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Solid State Physics. Philip Hofmann
Читать онлайн.Название Solid State Physics
Год выпуска 0
isbn 9783527837267
Автор произведения Philip Hofmann
Жанр Физика
Издательство John Wiley & Sons Limited
However, this simple approach is physically incorrect because such a wave function does not obey the Pauli principle. Since the electrons are fermions, the total wave function must be antisymmetric with respect to particle exchange, and the simple product wave function does not meet this requirement. The total wave function consists of a spatial part and a spin part, and thus there are two possibilities for forming an antisymmetric wave function – we can either combine a symmetric spatial part with an antisymmetric spin part or vice versa. The spatial part of the wave function can be constructed in two ways,
(2.9)
The plus sign in Eq. (2.9) returns a symmetric spatial wave function, which we can combine with an antisymmetric spin wave function with the total spin equal to zero (the so‐called singlet state); the minus in Eq. (2.10) results in an antisymmetric spatial wave function to be combined with a symmetric spin wave function with the total spin equal to 1 (the so‐called triplet state).
The antisymmetric wave function in Eq. (2.10) vanishes for
Figure 2.4 The energy changes
and for the formation of a hydrogen molecule. The dashed lines represent the approximation for long distances. The two insets show grayscale images of the corresponding electron probability density.An approximate way to calculate the eigenvalues of Eq. (2.3) was suggested by W. Heitler and F. London in 1927. Their approach was to use the known single‐particle 1s wave functions for atomic hydrogen for
(2.11)
According to the variational principle in quantum mechanics, the resulting energy will always be higher than the correct ground‐state energy, but it will approach it for a good choice of the trial wave functions.
The detailed calculation is quite lengthy and shall not be given here.1 The resulting ground‐state energies for the singlet and triplet states can be written as
(2.12)
(2.13)