Wave function

Where ∣ψ(t)⟩\left\lvert \psi(t) \right\rangle is a state in the Hilbert space, ψ(x,t)\psi(x,t) is the wavefunction w.r.t time tt and position xx.

This is like a Projection of ∣ψ⟩\ket{\psi} onto ∣x⟩\ket{x} -- the component of ∣ψ⟩\ket{\psi} along ∣x⟩\ket{x}.

Let

ψ(x,t)≜⟨x∣ψ(t)⟩\psi(x,t)\triangleq\left\langle x|\psi(t) \right\rangle ψ(x)≜⟨x∣ψ⟩\psi(x)\triangleq \braket{x|\psi}

Note

  1. Note If an operator A^\hat{A} acts on a wavefunction ψ(x)\psi(x), evaluated at xx, the notation looks like this
(A^ψ)(x)(\hat{A}\psi)(x)
  1. Note (where ψn\psi_n is the Probability amplitude, ∣ψ⟩,∣n⟩\ket{\psi},\ket{n} is a state)
ψn=∑mψmδnm=∑mψm⟨n∣m⟩=⟨n∣(∑mψm∣m⟩)=⟨n∣ψ⟩\psi_n = \sum_m \psi_m \delta_{nm} = \sum_m \psi_m \braket{n|m} = \bra{n}\left(\sum_m \psi_m \ket{m}\right) = \braket{n|\psi}