Wavefunction
A wavefunction is the component representation of a quantum state in a chosen basis. In the position basis it becomes the familiar function ψ(x).
Discrete version
Given a basis {|j⟩}, the wavefunction is the list of components:
ψ(j) = ⟨j|Ψ⟩
|Ψ⟩ = Σ_j ψ(j)|j⟩Continuous position version
For a particle on a line:
ψ(x) = ⟨x|Ψ⟩
∫ |ψ(x)|² dx = 1Here |ψ(x)|² is a probability density. The probability of finding the particle in a small interval is approximately |ψ(x)|² dx.
Basis dependence
The same ket |Ψ⟩ can have a position-space wavefunction ψ(x), a momentum-space wavefunction φ(p), or a finite list of spin components. The representation changes with the observable used as the basis.
Why waves appear
Momentum eigenstates in the position basis look like oscillatory complex exponentials. Wave packets are superpositions of these momentum waves. That is the concrete connection between particles and waves in Susskind’s route.
Common pitfalls
- Do not treat ψ(x) as basis-free.
- Do not treat |ψ(x)|² as the probability at an exact mathematical point; it is a density.
- Do not forget that complex phase can affect later interference even when |ψ|² is unchanged.