Chem 3240 · Lecture 3.6
Every measurable quantity has a corresponding operator (the hat \(\hat{}\)).
Position multiplies by \(x\): \[\hat{x} = x\]
Momentum differentiates, then scales by \(-i\hbar\): \[\hat{p}_x = -i\hbar\frac{d}{dx}\]
Recipe: take the classical expression, replace \(x\) and \(p\) by their operators.
Inherited from the linear Schrodinger equation.
\[\hat{A}(\psi_1 + \psi_2) = \hat{A}\psi_1 + \hat{A}\psi_2\]
\[\hat{A}(c\psi) = c\hat{A}\psi\]
\(\hat{x}\), \(\hat{p}_x\), and \(\hat{H}\) all satisfy this.
Order of operations matters, just like matrices (\(AB \neq BA\)).
\[\left[\hat{A},\hat{B}\right]f = \left(\hat{A}\hat{B} - \hat{B}\hat{A}\right)f\]
\[\left[A,A^n\right] = 0, \qquad \left[A,B\right] = -\left[B,A\right]\]
Position and momentum do not commute.
\[\hat{p}_x\hat{x}\psi = \frac{\hbar x}{i}\frac{d\psi}{dx} + \frac{\hbar}{i}\psi, \qquad \hat{x}\hat{p}_x\psi = \frac{\hbar x}{i}\frac{d\psi}{dx}\]
\[\left[\hat{p}_x,\hat{x}\right] = \frac{\hbar}{i} \qquad\Longleftrightarrow\qquad \left[\hat{x},\hat{p}_x\right] = i\hbar\]
This single line drives the uncertainty principle.
For any non-commuting \(\hat{A}\) and \(\hat{B}\):
\[\Delta A\,\Delta B \ge \frac{1}{2}\left|\left\langle\left[\hat{A},\hat{B}\right]\right\rangle\right|\]
Apply it to \(\hat{x}\) and \(\hat{p}_x\): \[\Delta x\,\Delta p_x \ge \frac{\hbar}{2}\]
Position and momentum cannot be measured precisely at once.
\[[\hat{A},\hat{B}]=0 \;\Longrightarrow\; \hat{A}\phi_k = a_k\phi_k,\quad \hat{B}\phi_k = b_k\phi_k\]
Both observables measured simultaneously in one experiment.
\(\hat{T}\) and \(\hat{p}_x\) commute (measure together): \[\left[\hat{T},\hat{p}_x\right] = \left[\frac{\hat{p}_x^2}{2m},\hat{p}_x\right] = 0\]
But \(\hat{x}\) and \(\hat{p}_x\) do not.
Average outcome of many measurements.
\[\langle A \rangle = \int \psi^* \hat{A}\,\psi \, d\tau\]
If \(\psi\) is an eigenfunction, \(\hat{A}\psi = a\psi\), then \[\langle A \rangle = a\int \psi^*\psi\,d\tau = a\]
The expectation value collapses to the eigenvalue.
A compact, representation-free language.
Ket \(|\psi\rangle\), bra \(\langle\psi|\), inner product \[\langle \phi | \psi \rangle = \int \phi^*(r)\,\psi(r)\,d\tau\]
Expectation value becomes one symbol: \[\langle A \rangle = \langle \psi | \hat{A} | \psi \rangle\]
The adjoint \(\hat{A}^\dagger\) is defined through the inner product:
\[\langle \phi | \hat{A}\psi \rangle = \langle \hat{A}^\dagger \phi | \psi \rangle\]
For matrices, the adjoint is the conjugate transpose, \((A^\dagger)_{jk} = A_{kj}^*\).
An operator is Hermitian (self-adjoint) when \[\hat{A} = \hat{A}^\dagger, \qquad \langle \phi | \hat{A}\psi \rangle = \langle \hat{A}\phi | \psi \rangle\]
Observables are represented by Hermitian operators.
Real eigenvalues (only real numbers can be measured): \[\hat{A}|\psi\rangle = a|\psi\rangle \implies a \in \mathbb{R}\]
Orthogonal eigenfunctions for distinct eigenvalues: \[\langle \psi_m | \psi_n \rangle = 0 \quad (m \ne n)\]
The quantum analog of symmetric matrices.
Observables are linear Hermitian operators with real eigenvalues, and their commutators decide which quantities can be known at once: \([\hat{x},\hat{p}_x]=i\hbar\) forbids sharp position and momentum together.