Observables & Eigenfunctions

In quantum mechanics, every measurable quantity (observable) is represented by a Hermitian operator. The possible outcomes of a measurement are the eigenvalues of this operator.

3.2.1 Hermitian Operators: An operator $\hat{Q}$ is Hermitian if $\hat{Q}^\dagger = \hat{Q}$. This means $\langle f|\hat{Q}|g\rangle = \langle g|\hat{Q}|f\rangle^*$ for all states $|f\rangle$ and $|g\rangle$. Hermitian operators have real eigenvalues, which is essential because measurement outcomes must be real numbers.

3.2.2 Determinate States: A determinate state is one for which a measurement of the observable $Q$ always yields the same result $q$. Such a state must be an eigenstate of $\hat{Q}$: $\hat{Q}|\psi\rangle = q|\psi\rangle$. In an eigenstate, the uncertainty $\Delta Q = 0$.

The eigenfunctions of a Hermitian operator form the mathematical backbone of quantum measurement theory.

3.3.1 Discrete Spectra: For bound states and other confined systems, the eigenvalues form a discrete set $\{q_n\}$. The eigenfunctions are normalizable and can be chosen to be orthonormal: $\langle q_m|q_n\rangle = \delta_{mn}$. They form a complete set: any state can be expanded as $|\psi\rangle = \sum_n c_n |q_n\rangle$.

3.3.2 Continuous Spectra: For unconfined systems (like the free particle), the eigenvalues form a continuous spectrum. The eigenfunctions are not normalizable in the usual sense. Instead, they are delta-function normalized: $\langle q'|q\rangle = \delta(q - q')$. The expansion of a state becomes an integral: $|\psi\rangle = \int c(q)|q\rangle\,dq$.

The distinction between discrete and continuous spectra is crucial. Bound states have discrete spectra; scattering states have continuous spectra. Some systems have both.

Given a quantum state $|\psi\rangle$ and an observable $\hat{Q}$ with eigenvalues $\{q_n\}$ and eigenstates $\{|q_n\rangle\}$, the generalized statistical interpretation states:

(1) The possible outcomes of measuring $Q$ are the eigenvalues $q_n$.

(2) The probability of obtaining $q_n$ is $P(q_n) = |\langle q_n|\psi\rangle|^2$.

(3) After measurement yielding $q_n$, the state collapses to $|q_n\rangle$.

(4) The expectation value is $\langle Q\rangle = \langle\psi|\hat{Q}|\psi\rangle = \sum_n q_n |\langle q_n|\psi\rangle|^2$.

(5) The uncertainty is $\Delta Q = \sqrt{\langle Q^2\rangle - \langle Q\rangle^2}$.

For continuous spectra, probabilities become probability densities: $\rho(q)\,dq = |\langle q|\psi\rangle|^2\,dq$ is the probability of finding the outcome in $[q, q+dq]$.

Two observables $A$ and $B$ are compatible if their operators commute: $[\hat{A}, \hat{B}] = 0$. Compatible observables share a complete set of simultaneous eigenstates: $\hat{A}|a,b\rangle = a|a,b\rangle$ and $\hat{B}|a,b\rangle = b|a,b\rangle$.

If $[\hat{A}, \hat{B}] = 0$, we can measure both $A$ and $B$ simultaneously with arbitrary precision. The order of measurement does not matter.

If $[\hat{A}, \hat{B}] \neq 0$, the observables are incompatible. There is no state that is simultaneously an eigenstate of both. Measuring $A$ disturbs $B$ and vice versa. This is the origin of the uncertainty principle.

The most famous incompatible pair is position and momentum: $[\hat{x}, \hat{p}] = i\hbar$. No quantum state has simultaneously definite position and definite momentum.

A spin-1/2 particle is in the state $|\psi\rangle = (3|\uparrow\rangle + 4i|\downarrow\rangle)/5$. We measure $S_z$. Find the possible outcomes, their probabilities, the expectation value, and the uncertainty.

Step 1 — Verify normalization. $|3/5|^2 + |4i/5|^2 = 9/25 + 16/25 = 1$. The state is normalized.

Step 2 — Possible outcomes. The eigenvalues of $S_z$ are $\pm\hbar/2$. These are the only possible measurement outcomes.

Step 3 — Probabilities. $P(+\hbar/2) = |\langle\uparrow|\psi\rangle|^2 = |3/5|^2 = 9/25$. $P(-\hbar/2) = |\langle\downarrow|\psi\rangle|^2 = |4i/5|^2 = 16/25$.

Step 4 — Expectation value. $\langle S_z\rangle = (\hbar/2)(9/25) + (-\hbar/2)(16/25) = -7\hbar/50$.

Step 5 — Uncertainty. $\langle S_z^2\rangle = (\hbar^2/4)(9/25 + 16/25) = \hbar^2/4$. Thus $\Delta S_z = \sqrt{\hbar^2/4 - 49\hbar^2/2500} = \hbar\sqrt{576/2500} = 12\hbar/50 = 6\hbar/25$.