(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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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
And the number of states between the energy range and is
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
Then the number of states will be
So the number of microstates is
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