# 9.2: Fundamental Equations

- Page ID
- 1239

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 ,

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

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

where

(915) |

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

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

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

We can write

(921) |

Thus, the integral equation (919) simplifies to

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

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).

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)