121
5.1. The quantum field: (i) descriptive
it is possible to write E as a sum of N separate terms, just as in (5.23):
N
∑
E =
[
1 mQ ˙ 2 +
1 mω
2 Q
2 ].
(5.27)
r
r r
r=1
2
2
The Q r ’s are the normal coordinates and the ω r ’s are the normal frequencies,
and there are N of them. If only one of the Q r ’s is non-zero, the N atoms are
moving in a single mode. The fact that the total energy in (5.27) is a sum of
N single-mode energies allows us to say that our N -atom solid behaves as if
it consisted of N separate and free harmonic oscillators – which, however, are
not to be identified with the coordinates of the original atoms. Once again,
and now much more crucially, it is the mode coordinates that are the relevant
degrees of freedom rather than those of the original particles.
The second stage in our programme is to treat such systems quantum
mechanically, as we should certainly have to for a real solid. It is still true
that – if the potential energy is a quadratic function of the displacements –
the transformation (5.26) allows us to write the total energy as a sum of N
mode energies, each of which has the form of a harmonic oscillator. Now,
however, these oscillators obey the laws of quantum mechanics, so that each
mode oscillator exists only in certain definite states, whose energy eigenvalues
are quantized. For each mode of frequency ω r , the allowed energy values are
1
∈ r = (n r + 2 )ħω r
(5.28)
where n r is a positive integer or zero. This is in sharp contrast to the classical
case, of course, in which arbitrary values are allowed for the oscillator energies.
The total energy eigenvalue then has the form
N
∑
1
E =
(n r + )ħω r .
(5.29)
2
r=1
The frequencies ω r are determined by the interatomic forces and are common
to both the classical and quantum descriptions; in quantum theory, though,
the states of definite energy of the vibrating N-body system are characterized by
the values of a set of integers (n 1 , n 2 , . . . , n N ), which determine the energies
of each mode oscillator.
For each mode oscillator, ħω r measures the quantum of vibrational energy;
the energy of an allowed mode state is determined uniquely by the number n r
of such quanta of energy in the state. We now make a profound reinterpretation of this result (first given, almost en passant by Born, Heisenberg and
Jordan (Born et al. 1926) in one of the earliest papers on quantum mechanics). We forget about the original N degrees of freedom q 1 , q 2 , . . . , q N and the
original N ‘atoms’, which indeed are only remembered in (5.29) via the fact
that there are N different mode frequencies ω r . Instead we concentrate on
the quanta and treat them as ‘things’ which really determine the behaviour
1
of our quantum system. We say that ‘in a state with energy (n r + )ħω r there
2
5.1. The quantum field: (i) descriptive
it is possible to write E as a sum of N separate terms, just as in (5.23):
N
∑
E =
[
1 mQ ˙ 2 +
1 mω
2 Q
2 ].
(5.27)
r
r r
r=1
2
2
The Q r ’s are the normal coordinates and the ω r ’s are the normal frequencies,
and there are N of them. If only one of the Q r ’s is non-zero, the N atoms are
moving in a single mode. The fact that the total energy in (5.27) is a sum of
N single-mode energies allows us to say that our N -atom solid behaves as if
it consisted of N separate and free harmonic oscillators – which, however, are
not to be identified with the coordinates of the original atoms. Once again,
and now much more crucially, it is the mode coordinates that are the relevant
degrees of freedom rather than those of the original particles.
The second stage in our programme is to treat such systems quantum
mechanically, as we should certainly have to for a real solid. It is still true
that – if the potential energy is a quadratic function of the displacements –
the transformation (5.26) allows us to write the total energy as a sum of N
mode energies, each of which has the form of a harmonic oscillator. Now,
however, these oscillators obey the laws of quantum mechanics, so that each
mode oscillator exists only in certain definite states, whose energy eigenvalues
are quantized. For each mode of frequency ω r , the allowed energy values are
1
∈ r = (n r + 2 )ħω r
(5.28)
where n r is a positive integer or zero. This is in sharp contrast to the classical
case, of course, in which arbitrary values are allowed for the oscillator energies.
The total energy eigenvalue then has the form
N
∑
1
E =
(n r + )ħω r .
(5.29)
2
r=1
The frequencies ω r are determined by the interatomic forces and are common
to both the classical and quantum descriptions; in quantum theory, though,
the states of definite energy of the vibrating N-body system are characterized by
the values of a set of integers (n 1 , n 2 , . . . , n N ), which determine the energies
of each mode oscillator.
For each mode oscillator, ħω r measures the quantum of vibrational energy;
the energy of an allowed mode state is determined uniquely by the number n r
of such quanta of energy in the state. We now make a profound reinterpretation of this result (first given, almost en passant by Born, Heisenberg and
Jordan (Born et al. 1926) in one of the earliest papers on quantum mechanics). We forget about the original N degrees of freedom q 1 , q 2 , . . . , q N and the
original N ‘atoms’, which indeed are only remembered in (5.29) via the fact
that there are N different mode frequencies ω r . Instead we concentrate on
the quanta and treat them as ‘things’ which really determine the behaviour
1
of our quantum system. We say that ‘in a state with energy (n r + )ħω r there
2