Checkpoint: Working with ψ

This checkpoint brings together the two core skills of module 7: normalizing a wavefunction and computing a probability from the Born rule. Both steps are needed for virtually every quantitative problem in quantum mechanics.

The tools you need

Normalization. A wavefunction $\psi(x)$ is physically valid only when

\int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1.

If the wavefunction is written as $\psi(x) = A \tilde\psi(x)$ with an undetermined constant $A$ , substituting gives

A^2 \int_{-\infty}^{\infty} |\tilde\psi(x)|^2 \, dx = 1 \quad\Longrightarrow\quad A = \frac{1}{\sqrt{\displaystyle\int_{-\infty}^{\infty} |\tilde\psi(x)|^2 \, dx}}.

Probability in an interval. Once $\psi$ is normalized, the Born rule gives the probability of finding the particle in the interval $[a, b]$ as

P(a \le x \le b) = \int_a^b |\psi(x)|^2 \, dx.

A worked walkthrough

Consider the one-sided exponential wavefunction

\tilde\psi(x) = e^{-x} \quad (x \ge 0), \qquad \tilde\psi(x) = 0 \quad (x < 0).

Step 1 — normalize. We need $\int_0^\infty |\tilde\psi(x)|^2 \, dx$ . Since $\tilde\psi$ is real,

\int_0^\infty e^{-2x} \, dx = \left[-\frac{e^{-2x}}{2}\right]_0^\infty = 0 - \left(-\frac{1}{2}\right) = \frac{1}{2}.

So $A^2 \cdot \tfrac{1}{2} = 1$ , giving $A = \sqrt{2}$ . The normalized wavefunction is

\psi(x) = \sqrt{2}\, e^{-x} \quad (x \ge 0).

Step 2 — check normalization. As a sanity check,

\int_0^\infty 2 e^{-2x} \, dx = 2 \cdot \frac{1}{2} = 1. \checkmark

Step 3 — probability density. The Born rule says the probability density is

|\psi(x)|^2 = 2 e^{-2x} \quad (x \ge 0).

This decays exponentially: the particle is most likely found near $x = 0$ and becomes progressively less likely to be found farther away.

Step 4 — probability in $[0, 1]$ . Integrating the density:

P(0 \le x \le 1) = \int_0^1 2 e^{-2x} \, dx = \left[-e^{-2x}\right]_0^1 = 1 - e^{-2}.

Numerically, $e^{-2} \approx 0.1353$ , so $P \approx 0.8647$ — about $86.5\%$ of the probability is concentrated in the first unit of length.

Key steps for any normalization problem

Write $\psi(x) = A\,\tilde\psi(x)$ and identify the integration region where $\tilde\psi \ne 0$ .
Compute $I = \int |\tilde\psi|^2 \, dx$ over that region.
Set $A = 1/\sqrt{I}$ (taking the positive real root).
Use $|\psi|^2 = A^2 |\tilde\psi|^2$ as the probability density to answer any probability question.

Try it

This is a numerical exercise — your code should return a number. Given $\psi(x) = A e^{-x}$ for $x \ge 0$ , first find $A$ so that $\psi$ is normalized, then return the probability $P(0 \le x \le 1)$ for the normalized wavefunction. Work it out analytically and return the exact expression.

Run your code to see the quantum state.