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A particle of mass m is placed above a rigid horizontal surface. In the presence of a gravitational field g its vertical motion is quantized. Find the asymptotic expression for the nth energy level for n » 1, with n = 0 being the ground state.
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Answer
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To get the requested accuracy, it is sufficient to use the standard WKB approximation,
(1)where En is the WKB value of the energy and H = En/mg is the classical turning point. The phase shift $\pi/4$ in the argument of the cosine is important: it ensures that (n matches onto an exponentially decaying solution at z > H. The boundary condition $\phi_n(0)=0$ is met if $\phi(0)=(n+3/4) \pi$, i.e.,
(2)so that
(3)This expression is accurate to $\mathcal{O}(1/n)$, and so keeping the term 3/4 on the background of n is still legitimate.
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