Cơ học lượng tử và vật lý nguyên tử

A box has no resting particle

• Confine an electron to a box and its lowest energy is not zero: the wavefunction must vanish at the walls, so it must curve.
• GRE quantum questions test normalisation 归一化, measurement probabilities, the infinite well and commutators 对易子.

Probabilities from coefficients

• $\int |\psi|^2 = 1$. For orthonormal $\phi_j$, $\psi = \sum c_j\phi_j$ has $\sum |c_j|^2 = 1$; measuring gives $E_j$ with probability $|c_j|^2$.
• $\psi = (\phi_1 + 2\phi_2)/\sqrt5$: $P(E_1) = \tfrac15$, $P(E_2) = \tfrac45$. Relative phases affect position densities, not these probabilities.

Observables and averages

• Observables are Hermitian 厄米的 operators: real eigenvalues. $\langle A\rangle = \int \psi^* A\psi\,dx$ is an average over many preparations, not one result.
• If $A^4 = I$ and $A$ is Hermitian, its eigenvalues are $\pm1$ — the complex fourth roots are excluded.

The infinite well

• $0 < x < L$: $\psi_n = \sqrt{2/L}\sin(n\pi x/L)$, $E_n = \dfrac{n^2\pi^2\hbar^2}{2mL^2} = \dfrac{n^2h^2}{8mL^2}$, $n = 1, 2, \dots$
• $E_n = n^2E_1$. Doubling $L$ divides every level by 4. $n = 0$ is not a state ($\psi \equiv 0$).

Commutators

• $[A, B] = AB - BA$. Commuting observables share an eigenbasis. $[x, p] = i\hbar$.
• $p$ commutes with $p^2/2m$ but not with $V(x)$. $[J_x, J_y] = i\hbar J_z$; $J^2$ commutes with each component.

Recap