / S 12 ∗ = S 22 / S 12 , M 21 = M 12 ∗ {\displaystyle M_{12}=-S_{11}^{*}/S_{12}^{*}=S_{22}/S_{12}{,}\ M_{21}=M_{12}^{*}} , whereby time-reversal symmetry is assumed. In the case of time-reversal symmetry, the transfer matrix M {\displaystyle \mathbf {M} } can be expressed by three real parameters:
M = 1 1 − r 2 ( e i φ − r ⋅ e − i δ − r ⋅ e i δ e − i φ ) {\displaystyle M={\frac {1}{\sqrt {1-r^{2}}}}{\begin{pmatrix}e^{i\varphi }&-r\cdot e^{-i\delta }\\-r\cdot e^{i\delta }&e^{-i\varphi }\end{pmatrix}}} with δ , φ ∈ [ 0 ; 2 π ] {\displaystyle \delta ,\varphi \in [0;2\pi ]} and r ∈ [ 0 ; 1 ] {\displaystyle r\in [0;1]} (in case r = 1 there would be no connection between the left and the right side)
Finite square well The one-dimensional, non-relativistic problem with time-reversal symmetry of a particle with mass m that approaches a (static) finite square well, has the potential function V with
V ( x ) = { − V 0 for | x | ≤ a ( V 0 > 0 ) and 0 for | x | > a {\displaystyle V(x)={\begin{cases}-V_{0}&{\text{for}}~~|x|\leq a~~(V_{0}>0)\quad {\text{and}}\\[1ex]0&{\text{for}}~~|x|>a\end{cases}}}
The scattering can be solved by decomposing the wave packet of the free particle into plane waves A k exp ( i k x ) {\displaystyle A_{k}\exp(ikx)} with wave numbers k > 0 {\displaystyle k>0} for a plane wave coming (faraway) from the left side or likewise D k exp ( − i k x ) {\displaystyle D_{k}\exp(-ikx)} (faraway) from the right side. The S-matrix for the plane wave with wave number k has the solution:
S 12 = S 21 = exp ( − 2 i k a ) cos ( 2 l a ) − i sin ( 2 l a ) l 2 + k 2 2 k l {\displaystyle S_{12}=S_{21}={\frac {\exp(-2ika)}{\cos(2la)-i\sin(2la){\frac {l^{2}+k^{2}}{2kl}}}}}
and S 11 = S 12 ⋅ i sin ( 2 l a ) l 2 − k 2 2 k l {\displaystyle S_{11}=S_{12}\cdot i\sin(2la){\frac {l^{2}-k^{2}}{2kl}}} ; hence e i δ = ± i {\displaystyle e^{i\delta }=\pm i} and therefore − e − i δ = e i δ {\displaystyle -e^{-i\delta }=e^{i\delta }} and S 22 = S 11 {\displaystyle S_{22}=S_{11}} in this case. Whereby l = k 2 + 2 m V 0 ℏ 2 {\displaystyle l={\sqrt {k^{2}+{\frac {2mV_{0}}{\hbar ^{2}}}}}} is the (increased) wave number of the plane wave