Susskind QM Lecture 10 - Harmonic Oscillator
One-line takeaway
The quantum harmonic oscillator is the universal model of small vibrations, with energy levels stacked like a ladder.
Plain-language map
A harmonic oscillator is any system whose potential energy is approximately quadratic near a stable equilibrium. That includes springs, small vibrations in solids, circuit oscillations, sound modes, and electromagnetic modes.
Classically, the oscillator potential is:
V(x) = (1/2)mω²x²Quantum mechanically, x and p become operators and the Hamiltonian is:
H = P²/(2m) + (1/2)mω²X²The oscillator is central because many complicated systems can be approximated as collections of independent oscillator modes.
Ladder picture
Instead of solving only by differential equations, Susskind emphasizes operators that raise and lower the energy level. The number operator labels energy states |n⟩, and the raising/lowering operators move between neighboring levels.
E_n = ħω(n + 1/2), n = 0, 1, 2, ...The ground state has nonzero energy. That is not an accident; a state with exactly zero position spread and zero momentum spread would violate the uncertainty principle.
Core links
Common pitfalls
- Do not expect the ground-state energy to be zero.
- Do not treat ladder operators as physical elevators; they are algebraic tools that map one energy eigenstate to another.
- Do not forget why the oscillator appears everywhere: smooth potentials look quadratic near minima.