The one-dimensional harmonic oscillator in thermodynamic equilibrium
An important case of a statistical mixture is that of a system in thermodynamic equilibrium with a heat reservoir at temperature T. The various possible dynamical states are the eigenstates of the Hamiltonian H. The statistical weight of a given eigenstate depends upon the corresponding eigenvalue of H. It is proportional to the Boltzmann factor exp(-E/(kT)), in which E is the eigenvalue of H and k is the Boltzmann constant. (k = 1.38*10-23J/K)
A system in thermodynamic equilibrium can be represented by the density operator
ρ = ∑npnρn = ∑npn|n><n|,
where {|n>} are the eigenstates of H,
H|n> = En|n>.
Then pn = Nexp(-En/(kT)), where N is a
normalization constant to make the total probability equal to one.
ρ = ∑nNexp(-En/(kT))|n><n| = ∑nNexp(-H/(kT))|n><n|
= Nexp(-H/(kT)),
since ∑n|n><n| = I.
We need Tr(ρ) = 1, Tr(Nexp(-H/(kT)) = 1, Tr(exp(-H/(kT))
= N-1,
N-1 is called the partition function Z,
and we write ρ = Z-1exp(-H/(kT)).
Let us calculate the partition function for the harmonic oscillator.
Z = ∑n<n|exp(-En/(kT)|)n> = ∑nexp(-(n+½)ħω/(kT))
= exp(-ħω/(2kT)) ∑nexp(-(nħω/kT).
(1 - x)-1 = 1 + x + x2 + x3 + ... =
∑nxn.
Therefore ∑nexp(-(nħω/(kT) ) = ∑nexp(-(ħω/(kT))n =
(1 - exp(-(ħω/(kT))-1,
and
Z = exp(-ħω/(2kT))/[1 - exp(-ħω/(kT))].
Now let us calculate the average energy.
<H> = Tr(ρH) = Z-1Tr(exp(-H/(kT)) H) = Z-1∑n(n+½)ħω exp(-(n+½)ħω/(kT))
= kT2(1/Z)dZ/dT,
since
dZ/dT = d/dT ∑nexp(-(n+½)ħω/(kT)) = (1/(kT2))∑n(n+½)ħω exp(-(n+½)ħω/(kT)).
Using Z = exp(-ħω/(2kT))/[1 - exp(-ħω/(kT))]
we
find a simpler expression for dZ/dT.
dZ/dT = (ħω/(2kT2)) exp(-ħω/(2kT))/[1 - exp(-ħω/(kT))]
+
(ħω/(kT2))exp(-ħω/(kT))exp(-ħω/(2kT))/[1 - exp(-ħω/(kT))]2
= Z(ħω/(2kT2)) + Z(ħω/(kT2)))exp(-ħω/(kT))/[1 - exp(-ħω/(kT))].
We now have <H> = ½ħω + ħω/[exp(ħω/(kT)) - 1].
This is Planck’s formula (to within a constant ½ħω) for the average energy of a quantized
oscillator.
The energy of a classical one dimensional oscillator is E(x,p) = ½p2/m
+ ½mω2x2.
The mean energy of such an oscillator in
thermodynamic equilibrium at temperature T is
<E> = ∫∞∞∫∞∞E(x,p)exp(-E(x,p)/(kT))dxdp/∫∞∞∫∞∞exp(-E(x,p)/(kT))dxdp
= kT.
| Temperature | QM oscillator <H> | Classical oscillator <E> |
| T --> 0 | ½ħω | 0 |
| kT << ħω | ½ħω + ħω exp(-ħω/(kT)) | kT |
| kT >> ħω | ½ħω + ħω/(ħω/(kT)) ≈ kT | kT |
Note: For the three-dimensional harmonic oscillator in thermodynamic equilibrium the mean energy <H> is three times that of a one-dimensional oscillator with the same frequency.