Susskind QM Lecture 8 - Particles and Waves

One-line takeaway

For particles, the finite component list becomes a continuous wavefunction, and position and momentum become operators on functions.

Plain-language map

After spin and entanglement, Susskind finally turns to the familiar “particles and waves” story. The important move is not mystical duality; it is the same state-vector logic applied to a continuous observable.

For a position measurement, the eigenvalue x can range continuously. The state components in the position basis form a function:

ψ(x) = ⟨x|Ψ⟩

The probability density for finding the particle near x is |ψ(x)|², and normalization becomes an integral:

∫ |ψ(x)|² dx = 1

Functions can be vectors when they obey the vector-space rules and have a suitable inner product. This is the infinite-dimensional Hilbert-space version of the finite spin story.

Position and momentum

In the x-basis, position multiplies the wavefunction and momentum differentiates it:

X ψ(x) = x ψ(x)
P ψ(x) = -iħ ∂ψ/∂x
[X, P] = iħ

Momentum eigenfunctions look wave-like, which is where the particle-wave connection becomes concrete.

Core links

• Wavefunction
• Hilbert space
• Observables and eigenvalues
• Uncertainty principle

Common pitfalls

• Do not think the wavefunction is basis-free; ψ(x) is the state in the position basis.
• Do not treat |ψ(x)|² as a probability at an exact point; in continuous systems it is a probability density.
• Do not forget that non-normalizable plane waves are idealized basis objects, not ordinary physical states.

Links

• Susskind QM Lecture 7 - More on Entanglement
• Susskind QM Lecture 9 - Particle Dynamics