Ehrenfest theorem

Consider two cases, A=r and A=p: d 1 1 p2 1 < r > = < [r,H] > = < [r, + V(r)] > = < [r,p2 ] > dt ih ih 2m 2imh 2 [r,p ] = [r.p]p + p[r,p] = 2ihp d 1 d < p > ∴ < r > = < 2ihp > ⇒ < r > = dt 2imh dt m d 1 1 p2 1 < p > = < [p,H] > = < [p, + V(r)] > = < [p,V(r)] > dt ih ih 2m ih [p,V(r)] ψ = − ih∇V(r)ψ − ()− ihV(r)∇ ψ = −ihψ∇V(r) ⇒ [p,V(r)] = −ih∇V(r) d 1 d ∴ < p > = < −ih∇V(r) > ⇒ < p > = < −∇V(r) > dt ih dt d d Compare this with classical result: m vr = pv and pv = − ∇V dt dt We see the correspondence between expectation of an operator and classical variable. Poisson brackets and commutators

Poisson brackets in classical mechanics: ⎛ ∂A ∂B ∂A ∂B ⎞ {A, B} = ⎜ − ⎟ ∑ ⎜ ⎟ j ⎝ ∂q j ∂p j ∂p j ∂q j ⎠ dA ⎛ ∂A ∂A ⎞ ∂A ⎛ ∂A ∂H ∂A ∂H ⎞ ∂A = ⎜ q + p ⎟ + = ⎜ − ⎟ + ∑ ⎜ & j & j ⎟ ∑ ⎜ ⎟ dt j ⎝ ∂q j ∂p j ⎠ ∂t j ⎝ ∂q j ∂q j ∂p j ∂p j ⎠ ∂t dA ∂A ∴ = {A,H}+ Classical equation of motion dt ∂ t Compare this with quantum mechanical result: d 1 ∂A < A > = < [A,H] > + dt ih ∂ t We see the correspondence between classical mechanics and quantum mechanics: 1 [A,H] → {A, H} classical ih Quantum mechanics and classical mechanics 1. Quantum mechanics is more general and cover classical mechanics as a limiting case.

2. Quantum mechanics can be approximated as classical mechanics when the action is much larger than h. ∫ pdq = nh >> h This is the same as saying a classical dynamic system is a quantum state with n→∞ (correspondence principle). →0 3. Another way to look at the above requirement is that classical mechanics is a limiting case of quantum mechanics when h→0, or equivalently, when λ→0 since λ=h/p. Lim Quantum mechanics → Classical mechanics h→0 Lim Quantum mechanics → Classical mechanics h→0 4. The probabilistic (uncertainty) nature in quantum mechanics is a result of quantum fluctuation. H is a measure of the quantum fluctuation and quantum fluctuation approaches 0 as h →0. Uncertainty principle becomes ∆x∆p~0 as the classical limit. 1D - Time independent Schrödinger equation

Given 1D potention V(x), we want to determine the energy eigenstates .

Time independent Schrödinger equation: 2 d2 − h ψ(x) + V(x)ψ(x) = Eψ(x) 2m dx 2

We will first study some general properties of this equation. Bound states ⇔ discrete states

V(x) Bound state occurs whenever the particle cannot move to infinity. The energy spectra of bound E states are discrete. In a one dimension problem, x bound states are NOT degenerate. Bound state are usually labeled with index n, the larger the n, the higher the energy. Hψn(x)=Enψn(x) The state with the smallest n (usually either 0 or 1) has the lowest energy. This state is called the ground state.

Node is defined as the point where the eigenfunction ψn(x) vanishes. The (n+1)th state has one more node than the n-th state. In normal case, ground state has 0 node. Unbound states ⇔ continuous states V(x) Unbound state occurs whenever Case 2 E the particle motion is not Case 1 confined. The energy spectra of unbound states are continuous. x Case 1: The continuous states are NOT degenerate.

Case 2: The continuous states are doubly degenerate, i,e, two enigen states for one energy value. The unbound states cannot be normalized. Parity Parity symmetry gives rise to an operator called parity operator P: Pψ(x) = ψ(-x)

General properties of parity operator:

The eigenvalues of parity operator can only be +1 or -1 because two inversion should bring back the original wavefunction: Pˆ 2 λu(x) = u(x) ˆ P u(x) =λ u(x) where u(x) is the eigenfunction ˆ 2 ˆ ∴ P u(x)λ = P[ u(x)] λand is the eigenvalue ˆ =λ Pu(x) λ = 2u(x) ∴u(x) = 2u(x) ⇒ λ = ±1 ψ Any arbitrary function can beψ written as a sum of an even and an odd function: 1 1 ψ ψ (x) = [](x) + (−x) + [](x) −ψ (−x) 2 2 1424 434 1424 434 : Even Odd Hamiltonian with parity symmetry

If [H, P] = 0, H and P

H and P share the same set of eigenstate basis.

Case 1. If H is non-degenerate, all energy eigenstates have either even or odd parity.

Case 2. If H is degenerate (e.g. unbound states), an energy eigenstate may not have a parity.