Tamm-Dancoff approximation
With the Tamm-Dancoff approximation (TDA) it is possible to describe collective behavior in the many-particle theory of fermions using 1-particle-1-hole excitations (1p1h excitation). It is named after Igor Tamm (1945) and Sidney Dancoff (1950)
Description using an example from nuclear physics
Here the TDA is explained using the example of nuclear physics. In the shell model of the theory of atomic nuclei , the shell structure is obtained by solving the Schrödinger equation for an average nuclear potential. Analytical expressions for the shell energies are obtained e.g. B. for a box potential or the spherical harmonic oscillator . Better results are obtained for a Woods-Saxon potential (box potential with a fuzzy edge). The correct magic numbers can only be obtained by taking into account the spin-orbit interaction . This level of the model takes into account only averaged effects of the relatively complicated nature of the interaction of the nucleons with one another, which is expressed, for example, in the Woods-Saxon potential plus spin-orbit interaction.
If the shells obtained in this way are simply filled with nucleons, the real atomic nucleus is only obtained in a very rough approximation . The simplest excitations in the shell model are 1-particle-1-hole excitations. A nucleon is lifted from a shell into a higher, as yet unoccupied shell. This procedure delivers good results for nuclei with few valence nucleons, i.e. few nucleons outside the last closed shell. With a larger number of valence nucleons, this approach becomes problematic, since an excitation z. B. can also be a quantum mechanical superposition of many 1-nucleon excitations. One then speaks of correlations and collectivity .
This is exactly where the Tamm-Dancoff approximation begins. The approximation is that only 1-particle-1-hole excitations are taken into account. The wave function of the excited states is calculated on the basis of the shell model ground state. Excited states are described by a superposition of all 1p1h excitations and use the Hartree-Fock method . One calculates the energetically lowest state, reduces the Hilbert space of the system by removing this state and only taking into account orthogonal states, calculates again the energetically lowest state in the new smaller Hilbert space, etc. In practice, however, one often cannot find all possible 1p1h- Consider excitations, but cuts off the Hilbert space at energetically higher 1p1h excitations, e.g. B. after the next or the next unoccupied shell.
The states calculated in this way can have a collective character and B. represent vibrations of the core surface. Such states are often referred to as phonons, analogous to lattice excitations in solid-state physics . However, the analogy is not exact.
The TDA is relatively good, so that you can also couple different phonons - which goes beyond the actual scope of the approximation. This leads to more highly excited states that are almost exactly described by the TDA, but represent highly complex configurations in the pure shell model. An example: In most gg nuclei (even number of protons and neutrons) with little / no deformation (i.e. near the shell) one finds the low lying states 2+ and 3− (the numbers mean angular momentum values and parity ). These states can be interpreted as quadrupole and octupole deformation vibrations of the nuclear surface. The sum energy of these two states is a triplet state made up of 1−, 3− and 5−, which is generated by coupling the two phonons. If the coupling were perfect, these three states would have exactly the same energy, but due to a slightly anharmonic coupling these three states split energetically.
A disadvantage of the TDA is that the ground state does not explicitly contain any correlations. However, this is eliminated in one of the TDA-related methods, the Random Phase Approximation (RPA).