Susskind Quantum Mechanics Examples

Small examples make the Susskind volume usable. The goal is to keep each example close enough to the equations that it can be replayed by hand.

Spin prepared up, then measured sideways

Prepare a spin in |u⟩, meaning it is definitely +1 for a z-axis measurement. If the apparatus is rotated to the x-axis, the same state is a superposition in the x-basis, so the result is not determined. The probabilities come from the x-basis amplitudes.

Use this for: Susskind QM Lecture 1 - Systems and Experiments, Qubit and spin, Measurement and state preparation.

Right spin as a superposition

A useful two-state model is:

|r⟩ = (|u⟩ + |d⟩)/√2
|l⟩ = (|u⟩ - |d⟩)/√2

Measuring |r⟩ along z gives up or down with equal probability. Measuring it along x gives the definite right outcome. This is the cleanest antidote to thinking that the spin secretly had both z and x values beforehand.

Use this for: Bra-ket notation, Quantum state, Observables and eigenvalues.

Compatible observables

For two separate spins, Alice’s z-spin and Bob’s z-spin can be known together because they act on different parts of the tensor product space. By contrast, σx and σz for the same spin are incompatible.

Use this for: Commutators and compatible observables, Tensor product states.

Singlet-style entanglement

A maximally correlated two-spin state can be written schematically as:

|singlet⟩ = (|ud⟩ - |du⟩)/√2

The composite state is sharp, but neither subsystem has its own pure spin direction. Measurements are correlated without allowing faster-than-light messaging.

Use this for: Entanglement, Density matrix, Susskind QM Lecture 6 - Combining Systems Entanglement.

Moving wave packet

With the simple Hamiltonian H = cP, the Schrödinger equation has solutions shaped like ψ(x - ct). The packet moves at speed c without changing shape. This is a warm-up before the usual nonrelativistic Hamiltonian P²/(2m) + V(X).

Use this for: Susskind QM Lecture 9 - Particle Dynamics, Schrödinger equation, Wavefunction.

Particle in a potential

The usual one-dimensional particle Hamiltonian is kinetic energy plus potential energy:

H = P²/(2m) + V(X)

In the position basis this becomes a differential equation for ψ(x,t). The potential multiplies the wavefunction; the kinetic term differentiates it twice.

Use this for: Susskind QM Lecture 9 - Particle Dynamics.

Harmonic oscillator ladder

The oscillator Hamiltonian has evenly spaced energy levels. The raising operator moves |n⟩ to a multiple of |n+1⟩; the lowering operator moves |n⟩ to a multiple of |n-1⟩. The ground state |0⟩ is special because lowering it gives zero, not a negative-energy state.

Use this for: Quantum harmonic oscillator, Susskind QM Lecture 10 - Harmonic Oscillator.