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# Fundamental Equations

Consider time-independent scattering theory, for which the Hamiltonian of the system is written

 (910)

where is the Hamiltonian of a free particle of mass ,

 (911)

and represents the non-time-varying source of the scattering. Let be an energy eigenket of ,

 (912)

whose wavefunction is . This state is assumed to be a plane wave state or, possibly, a spherical wave state. Schrödinger's equation for the scattering problem is

 (913)

where is an energy eigenstate of the total Hamiltonian whose wavefunction is . In general, both and have continuous energy spectra: i.e., their energy eigenstates are unbound. We require a solution of Equation (913) that satisfies the boundary condition as . Here, is a solution of the free particle Schrödinger equation, (912), corresponding to the same energy eigenvalue.

Adopting the Schrödinger representation, we can write the scattering problem (913) in the form

 (914)

where

 (915)

Equation (914) is called the Helmholtz equation, and can be inverted using standard Green's function techniques. Thus,

 (916)

where

 (917)

Note that the solution (916) satisfies the boundary condition as . As is well-known, the Green's function for the Helmholtz problem is given by

 (918)

Thus, Equation (916) becomes

 (919)

Let us suppose that the scattering Hamiltonian, , is only a function of the position operators. This implies that

 (920)

We can write

 (921)

Thus, the integral equation (919) simplifies to

 (922)

Suppose that the initial state is a plane wave with wavevector (i.e., a stream of particles of definite momentum ). The ket corresponding to this state is denoted . The associated wavefunction takes the form

 (923)

The wavefunction is normalized such that

 (924)

Suppose that the scattering potential is only non-zero in some relatively localized region centered on the origin ( ). Let us calculate the wavefunction a long way from the scattering region. In other words, let us adopt the ordering . It is easily demonstrated that

 (925)

to first order in , where

 (926)

is a unit vector that points from the scattering region to the observation point. Here, and . Let us define

 (927)

Clearly, is the wavevector for particles that possess the same energy as the incoming particles (i.e., ), but propagate from the scattering region to the observation point. Note that

 (928)

In the large- limit, Equation (922) reduces to

 (929)

The first term on the right-hand side is the incident wave. The second term represents a spherical wave centred on the scattering region. The plus sign (on ) corresponds to a wave propagating away from the scattering region, whereas the minus sign corresponds to a wave propagating towards the scattering region. It is obvious that the former represents the physical solution. Thus, the wavefunction a long way from the scattering region can be written

 (930)

where

 (931)

Let us define the differential cross-section, , as the number of particles per unit time scattered into an element of solid angle , divided by the incident flux of particles. Recall, from Chapter 3, that the probability current (i.e., the particle flux) associated with a wavefunction is

 (932)

Thus, the probability flux associated with the incident wavefunction,

 (933)

is

 (934)

Likewise, the probability flux associated with the scattered wavefunction,

 (935)

is

 (936)

Now,

 (937)

giving

 (938)

Thus, gives the differential cross-section for particles with incident momentum to be scattered into states whose momentum vectors are directed in a range of solid angles about . Note that the scattered particles possess the same energy as the incoming particles (i.e., ). This is always the case for scattering Hamiltonians of the form specified in Equation (920).

Next: Born Approximation Up: Scattering Theory Previous: Introduction
Richard Fitzpatrick 2013-04-08