The energy E of a system of three independent harmonic oscillators is given by 1 1 E = (nx + ½ )ħw + (ny + ½⁄3)ħw + (nz + 12 )ħw. (20.49) Show that the partition function Z is given by Z = ZŠHO, (20.50) where ZSHO is the partition function of a simple harmonic oscillator given in eqn 20.3. Hence show that the Helmholtz function is given by 3 F = =ħw + 3kpT ln(1 -Bħw) (20.51)
The energy E of a system of three independent harmonic oscillators is given by 1 1 E = (nx + ½ )ħw + (ny + ½⁄3)ħw + (nz + 12 )ħw. (20.49) Show that the partition function Z is given by Z = ZŠHO, (20.50) where ZSHO is the partition function of a simple harmonic oscillator given in eqn 20.3. Hence show that the Helmholtz function is given by 3 F = =ħw + 3kpT ln(1 -Bħw) (20.51)
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Transcribed Image Text:The energy E of a system of three independent
harmonic oscillators is given by
1
E = (Nx + ½)ħw + (ny + ½ )ħw + (nz + 1⁄)ħw.
2
(20.49)
Show that the partition function Z is given by
Z = ZSHO,
(20.50)
where ZSHO is the partition function of a simple
harmonic oscillator given in eqn 20.3. Hence show
that the Helmholtz function is given by
3
F = = ħw + 3k³Tln(1 – e¯ßhw), (20.51)
and that the heat capacity tends to 3kB at high
temperature.
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