Thermodynamics, Statistical Physics, and Quantum Mechanics

Since we know that Anticipating the result, let Form the raising and lowering operators and

Find the commutators

Frompart (a) we know that We now ask what is the eigenvalue of for the states

So, these states have the same eigenvalue of Now, examine the eigenvalue of for these states:

In (S.5.23.3) we see that has the effect of raising or lowering the
of the states so that

where are the corresponding coefficients. As determined above, we
knowthat so cannot be applied indefinitely to the state
there must be an such that

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