Elements of Modern Physics
132
In Chapter 4, it was shown, that the quantum-mechanical framework provides a
detailed and accurate description of the energy levels of the one-electron atom.
In this chapter, the attempts to understand the properties of many-electron atoms
and molecules will be discussed.
It is generally difficult to obtain accurate solutions to the problem of
N interacting particles, where N ≥ 3, whether in classical mechanics or in quantum
mechanics (simple harmonic potential is an exception). However, there is a
special symmetry property for spin 1/2 particle sin quantum mechanics, called
the Pauli exclusion principle, which gives a very satisfactory qualitative and
often quantitative understanding of many-electron atoms and molecules.
5.1 EXCHANGE SYMMETRY OF WAVE FUNCTIONS
In the treatment of two or more identical particles, such as electrons, within the
framework of quantum mechanics, the uncertainty principle limits our ability to
follow the motion of the particles without disturbing the system (in classical
mechanics this disturbance can be made indefinitely small). Therefore, in general,
it cannot be ascertained as to which of the identical particles has been found at
a place.
Consider the Hamiltonian H (1, 2, ...N) of N identical interacting particles,
where the numbers represent all the variables of the particles, i.e. spatial
coordinates and spin variables. Since the particles are identical, the following is
true:
H (1, 2 ..., i ..., j, ... N) = H (1, 2, ..., j, ..., i, ...N)
(5.1)
Now, an operator P ij which interchanges all the coordinates of articles i and
j is defined as:
P ij ψ (1, 2, ... i, ..., j,...N) = H (1, 2, ..., j, ..., i, ...N)
(5.2)
It is easy to show that P ij is a hermitian operator (See Eq. 3.35). Furthermore, in view of Eq. (5.1), it is seen that
P ij H (1, 2, ... i, ..., j, ... N) = H (1, 2, ..., i, ..., j, ...N) P ij
(5.3)
This means that states which are simultaneous eigenstates of H and P ij can
be chosen (see Sec. 3.4).
The eigenvalues of P ij can be deduced from Eq. (5.2) by noting that
2
ij
P = 1
(5.4)
which implies that the possible eigenvalues of P ij are + 1 or –1. It is experimentally
observed that the physical states indeed are (not just ‘can be’) eigenstates of P ij
and the eigenvalues are characteristic of the nature of the particles. This result
is stated in terms of the following rules:
132
In Chapter 4, it was shown, that the quantum-mechanical framework provides a
detailed and accurate description of the energy levels of the one-electron atom.
In this chapter, the attempts to understand the properties of many-electron atoms
and molecules will be discussed.
It is generally difficult to obtain accurate solutions to the problem of
N interacting particles, where N ≥ 3, whether in classical mechanics or in quantum
mechanics (simple harmonic potential is an exception). However, there is a
special symmetry property for spin 1/2 particle sin quantum mechanics, called
the Pauli exclusion principle, which gives a very satisfactory qualitative and
often quantitative understanding of many-electron atoms and molecules.
5.1 EXCHANGE SYMMETRY OF WAVE FUNCTIONS
In the treatment of two or more identical particles, such as electrons, within the
framework of quantum mechanics, the uncertainty principle limits our ability to
follow the motion of the particles without disturbing the system (in classical
mechanics this disturbance can be made indefinitely small). Therefore, in general,
it cannot be ascertained as to which of the identical particles has been found at
a place.
Consider the Hamiltonian H (1, 2, ...N) of N identical interacting particles,
where the numbers represent all the variables of the particles, i.e. spatial
coordinates and spin variables. Since the particles are identical, the following is
true:
H (1, 2 ..., i ..., j, ... N) = H (1, 2, ..., j, ..., i, ...N)
(5.1)
Now, an operator P ij which interchanges all the coordinates of articles i and
j is defined as:
P ij ψ (1, 2, ... i, ..., j,...N) = H (1, 2, ..., j, ..., i, ...N)
(5.2)
It is easy to show that P ij is a hermitian operator (See Eq. 3.35). Furthermore, in view of Eq. (5.1), it is seen that
P ij H (1, 2, ... i, ..., j, ... N) = H (1, 2, ..., i, ..., j, ...N) P ij
(5.3)
This means that states which are simultaneous eigenstates of H and P ij can
be chosen (see Sec. 3.4).
The eigenvalues of P ij can be deduced from Eq. (5.2) by noting that
2
ij
P = 1
(5.4)
which implies that the possible eigenvalues of P ij are + 1 or –1. It is experimentally
observed that the physical states indeed are (not just ‘can be’) eigenstates of P ij
and the eigenvalues are characteristic of the nature of the particles. This result
is stated in terms of the following rules: