15.3 Wave functions and probability distributions

Wave function interpretation

Physical meaning and Born interpretation

A wave function, written as $\Psi(x, t)$, is a complex-valued function that fully describes the quantum state of a particle as a function of position and time. On its own, the wave function doesn't correspond to anything you can directly measure. What gives it physical meaning is the Born interpretation.

The Born interpretation states that $|\Psi(x, t)|^2$ gives the probability density for finding the particle at position $x$ at time $t$. More precisely, $|\Psi(x,t)|^2 \, dx$ is the probability of finding the particle in an infinitesimal interval $dx$ around $x$. This is what bridges the abstract math of wave functions to measurable outcomes.

In the double-slit experiment, for instance, the wave function passes through both slits simultaneously and interferes with itself. The resulting $|\Psi|^2$ on the detection screen produces the characteristic interference pattern, with alternating bands of high and low probability.

Probability density and its relation to the wave function

Because $\Psi$ is generally complex-valued, you obtain the probability density by multiplying the wave function by its complex conjugate:

$|\Psi(x,t)|^2 = \Psi^*(x,t)\,\Psi(x,t)$

This guarantees a real, non-negative result, which is necessary for something representing a probability.

The wave function itself can be positive, negative, or complex, and those signs/phases matter for interference effects. But only the squared modulus has a direct physical interpretation as probability density.

Probability density and distribution

Calculating probability density

To find the probability that a particle is located within a specific region $[a, b]$, you integrate the probability density over that region:

$P(a \leq x \leq b) = \int_a^b |\Psi(x,t)|^2 \, dx$

Step-by-step process:

1. Start with the wave function $\Psi(x, t)$ for the system.
2. Compute the probability density: $|\Psi(x,t)|^2 = \Psi^* \Psi$.
3. Integrate $|\Psi|^2$ from $a$ to $b$ to get the probability of finding the particle in that interval.

For example, to find the probability of locating an electron in the left half of a one-dimensional box of length $L$, you'd integrate $|\Psi(x)|^2$ from $0$ to $L/2$.

Probability distribution and its properties

The probability distribution across all space must satisfy two key properties:

• Non-negativity: $|\Psi(x,t)|^2 \geq 0$ everywhere, since probabilities can't be negative.
• Normalization: The total probability of finding the particle somewhere must equal one:

$\int_{-\infty}^{\infty} |\Psi(x,t)|^2 \, dx = 1$

Different potentials produce different probability distributions. For a quantum harmonic oscillator in its ground state, the probability distribution is a Gaussian centered at the equilibrium position. The width of that Gaussian depends on the oscillator frequency $\omega$ and the particle mass $m$, specifically through the characteristic length scale $\sqrt{\hbar / (m\omega)}$.

Normalization of wave functions

Purpose of normalization

Normalization ensures that the total probability of finding the particle somewhere in all of space equals exactly one. Without this, the Born interpretation breaks down because the probabilities extracted from $|\Psi|^2$ wouldn't be meaningful.

Any valid solution to the Schrödinger equation can be multiplied by a constant and still be a solution. Normalization pins down that constant so the probabilistic interpretation holds. For example, in a two-state system like a spin-1/2 particle, normalization guarantees that the probabilities of measuring spin-up and spin-down sum to one.

Procedure for normalizing wave functions

1. Start with an unnormalized wave function $\Psi(x)$ (possibly containing an unknown constant $A$).
2. Compute the integral $\int_{-\infty}^{\infty} |\Psi(x)|^2 \, dx$.
3. Set that integral equal to 1 and solve for the normalization constant $A$.
4. Multiply the original wave function by $A$ to get the normalized version.

Worked example: For a Gaussian wave function $\Psi(x) = A\,e^{-x^2/(2\sigma^2)}$:

• Compute $\int_{-\infty}^{\infty} |A|^2 e^{-x^2/\sigma^2} dx = |A|^2 \sqrt{\pi}\,\sigma$
• Set equal to 1: $|A|^2 \sqrt{\pi}\,\sigma = 1$
• Solve: $A = \left(\frac{1}{\pi^{1/2}\,\sigma}\right)^{1/2}$

This value of $A$ ensures $\int_{-\infty}^{\infty} |\Psi(x)|^2 dx = 1$.

Expectation values from wave functions

Definition and calculation of expectation values

The expectation value of an observable is the average result you'd obtain if you measured that observable on a large number of identically prepared systems. It's not necessarily a value you'd get in any single measurement.

For an observable with corresponding operator $\hat{O}$, the expectation value is:

$\langle O \rangle = \int_{-\infty}^{\infty} \Psi^*(x)\,\hat{O}\,\Psi(x)\, dx$

Note the order: the complex conjugate $\Psi^*$ goes on the left, the operator acts on $\Psi$ on the right, and you integrate over all space.

For energy, the relevant operator is the Hamiltonian:

$\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$

So the expectation value of energy is $\langle E \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{H}\,\Psi \, dx$.

Expectation values of position and momentum

Position: The position operator is simply $\hat{x} = x$, so:

$\langle x \rangle = \int_{-\infty}^{\infty} \Psi^*(x)\, x \,\Psi(x)\, dx$

Momentum: The momentum operator in the position representation is $\hat{p} = -i\hbar\frac{d}{dx}$, so:

$\langle p \rangle = \int_{-\infty}^{\infty} \Psi^*(x)\left(-i\hbar\frac{d}{dx}\right)\Psi(x)\, dx$

Here $\hbar$ is the reduced Planck constant ($h/2\pi$).

These expectation values connect quantum theory to experiment. For a quantum harmonic oscillator in any energy eigenstate, both $\langle x \rangle = 0$ and $\langle p \rangle = 0$ due to the symmetry of the potential about the equilibrium position. This doesn't mean the particle is stationary; it means the average position and average momentum are zero because the particle spends equal time on either side and moving in either direction.