Read these examples as applications of one recipe, not as a list of orange formulas. First identify the classically allowed region. Then write \(p(x)=\sqrt{2m[E-V(x)]}\). Finally integrate over a complete classical cycle and impose \(\oint p\,dx=nh\).
For the harmonic oscillator, the closed curve in phase space is an ellipse whose area is \(2\pi E/\omega\), so SCQR gives \(E_n=n\hbar\omega\). For the infinite well, the particle moves with constant momentum across the box and back, giving \(2pa=nh\) and \(E_n=n^2h^2/(8ma^2)\). The method is therefore a logical chain from classical motion to allowed energies.
Start with the classical oscillator energy. At fixed \(E\), position and momentum cannot vary independently; they trace a closed ellipse in phase space:
The closed action \(\oint p\,dx\) is the area enclosed by this ellipse:
The SCQR result explains the equal spacing. Full quantum mechanics gives the exact spectrum, \(E_n=(n+1/2)\hbar\omega\).
Inside the box, \(V=0\), so the momentum is constant. The closed classical path goes from one wall to the other and back.
Applying the quantization rule gives the allowed momenta:
The same result can be read as a standing-wave condition, \(a=n\lambda/2\). The walls select wavelengths, and the wavelengths select energies.
The action integral changes with the turning points, determined by \(E=V(x)\). Between those points,
As \(q\) becomes large, the potential approaches the infinite-well limit, so the allowed spectrum gradually resembles the box spectrum.
- Find the classically allowed region.
- Write \(p(x)=\sqrt{2m[E-V(x)]}\).
- Integrate over one complete classical cycle.
- Set \(\oint p\,dx=nh\) and solve for \(E_n\).
The equations make sense only in this order. The orange boxes are the calculation path, not isolated formulas.
This page is designed to support short guided exercises on: Harmonic oscillator, infinite well and power-law potentials as compact examples of semi-classical reasoning.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
- Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
- Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: Harmonic oscillator, infinite well and power-law potentials as compact examples of semi-classical reasoning.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation (Harmonic Oscillator): \[E=\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}.\]
- Equation (Harmonic Oscillator): \[1=\frac{p^2}{2mE}+\frac{x^2}{2E/(m\omega^2)}\]
- Equation (Harmonic Oscillator): \[\oint p\,dx=A=\pi\sqrt{2mE}\sqrt{\frac{2E}{m\omega^2}}=\frac{2\pi E}{\omega}\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
- Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.