Scattering theory

In quantum mechanics, the process of scattering of quantum objects must always be described differently than in classical mechanics, as the concept of a trajectory is missing. The quantum- mechanical description is called scattering theory.

Mathematical concept

In quantum mechanics, an object (such as an electron) will always be described by a state. The state is made up of a local state of a spin state, and many other physical quantities (eg the isospin ):

In the following, only the spatial wave function is considered. This is justified under the following assumptions:

• No internal degrees of freedom ( spin, excited state, ...)
• Distinguishable particles ( symmetry of the wave function for bosons and fermions is not considered here )
• No multiple scattering

Similarly as in classical mechanics the Zweiteilchenproblem can first be reduced to an equivalent Einteilchenproblem, in which a single quantum object converge at a still in the original powerhouse. Starting point of scattering theory is the description of the interaction by a potential and the derived Hamiltonian.

The wave function of the incident particle is described at the start of the spreading process by a wave packet:

This Fourier representation of the particle by plane waves can also be made via the steady states ( the eigenstates of the Hamiltonian ):

Where the eigenvalues ​​of the wave vector on

Related. These states are called scattering states, since a state with positive energy is unbound and also out of the reach of the potential has a finite probability. A single scattering state of a first bit corresponds physically plausible situations because the probability of the current density

Disappears, thus always the same amount of the particle of the scattering center - to as draining. However, this is necessary because a steady state of a standing wave is analogous, as they are known in acoustics, for example. Only by superimposing one arrives at the ideological situation of a first incoming and then scattered wave packet. The stationary Schrödinger equation leads to the Helmholtz equation and the inhomogeneous solution to an implicit integral equation, which leads to the asymptotic form of the scattering states:

This asymptotic behavior that make up the wave function of an undisturbed continuous plane wave and an outgoing spherical wave at large distances from the scattering center, is also known as Sommerfeld boundary condition. The physical information about the scattering potential is the scattering amplitude in more detail in amount, the -reach by scattering experiments differential cross section

In the case of a central potential of the angular momentum is conserved, and you get the wave function after simultaneous eigenstates of and the scattering states are then called partial waves and can be, as well as now only develop on the angle -dependent scattering amplitude and the scattering cross section, according to Legendre polynomials, which is also referred to as Partialwellenentwicklung. Another method to calculate the scattering amplitude is the Born approximation.

• Quantum mechanics
• Quantum field theory
• Scattering theory