KTU S1
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The Wave Function and Its Physical Interpretation

By the end you should be able to: State Born's probability interpretation of the wave function, list the conditions a physically acceptable wave function must satisfy, and normalise a given wave function.

Quantum mechanics describes a particle by a wave function ψ(x,t)\psi(x,t). The immediate question is what it means, and the answer was not obvious even to the people who wrote the equation.

What it is not

ψ\psi is not a physical wave in space, like a sound wave or a ripple. It is generally complex, and a complex number is not something an instrument reads. ψ\psi itself is not measurable.

Born's interpretation

Max Born supplied the interpretation that survived. The measurable quantity is ∣ψ∣2|\psi|^2, and it is a probability density:

∣ψ(x,t)∣2 dx=probability of finding the particle between x and x+dx\boxed{|\psi(x,t)|^{2}\,dx = \text{probability of finding the particle between } x \text{ and } x+dx}

where ∣ψ∣2=ψ∗ψ|\psi|^2 = \psi^{*}\psi and ψ∗\psi^{*} is the complex conjugate. Taking the modulus squared turns a complex function into a real, non-negative one — exactly what a probability needs to be.

In three dimensions ∣ψ∣2 dV|\psi|^2\,dV is the probability of finding the particle in the volume element dVdV.

The particle is not spread out. When you look, you find one whole particle at one place. ∣ψ∣2|\psi|^2 tells you the probability of that place, not how much of the particle is there.

Normalisation

The particle must be found somewhere, so the total probability over all space is 1:

∫−∞∞∣ψ∣2 dx=1\int_{-\infty}^{\infty} |\psi|^{2}\,dx = 1

A wave function satisfying this is normalised. Since the Schrödinger equation is linear, any solution can be multiplied by a constant chosen to make this true — that constant is the normalisation constant, and finding it is a standard exam calculation.

Conditions on an acceptable wave function

Not every mathematical solution describes a real particle. A physically acceptable ψ\psi must be:

  1. Single valued. One value at each point. Two different values would mean two different probabilities of finding the particle in the same place, which is meaningless.
  2. Finite everywhere. An infinite ψ\psi would give infinite probability density.
  3. Continuous. ψ\psi must have no jumps.
  4. With a continuous first derivative dψ/dxd\psi/dx, except where the potential is infinite. This is needed because the Schrödinger equation contains d2ψ/dx2d^2\psi/dx^2, and a discontinuous first derivative would make the second derivative infinite.
  5. Normalisable, that is ∫∣ψ∣2dx\int|\psi|^2 dx finite. This requires ψ→0\psi \to 0 as x→±∞x \to \pm\infty.

Together these are called the conditions for a well-behaved wave function. The exception in (4) matters: the infinite square well of the next topic has an infinite potential at its walls, and there the derivative is allowed to be discontinuous. That is not a flaw in the model — it is the condition being correctly applied.

Expectation values

If ∣ψ∣2|\psi|^2 is a probability density then averages follow the usual rule. The expectation value of position is

⟨x⟩=∫−∞∞ψ∗ x ψ dx\langle x \rangle = \int_{-\infty}^{\infty} \psi^{*}\,x\,\psi\,dx

This is the mean of many measurements on identically prepared systems, not the result of one measurement.