Quantum Mechanics · Lesson 10

Chapter practice

Bring the chapter together: the time-independent equation and its allowed and forbidden regions, matching conditions, the infinite, finite and delta wells, parity and nodes, the step, tunnelling and resonances. No worked example sits above the answers.

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- \frac{ℏ ^{2}}{2 m} ψ^{''} + V ψ = E ψ

E_{n} = \frac{n ^{2} π ^{2} ℏ ^{2}}{2 m L ^{2}}

T + R = 1

κ = \frac{2 m ( V - E )}{ℏ}

Chapter navigation

Complete Transmission resonances first.

Intuition

Bring the chapter together: the time-independent equation and its allowed and forbidden regions, matching conditions, the infinite, finite and delta wells, parity and nodes, the step, tunnelling and resonances. No worked example sits above the answers.

A cook with a few basic sauces can make a hundred dishes. The one-dimensional problems are those sauces; every later chapter uses them.

Before you practise

Solve piece by piece: oscillating where $E>V$ , exponential where $E<V$ , matched by continuity of $\psi$ and, where $V$ is finite, of $\psi'$ . Bound states decay at both ends and come in a discrete, non-degenerate ladder; scattering states are read through currents.

\begin{aligned}&E_{n}=\frac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}},\qquad R+T=1\\ &T\approx16\tfrac{E}{V_{0}}\bigl(1-\tfrac{E}{V_{0}}\bigr)e^{-2\kappa a}\end{aligned}

Reminders

Delta well: one state, $\kappa=m\alpha/\hbar^{2}$ , $E=-m\alpha^{2}/2\hbar^{2}$ .
Finite well: at least one bound state; about $2z_{0}/\pi$ in all.
Step: $R=\bigl(\tfrac{k_{1}-k_{2}}{k_{1}+k_{2}}\bigr)^{2}$ , with reflection even above the step.
Resonance: $T=1$ when $k'a=n\pi$ .

Applications