Wavefunction and Schrödinger Equation

Quantum mechanics describes a particle using a wavefunction (rather than a single trajectory). The Schrödinger equation is the wave equation that tells you what wavefunctions are allowed in a given potential energy V(x).

1. Meaning of the wavefunction

• The wavefunction Ψ(x,t) is a (generally complex) function that encodes the state of the particle.
• The wavefunction itself is not measured directly. Predictions come from its modulus squared.
• In one dimension, dP = |Ψ(x,t)|² dx; in three dimensions, dP = |Ψ(r,t)|² dV.
• Normalisation: total probability is 1: ∫_(-∞)^∞ |Ψ(x,t)|² dx = 1

The diagram below separates the sign-changing wavefunction from the measurable density and shows interval probability as an area under |ψ|².

2. Matter-wave link

Particles show wave behaviour. For a non-relativistic particle with momentum p:

λ = h/p

It is often useful to write momentum in “wave” form:

p = ħ k ⇒ k = 2π/λ

This connection is why quantised wavelengths in a bound system lead to quantised momenta/energies.

3. Schrödinger equations

Time-dependent equation in one dimension

iħ (∂ Ψ)/(∂ t) = -(ħ²/2m)(∂² Ψ)/(∂ x²) + V(x)Ψ

It tells you how the state evolves in time for a given potential energy V(x).

Time-independent equation for stationary states

If V(x) does not depend on time, many exam problems focus on stationary states with:

Ψ(x,t) = ψ(x)e^(-iEt/ħ)

Substituting this into the time-dependent equation gives the time-independent form:

-(ħ²/2m)d²ψ/dx² + V(x)ψ = Eψ

This is an eigenvalue equation: for bound systems, only certain values of E produce acceptable (normalisable) solutions.

4. Physical constraints and boundary conditions

In typical 1D H3 problems, acceptable wavefunctions must satisfy:

• ψ is finite and single-valued.
• ψ is continuous.
• dψ/dx is continuous wherever V(x) is finite.
• ψ is normalisable: ∫ |ψ|² dx is finite.
• At an infinite wall, the wavefunction must vanish at the boundary (e.g., ψ = 0 at the wall).

5. Problem workflow

1. Sketch V(x) and split space into regions where V is simple/constant.
2. Solve the differential equation in each region:
   • If E > V: solutions are oscillatory (sin/cos).
   • If E < V: solutions are exponential (decaying/growing).
3. Apply boundary conditions (continuity + wall conditions) to determine constants and quantise allowed E.
4. Normalise (if asked).
5. Interpret results using |ψ|² and sanity-check units.

6. Quick check

A. If you scale the wavefunction by a constant A, what happens to the probability density?

B. In a region of constant potential, what kind of wavefunction do you expect for E > V vs E < V?