Chapter 02
The wavefunction and the Schrödinger equation
The first computational toolkit: wavefunctions, eigenvalue problems, exact 1D solutions.
- Sections
- 11
- Finalised
- 11/11
- Simulations
- 7
- Estimated time
- 4 hours
- Read first
- Chapter 01
- 2.1The probability interpretation of the wavefunction◈ 1 sim(s)The wavefunction is not the shape of the particle, nor a density of particle stuff. It is a complex function, and only its modulus squared connects to experiment.
- 2.2Normalisation and probability currentThe total probability has to stay 1 forever. Proving that also hands us a quantity describing where the probability is flowing.
- 2.3The time-independent Schrödinger equationSeparating variables splits the time-dependent problem in two: an eigenvalue problem that fixes the levels, and a trivial phase factor that handles the evolution.
- 2.4The time-dependent Schrödinger equationStationary states are only particular solutions. The general one is a linear superposition of them — and once the coefficients are fixed, the entire future is fixed with them.
- 2.5Stationary and non-stationary states◈ 1 sim(s)What is stationary is the probability distribution, not the particle. Get that straight and half the confusion in quantum mechanics disappears.
- 2.6Expectation values, variance and the uncertainty principle◈ 1 sim(s)Extracting numbers from a probability distribution: means, fluctuations, and one inequality that constrains them.
- 2.7The one-dimensional infinite square well◈ 1 sim(s)Quantum mechanics' "hydrogen atom before hydrogen": completely solvable, and almost every quantum feature already shows up.
- 2.8The one-dimensional finite square wellLower the walls to a finite height and three new things happen: the bound states become finite in number, the wavefunction leaks outside, and a transcendental equation enters the picture.
- 2.9The harmonic oscillator◈ 1 sim(s)The most reused model in physics. Solving it with ladder operators is ten times faster than grinding through the differential equation, and it shows the structure far more clearly.
- 2.10The free particle and Gaussian wave packets◈ 1 sim(s)The simplest potential (no potential at all) turns out to be the most awkward: the stationary states are not normalisable. The fix is to superpose them, and the price is that the packet spreads.
- 2.11Barrier penetration and quantum tunnelling◈ 1 sim(s)A particle gets through a wall it does not have the energy to climb. This is no theoretical curiosity: alpha decay, the scanning tunnelling microscope and flash memory all run on it.