Quantum Mechanics · Chapter 06

Motion in one dimension

The time-independent Schrödinger equation on a line, and what it gives in the problems every later chapter borrows from: the infinite and the finite square well, the delta-function well, the potential step, barriers that are tunnelled through and wells that let some energies pass untouched. Along the way, why one-dimensional bound states never share an energy and why the next one up always has one more node.

- \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}

Lessons

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05The delta-function wellThe thinnest possible well is a delta-function spike pulling downward. It is solvable in two lines, and it has exactly one bound state: a wavefunction decaying away from the spike on both sides, with a corner at the spike itself. It is the simplest model of anything that holds a particle over a very short range.Read lesson →

06Parity and nodesBound states on a line have two tidy properties. No two of them share an energy — the Wronskian of the Differential Equations course proves it. And if the potential is the same on both sides of a point, every bound state is either even or odd about it. Add the node theorem, that the $n$-th state crosses zero $n-1$ times, and the whole ladder of states can be sketched before a single equation is solved.Read lesson →

07The potential stepSend a particle at a step in the potential. Classically, if it has more energy than the step it rolls over without a hitch, and if it has less it bounces straight back. Quantum mechanically a wave partly reflects even when it has energy to spare, and when it does not, it still reaches a little way into the step before turning round.Read lesson →

08Barriers and tunnellingMake the step a barrier of finite width and something classically impossible happens: a particle without the energy to climb the barrier sometimes appears on the far side. Inside the barrier the wavefunction decays but does not vanish, and whatever is left at the far wall carries on as a wave. The chance falls off exponentially with the width and with the square root of the energy deficit — so steeply that it decides the rates of nuclear decay and the current in a tunnelling microscope.Read lesson →

09Transmission resonancesA particle passing over a well, or over a barrier with energy to spare, usually reflects a little. But at special energies the reflection vanishes completely and the particle passes as if the obstacle were not there. At those energies a whole number of half-wavelengths fits across the obstacle, and the waves reflected from its two edges cancel exactly.Read lesson →

10Chapter practiceBring 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.Read lesson →

Mapping the lesson.

Motion in one dimension

Build the foundation