The infinite well is the chapter's cleanest example. The particle is free inside the box, but the walls impose boundary conditions. Those boundary conditions force standing waves, and standing waves force discrete energies.
The potential is zero inside the interval and infinite outside:
The infinite walls force the wave function to vanish at the boundaries:
Inside the well, \(V=0\), so the TISE becomes
The general oscillatory solution is
The first boundary condition gives \(A+B=0\), so \(B=-A\) and the solution becomes a sine wave.
After using \(\psi(0)=0\), write \(\psi(x)=\tilde A\sin(kx)\). The second boundary condition gives
Therefore
Normalization fixes the remaining amplitude:
Each higher \(n\) adds nodes and raises the energy quadratically. The result is not a decorative standing wave: it is the first concrete example of an energy spectrum obtained from boundary conditions.
This result is worth comparing with Chapter 1 · Item 1.13. The semi-classical quantization rule treats the particle as moving classically from wall to wall and back, so the closed action is
That gives \(p_n=nh/(2a)\) and therefore the same energy spectrum,
The agreement is useful, but the wave-mechanics solution is richer: it gives \(\psi_n(x)\), nodes, normalization and the probability density \(|\psi_n(x)|^2\), not only the allowed energies.
The infinite well is an exercise in translating walls into boundary conditions. Inside the well, the particle is free; at the walls, the wave function must vanish:
The allowed spatial functions are sine waves that fit an integer number of half-wavelengths into the box:
The energy spectrum follows from the curvature of these functions. Higher \(n\) means more nodes, shorter wavelength, larger momentum scale and larger energy.
This page is designed to support short guided exercises on: Boundary conditions, allowed sine functions, normalized eigenstates, nodes, probability densities and the quadratic energy spectrum.
- 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: Boundary conditions, allowed sine functions, normalized eigenstates, nodes, probability densities and the quadratic energy spectrum.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[V(x)=0\quad(0\lt x\lt a),\qquad V(x)=+\infty\quad(x\le 0\ \mathrm{or}\ x\ge a).\]
- Equation: \[\psi_n(x)=\sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right),\qquad n=1,2,3,\ldots\]
- Equation: \[E_n=\frac{n^2\pi^2\hbar^2}{2ma^2}=\frac{n^2h^2}{8ma^2}.\]
- 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.