Lecture Notes#

Homework#

1. Harmonic oscillator. For the harmonic oscillator \(V(x) = \frac{1}{2}m\omega^{2}x^{2}\), the classical turning points at energy \(E\) are \(x = \pm a\) with \(a = \sqrt{2E/(m\omega^{2})}\).

(a) Compute the orbit integral \(\oint p\,\mathrm{d}x = 2\int_{-a}^{a}\sqrt{2m(E - \tfrac{1}{2}m\omega^{2}x^{2})}\,\mathrm{d}x\) and show that it equals \(2\pi E/\omega\).

(b) Apply the Bohr-Sommerfeld rule (114) to derive \(E_{n} = \hbar\omega(n + 1/2)\). Note that this is the exact quantum result.

2. Particle in a box. A particle of mass \(m\) is confined between hard walls at \(x = 0\) and \(x = a\). Inside the box \(V = 0\), so \(p = \sqrt{2mE}\).

(a) For hard walls, the Maslov index is \(\mu = 1\) per wall (versus \(\mu = 1/2\) at a soft turning point), so use the generalized rule (115) with \(\mu_{a} = \mu_{b} = 1\).

(b) Show that this gives \(E_{n} = (n+1)^{2}\pi^{2}\hbar^{2}/(2ma^{2})\) for \(n = 0, 1, 2,\ldots\), which is the exact spectrum (with the conventional relabeling \(n\to n-1\)).

(c) Explain physically why the hard-wall Maslov index differs from the soft case: what does the wave do at a rigid boundary versus at a smooth turning point?

3. Linear potential. A particle in a half-line gravitational-like potential \(V(x) = Fx\) for \(x \geq 0\), with a hard wall at \(x = 0\).

(a) At energy \(E\), find the soft turning point \(x_{a} = E/F\) and compute \(\int_{0}^{x_{a}}\sqrt{2m(E - Fx)}\,\mathrm{d}x = \tfrac{2}{3F}(2mE)^{3/2}\).

(b) Write the Bohr-Sommerfeld rule with the correct Maslov indices (\(\mu = 1\) at the hard wall, \(\mu = 1/2\) at the soft turning point) and solve for \(E_{n}\).

(c) Show that the energy spacing \(\Delta E_{n}\) decreases with \(n\), unlike the harmonic oscillator. Why does a linear potential produce non-uniform level spacing?

4. Anharmonic oscillator. Consider \(V(x) = \tfrac{1}{2}m\omega^{2}x^{2} + \lambda x^{4}\) with \(\lambda > 0\) small.

(a) Explain qualitatively why the quartic term steepens the potential at large \(\vert x\vert\), shrinking the classical turning points compared to the pure harmonic case at the same energy.

(b) Argue that the orbit integral \(\oint p\,\mathrm{d}x\) is therefore smaller than for the harmonic oscillator at the same energy, so Bohr-Sommerfeld predicts higher energy levels. Are the energy spacings larger or smaller than \(\hbar\omega\)?

(c) For which regime (low \(n\) or high \(n\)) is the quartic correction most important? Use the ratio \(\lambda x^{4}/(\tfrac{1}{2}m\omega^{2}x^{2})\) at the turning point.

5. Two-dimensional anisotropic oscillator. A particle moves in \(V(x,y) = \tfrac{1}{2}m\omega_{x}^{2}x^{2} + \tfrac{1}{2}m\omega_{y}^{2}y^{2}\) with \(\omega_{x} \neq \omega_{y}\).

(a) The motion separates: \(x\) and \(y\) each oscillate independently. Apply Bohr-Sommerfeld to each direction to derive \(E_{n_{x},n_{y}} = \hbar\omega_{x}(n_{x} + \tfrac{1}{2}) + \hbar\omega_{y}(n_{y} + \tfrac{1}{2})\).

(b) For \(\omega_{x} = 2\omega_{y} = 2\omega\), list the first five distinct energy levels in units of \(\hbar\omega\) and identify any degeneracies.

(c) The same construction fails for non-separable (in particular chaotic) potentials. In one sentence, explain why the rule “quantize each closed action variable independently” requires the system to be integrable (as many conserved quantities as degrees of freedom).

6. Hydrogen atom (circular orbits). For the Coulomb potential \(V(r) = -q_{e}^{2}/(4\pi\epsilon_{0}r)\), consider circular orbits with angular momentum \(L = n_{\varphi}\hbar\).

(a) From the force balance \(mv^{2}/r = q_{e}^{2}/(4\pi\epsilon_{0}r^{2})\) and \(L = mvr = n_{\varphi}\hbar\), show that \(r_{n_{\varphi}} = n_{\varphi}^{2}a_{0}\) where \(a_{0} = 4\pi\epsilon_{0}\hbar^{2}/(m q_{e}^{2})\) is the Bohr radius.

(b) Use the virial theorem (\(E = -T = V/2\) for the Coulomb potential) to derive \(E_{n_{\varphi}} = -13.6\,\text{eV}/n_{\varphi}^{2}\).

(c) Adding the radial Bohr-Sommerfeld quantization with a radial quantum number \(n_{r}\) promotes \(n_{\varphi}\) to \(n = n_{r} + n_{\varphi}\). Explain why this reproduces the same formula \(E_{n} = -13.6\,\text{eV}/n^{2}\) with additional degeneracy.

7. Correspondence principle. In the classical limit (\(n \gg 1\)), the energy spacing \(\Delta E = E_{n+1} - E_{n}\) should equal \(\hbar\omega_\text{cl}\), where \(\omega_\text{cl}\) is the classical orbital frequency.

(a) For the harmonic oscillator, verify \(\Delta E = \hbar\omega\) for all \(n\).

(b) For a power-law potential \(V(x) = C\vert x\vert^{\alpha}\), Bohr-Sommerfeld gives \(E_{n} \propto n^{2\alpha/(\alpha+2)}\). Show that \(\Delta E/E_{n}\to 0\) as \(n\to\infty\), so the spectrum becomes quasi-continuous in the classical limit.

(c) For hydrogen (\(E_{n}\propto -1/n^{2}\)), compute the classical orbital frequency \(\omega_\text{cl} = \Delta E/\hbar\) at large \(n\) and show it scales as \(n^{-3}\). This is the radiation frequency in transitions between adjacent high-\(n\) Rydberg levels.