Show that the "old quantum theory" (Problem 12.4) predicts...

Show that the "old quantum theory" (Problem 12.4) predicts that the energy levels of the harmonic oscillator are $n \hbar \omega$ rather than $\left(n+\frac{1}{2}\right) \hbar \omega$. Comment on the dependence on $n$ of the fractional error in $E_{n}$.

Key Concepts

Bohr-Sommerfeld Quantization

This is a semiclassical method originating from the old quantum theory where the action integral over a complete cycle is set equal to an integer multiple of h (or ?). It leads to quantization conditions that, while capturing some features of quantum behavior, miss subtle corrections (like the zero-point energy) present in a full quantum mechanical treatment.

Quantum Harmonic Oscillator

The harmonic oscillator is a central model in quantum mechanics featuring a quadratic potential. In the exact quantum solution, the energy levels are given by (n + 1/2)??. This formula reflects both the discrete energy structure and the inherent zero-point energy due to the uncertainty principle.

Zero-Point Energy

The exact quantum treatment of systems such as the harmonic oscillator reveals a nonzero minimum energy, (1/2)??, known as the zero-point energy. The omission of this term in the old quantum theory’s predictions leads to discrepancies between the semiclassical and full quantum results.

Semiclassical Approximation Errors

Due to the absence of the zero-point energy term, the old quantum theory predicts energy levels proportional to n??, which are lower than the true energies by (1/2)??. Consequently, the fractional error, defined as the error divided by the actual energy, behaves roughly as 1/(2n) for large n, indicating that the relative discrepancy decreases as the quantum number increases.

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