Discussion
Using WKB approximation, the allowed energies \(E_{n}\), (n = 1, 2, 3...) for a one-dimensional harmonic oscillator are :
Answer: A
With n = 1, 2, 3, ..., the WKB energies are \(\left(n - \frac{1}{2}\right)\hbar\omega\).
For a potential with two smooth turning points the WKB quantisation condition is
\[ \begin{aligned} \oint p\,dx &= 2\pi\hbar\left(n - \tfrac{1}{2}\right) \\ n &= 1, 2, 3, \ldots \end{aligned} \]
For the oscillator the phase-space orbit is an ellipse of area \(2\pi E/\omega\), so \(E = (n - \frac{1}{2})\hbar\omega\), the exact result.
The familiar form \((n + \frac{1}{2})\hbar\omega\) uses n = 0, 1, 2, ...; with n starting at 1, as the question specifies, it would miss the ground state \(\frac{1}{2}\hbar\omega\). The options with 2n give double the level spacing.
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