Logarithmic Potential in WKB

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The Bohr-Sommerfeld quantization condition for this problem is

Note that defines We may then rewrite the integral as

Let us now make the substitution, We then obtain

or, evaluating the integral,

Solving for we obtain

The energy spectrum is thus

If we now calculate the spacing between two adjacent energy levels, we obtain

We see that this spacing is indeed independent of mass (and, in fact, of as well).

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