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(a) you had a set of N quantum mechanical oscillators with total energy E =N hw + M hw , 2 and you found the number of states with energy between E and E + 8E is given by In N ~ (M + N) In(M + N) – M In M – N In N + In(§E/hw) . Write this in terms of E, not M, and show that a collection of N harmonic oscillators at temperature T has energy Nhw (ehw/kT ehw/kT +1 E = N + 2 1 2 ehw /kT (b) Show that when kT < hw, the oscillators are all just sitting in their ground states (state of lowest energy) to a very good approximation. (c) Show that E - NkT when kT > ħw. We will see, later in the course, that this is the answer one gets from classical physics.

(a) you had a set of N quantum mechanical oscillators with total energy E =N hw + M hw , 2 and you found the number of states with energy between E and E + 8E is given by In N ~ (M + N) In(M + N) – M In M – N In N + In(§E/hw) . Write this in terms of E, not M, and show that a collection of N harmonic oscillators at temperature T has energy Nhw (ehw/kT ehw/kT +1 E = N + 2 1 2 ehw /kT (b) Show that when kT < hw, the oscillators are all just sitting in their ground states (state of lowest energy) to a very good approximation. (c) Show that E - NkT when kT > ħw. We will see, later in the course, that this is the answer one gets from classical physics.

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(a) you had a set of N quantum mechanical oscillators with total energy 1 E = Nhw + Mhw, and you found the number of states with energy between E and E + 8E is given by In N~ (M + N) In(M + N) - M In M – N In N + In(§E/hw) . Write this in terms of E, not M, and show that a collection of N harmonic oscillators at temperature T has energy ehw/kT 1 = N hw E = 2 elw /kT – 1 ehw/kT – 1 (b) Show that when kT < hw, the oscillators are all just sitting in their ground states (state of lowest energy) to a very good approximation. (c) Show that E - NkT when kT > ħw. We will see, later in the course, that this is the answer one gets from classical physics.
Transcribed Image Text:(a) you had a set of N quantum mechanical oscillators with total energy 1 E = Nhw + Mhw, and you found the number of states with energy between E and E + 8E is given by In N~ (M + N) In(M + N) - M In M – N In N + In(§E/hw) . Write this in terms of E, not M, and show that a collection of N harmonic oscillators at temperature T has energy ehw/kT 1 = N hw E = 2 elw /kT – 1 ehw/kT – 1 (b) Show that when kT < hw, the oscillators are all just sitting in their ground states (state of lowest energy) to a very good approximation. (c) Show that E - NkT when kT > ħw. We will see, later in the course, that this is the answer one gets from classical physics.
Expert Solution
Step 1

Given,

The total energy of the N quantum mechanical oscillator is

E=12Nhω+Mhω

And the number of states between the energy range E and E+δE is

lnΩ~M+NlnM+N-MlnM-NlnN+lnδEhω=MlnM+N-MlnM+NlnM+N-NlnN+lnδEhω=MlnM+NM+NlnM+NN+lnδEhω

a)

And we have to write this in terms of E, and eliminate M. So to do this, first, we have to write M in terms of E and N from the total energy

M=1hωE-12Nhω

Then the number of states will be

lnΩ~MlnM+NM+NlnM+NN+lnδEhω=1hωE-12Nhωln1hωE-12Nhω+N1hωE-12Nhω+Nln1hωE-12Nhω+NN+lnδEhω=1hωE-12NhωlnE+12NhωE-12Nhω+NlnE+12NhωhωN+lnδEhω

So the number of microstates is

lnΩ~1hωE-12NhωlnE+12NhωE-12Nhω+NlnE+12NhωhωN+lnδEhω

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