Quantum Mechanics: Uncertainty Principle and Non-commuting Operators - Prof. M. W. Bromley, Study notes of Quantum Mechanics

Download Quantum Mechanics: Uncertainty Principle and Non-commuting Operators - Prof. M. W. Bromley and more Study notes Quantum Mechanics in PDF only on Docsity! Lecture 9 Outline - observables / S.E. • QM postulates [Section 4.1 and 4.2]: • Incompatible observables / Uncertainty principle • eg. Gaussians and Stern-Gerlach • Schrödinger Eqn. [Section 4.3] Generalised Uncertainty - eg. Gaussian σ2Ωσ 2 Λ ≥ ( 1 2i 〈[Ω,Λ]〉 ) • non-zero for non-commuting operaters eg. [x̂, p̂] = i~, • Last time: Gaussian 〈x|ψ〉 = ψ(x) = (πσ2)− 14 e−(x−a)2/2σ2 〈x〉 = 〈ψ|x̂|ψ〉 = ∫ ∞ −∞ 〈ψ|x〉〈x|x̂|ψ〉dx = ∫ ∞ −∞ ψ∗(x)xψ(x)dx = a • Recall K-defn of eigenfunctions ψk(x) = 〈x|k〉 = Aeikx φ(p) = 〈p|ψ〉 = ∫ ∞ −∞ 〈p|x〉〈x|ψ〉dx = ∫ ∞ −∞ ψ∗p(x)ψ(x)dx = ∫ ∞ −∞ e−ipx/~√ 2π~ e−(x−a) 2/2σ2 (πσ2) 1 4 dx = ( σ2 π~2 ) 1 4 e−ipa/~e−p 2σ2/2~2 • So... (lack of) Simultaneous Eigenfunctions • all are Hermitian with eigenvalues +~/2 and −~/2 Eigenbasis expansion Diagonalisation • ie. all are Hermitian (measure spin in x, y, z dirs) Schrödinger Eqn 4. The state vector evolves according to Schrödinger Eqn i~ d dt |ψ(t)〉 = Ĥ|ψ(t)〉 • Hamiltonian of a Simple Harmonic Oscillator potential Hclassical = p2 2m + 1 2 mω2x2 ⇒ Ĥ = p̂ 2 2m + 1 2 mω2x̂2 • Can choose whether to use |x〉 or |p〉 as basis Ĥ ≡ − ~ 2 2m d2 dx2 + 1 2 mω2x2 or Ĥ ≡ p 2 2m − ~ 2mω2 2 d2 dp2 • depends on which is easier to solve with a wavefunction • We (later on) solve the SHO problem with a 3rd basis. Schrödinger Eqn - General soln i~|dψ dt (t)〉 = Ĥ|ψ(t)〉 Ĥ|E〉 = E|E〉 • when Ĥ is Hermitian eigensolutions must exist! • Ĥ eigenproblem is time-independent Schrödinger Eqn. |ψ(t)〉 = ∑ |E〉〈E|ψ(t)〉 = ∑ aE(t)|E〉 • Now rewrite i~ d dt |ψ(t)〉 − Ĥ|ψ(t)〉 = |0〉 |0〉 = ( i~ d dt − Ĥ ) |ψ(t)〉 = ∑ ( i~ daE(t) dt − EaE(t) ) |En〉 • By linear independence defn (|0〉 = ∑ i ci|i〉 iff ai = 0) daE(t) dt = − iE ~ aE(t) solns aE(t) = aE(0)e −iEt/~ |ψ(t)〉 = ∑ aE(0)e −iEt/~|E〉 = ∑ |E〉〈E|ψ(0)〉e−iEt/~