Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum version of motion near a stable equilibrium. It is one of the two central building blocks in Susskind’s quantum mechanics volume, alongside the qubit.
Why it appears everywhere
Near a smooth minimum, many potentials look approximately quadratic. That makes the oscillator the local model for vibrations in molecules, solids, circuits, waves, and field modes.
Hamiltonian
H = P²/(2m) + (1/2)mω²X²The first term is kinetic energy; the second is quadratic potential energy.
Energy ladder
The oscillator has discrete, evenly spaced energy levels:
E_n = ħω(n + 1/2)Ladder operators raise or lower n by one step. The lowest state |0⟩ cannot be lowered further.
Ground state
The ground state has nonzero energy, often called zero-point energy. This reflects the incompatibility of making both position and momentum perfectly sharp.
Why it matters later
Quantum field theory can be understood as many oscillator modes, one for each mode of a field. That makes the oscillator much more than a toy spring problem.
Common pitfalls
- Do not expect a classical continuum of oscillator energies.
- Do not expect the ground state to have zero energy.
- Do not forget the oscillator is a framework, not only a literal mass on a spring.