THE UNIVERSITY OF BRITISH COLUMBIA

PHYSICS 455

Lecture # 6 :

Fri. 17 Jan. 1997

Two-State Systems

I. Recapitulate:

• Partition Function: Z = exp(-/)

• Mean Thermal Energy: U = exp(-/) = ² (log Z)/

II. Simplest Possible Case: TWO-STATE SYSTEM

1. Let the energy of the lower state be zero and that of the upper state be .

   Then Z = 1 + exp(-/)

   and U = e^-/. /Z = / (e^/ + 1)

2. Alternatively, we can set the zero of energy halfway in between the two states (as in the case of a spin-½ particle in a magnetic field), so that the lower state has an energy of -/2 and the upper state has an energy of +/2.

   Then Z = exp(+/2) + exp(-/2) = 2 cosh(/2)

   and U = -(/2) tanh(/2).

C_V = (/)^{2 .} e^/ / (e^/ + 1)².

LIMITING CASES:

• 0: / 1 so e^/ 1, leaving U e^-/ 0 and C_V (/)² e^-/ 0 as well. (The exponential always grows or shrinks faster than any power law.)

• : / 0 so e^/ 1 /, giving U /(2+/) /2 (i.e. no preference for either state) and C_V (/)² (1+/) / (2+/)² ½ (/)² 0. (Once both states are equally populated, there is no place to put any more energy on a statistical basis.)