QM · Quantum mechanics and atomic physics

A particle in a box cannot have zero ground-state kinetic energy. Its confinement changes what states are allowed.

Prerequisites: 24, 37, 38, 39.

• Use wavefunctions, operators and measurement probabilities
• Solve standard bound-state models and tunnelling reasoning
• Apply angular momentum, spin and atomic spectral principles

A state wavefunction 波函数 must satisfy the model’s boundary conditions and have total probability one: integrate |ψ(x)|² over the allowed domain. For orthonormal states φ_j, a superposition ψ=Σc_jφ_j has norm squared Σ|c_j|² because the cross terms integrate to zero. Thus an equal superposition of N orthonormal states has coefficient magnitude 1/sqrt(N). The probability of finding one constituent energy is |c_j|², not c_j. Relative phases may affect position probabilities even when energy probabilities are unchanged.

An observable is represented by a Hermitian operator 算符. Its eigenvalues are real, and a normalised state gives expectation value ⟨A⟩=∫ψ* Aψ dx. The expectation is an average over repeated preparations, not necessarily the result of one measurement. If A satisfies a polynomial identity, apply it to an eigenstate: for A⁴=I, a real eigenvalue must be +1 or −1. A complex fourth root is excluded by Hermiticity even though it solves the polynomial.

For an infinite well 0<x<L, ψ(0)=ψ(L)=0 allows sine modes sin(nπx/L), with n=1,2,… . Their energies are E_n=n²π²$\hbar$²/(2mL²), so the ground state is not zero and doubling L divides every energy by four. The equivalent h expression is n²h²/(8mL²); never substitute h for $\hbar$ without its 2π factor. A finite barrier instead permits exponential tails, and a travelling wave in a classically allowed region has oscillatory rather than exponentially decaying spatial dependence.

Operators commute when [A,B]=AB−BA=0; compatible observables can have a common eigenbasis. Since kinetic energy p²/(2m) is a function of p, it commutes with p. A position-dependent potential generally makes H=p²/(2m)+V(x) fail to commute with p. Angular momentum obeys [J_x,J_y]=i$\hbar$J_z and cyclic permutations, while J² commutes with each component. Atomic transitions exchange positive photon energy equal to a level difference; angular momentum and mechanism-specific selection rules restrict which transitions occur.

An infinite well extends from 0 to L. Vanishing wavefunction at both ends gives

Prepare $\psi=(\psi_1+i\psi_2)/\sqrt2$. Orthogonality makes its norm one. Energy measurements give E1 or 4E1 with equal probability: This average is not an allowed single energy result. The relative phase changes in time, so a superposition of different energies is not a stationary state.

A stationary state can have a nonzero energy even though its probability density is time-independent.

normalisation 归一化: Making total probability equal to one.

operator: A mathematical action representing an observable.