9.2: Fundamental
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider time-independent scattering theory, for which the Hamiltonian of the system is written
H0 is the Hamiltonian of a free particle of mass H0=p22m, ???and |ϕ⟩ be an energy eigenket of H0|ϕ⟩=E|ϕ⟩, ???
whose wavefunction ϕ(x′) . This state is assumed to be a plane wave state or, possibly, a spherical wave state. Schrödinger's equation for the scattering problem is
|ψ⟩ is an energy eigenstate of the total Hamiltonian whose wavefunction ψ(x′) . In general, both H0+H1 have continuous energy spectra: i.e., their energy eigenstates are unbound. We require a solution of Equation ??? that satisfies the boundary condition H1→0 . Here, (∇2+k2)ψ(x)=2mℏ2⟨x|H1|ψ⟩, ???where
ψ(x)=ϕ(x)+2mℏ2∫d3x′G(x,x′)⟨x′|H1|ψ⟩, ???where
|ψ⟩→|ϕ⟩ as G(x,x′)=−exp(±ik|x−x′|)4π|x−x′|. ???Thus, Equation ??? becomes

Let us suppose that the scattering Hamiltonian, ⟨x′|H1|x⟩=V(x)δ(x−x′). ???
We can write

Thus, the integral equation ??? simplifies to

Suppose that the initial state k (i.e., a stream of particles of definite momentum |k⟩ . The associated wavefunction takes the form
⟨x|k⟩=exp(ik⋅x)(2π)3/2. ???The wavefunction is normalized such that
![$ \langle {\bf k}\vert{\bf k}'\rangle =\int d^3 x\, \langle {\bf k}...
...\, {\bf x}\cdot({\bf k} -{\bf k}')]} {(2\pi )^3} = \delta ({\bf k} - {\bf k'}).$](http://farside.ph.utexas.edu/teaching/qm/lectures/img2159.png)
Suppose that the scattering potential x=0 ). Let us calculate the wavefunction r≫r′ . It is easily demonstrated that
r′/r , where r=|x| and k′=ker. ???Clearly, k′ is the wavevector for particles that possess the same energy as the incoming particles (i.e., exp(±ik|x−x′|)≃exp(±ikr)exp(∓ik′⋅x′). ???
In the large-r limit, Equation ??? reduces to

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 ψ(x)=1(2π)3/2[exp(ik⋅x)+exp(ikr)rf(k′,k)], ???
where

Let us define the differential cross-section, dΩ , divided by the incident flux of particles. Recall, from Chapter 3, that the probability current (i.e., the particle flux) associated with a wavefunction j=ℏmIm(ψ∗∇ψ). ???
Thus, the probability flux associated with the incident wavefunction,
jinc=ℏ(2π)3mk. ???Likewise, the probability flux associated with the scattered wavefunction,
jsca=ℏ(2π)3m|f(k′,k)|2r2ker. ???Now,
dσdΩ=|f(k′,k)|2. ???Thus, ℏk to be scattered into states whose momentum vectors are directed in a range of solid angles ℏk′ . Note that the scattered particles possess the same energy as the incoming particles (i.e., k′=k ). This is always the case for scattering Hamiltonians of the form specified in Equation ???.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)