A Guide to Physics Problems. Part 2. Thermodynamics, Statistical Physics, and Quantum Mechanics

as found in thefirst solution.

b) Thefirst-orderenergyshift from would be zero (no diagonal elements

in the matrix). Theleadingcorrectionwould be the second-ordershift

as defined by the formula

where means sum over From (S.5.44.3) and (S.5.44.4), we
have

As for any second-ordercorrection to theground state, it is negative. To

makethisexpression equal to the one inpart (a),we require that

So,

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