From Formalism to Real Systems: The Master Recipe

From the postulates to predictions

In Quantum Mechanics I you assembled the machinery: states as vectors in Hilbert space, measurable quantities as Hermitian operators, and the link to experiment through the Born rule. That framework is elegant but abstract. This track turns the crank. Given a real potential — a well, a barrier, an atom — we solve for the allowed energies and states and predict what a spectrometer or a microscope actually sees.

The equation you will solve again and again

The dynamics are set by the time-dependent Schrödinger equation, with the Hamiltonian operator \hat H standing for the total energy — for one particle, kinetic plus potential.

i\hbar\,\frac{\partial \Psi}{\partial t} = \hat H \Psi, \qquad \hat H = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf r)

The time-dependent Schrödinger equation for a single particle.

For a time-independent potential we look for stationary states by separation of variables: \Psi(\mathbf r,t)=\psi(\mathbf r)\,e^{-iEt/\hbar}. The time part is just a phase; the spatial part \psi obeys the time-independent Schrödinger equation.

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\,\psi = E\,\psi

The time-independent (energy eigenvalue) equation in one dimension.

Written as \hat H\psi = E\psi, this is an eigenvalue problem: its solutions are the energy eigenstates, each with a definite energy E. Any general state is a superposition of them, and each piece simply carries its own phase e^{-iE_n t/\hbar}. Solve this once for a given V(x) and you know the system's whole future.

Why energy comes in lumps

Quantization is not something we impose system by system. It emerges from demanding a physically acceptable wavefunction: single-valued, continuous, and normalizable (its probability must total one). For a bound particle, only special values of E yield a solution that stays finite and integrable. All the others blow up and are thrown away.

\int_{-\infty}^{\infty} |\psi(x)|^2\,dx = 1

Normalization: the total probability of finding the particle somewhere is one.

A first look: the box, the barrier, the atom

Three archetypes carry this whole track. The particle in a box shows how confinement quantizes energy. The tunnelling barrier shows a particle passing through a region it could never enter classically. The hydrogen atom adds three dimensions, angular momentum and spin, and reproduces the spectrum of real atoms. Everything else is these ideas recombined.

Hold the contrasts with classical intuition firmly in mind: classically forbidden regions can be occupied, energy comes in discrete levels, and even the lowest state is never perfectly at rest. These are not paradoxes to be explained away — they are the honest content of the theory, and by the end of this track you will compute them.

Your roadmap

Guide 2 solves bound states in one dimension — the infinite and finite wells — and reads off zero-point energy and confinement. Guide 3 tackles tunnelling. Guide 4 goes to three dimensions: angular momentum, spin, the hydrogen atom and the Pauli principle. Guide 5 hands you the working toolkit — perturbation theory, the variational method, identical particles and scattering — the methods you use when nothing is exactly solvable.