Wave function and time dependent Schrödinger equation

Significance of the wave function

The wave function Ψ(x, t) describes the quantum state of a particle.

• Ψ itself is not measurable. It can be complex.
• |Ψ|² = Ψ*Ψ is the probability density: |Ψ|² dx is the probability of finding the particle between x and x + dx at time t.
• Normalisation: ∫|Ψ|² dx = 1 (the particle is somewhere).
• Ψ must be single-valued, finite and continuous, and its derivative must be continuous (where V is finite).
• It contains all information about the system, such as the expectation value ⟨x⟩ = ∫Ψ* x Ψ dx.

Time-dependent Schrödinger equation

Take a free particle travelling along x as a plane wave (de Broglie):

Ψ(x, t) = A e^{i(kx − ωt)}, with p = ħk and E = ħω.

1. Differentiate with respect to t: ∂Ψ/∂t = −iωΨ. Multiply by iħ: iħ ∂Ψ/∂t = ħωΨ = EΨ.

2. Differentiate twice with respect to x: ∂²Ψ/∂x² = −k²Ψ. Multiply by −ħ²: −ħ² ∂²Ψ/∂x² = ħ²k²Ψ = p²Ψ.

3. For a particle with potential energy V(x), total energy E = p²/2m + V. Multiply by Ψ:

   EΨ = (p²/2m) Ψ + VΨ

4. Substitute from steps 1 and 2:

iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + V(x) Ψ

In three dimensions: iħ ∂Ψ/∂t = −(ħ²/2m)∇²Ψ + VΨ, that is, iħ∂Ψ/∂t = ĤΨ, with Ĥ the Hamiltonian (energy) operator.

The equation is first order in time, so Ψ at one instant fixes its later evolution. It is a postulate, justified by its agreement with experiments.