MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Course Instructors: Professor Robert Field and Professor Andrei Tokmakoff 5.74, Problem Set #1 Spring 2004 Due Date: February 18, 2003 1. Let the eigenfunctions and eigenvalues of an operator Aˆ be ϕn and an respectively: Aˆϕ= anϕ . If fx( ) is a function that we can expand in powers n nAˆof x, show that ϕ is an eigenfunction of f ( ) with eigenvalue fan():nf ( Aˆ )ϕ= f (an )ϕn n εa Vab 2. For a two-level system with an Hamiltonian H =Vba εb 22a) Show that the eigenvalues are ε=E ± ∆ + ±Vab ε −ε b ε +ε ba awhere ∆= and E = . 2 2 Vabb) If we define a transformation tan 2θ= , find the form of the ∆eigenvectors of the coupled states ϕ+ , ϕ− . What is the similarity , ϕ ,transformation that takes you from the { } to the { }ϕ+ ϕ− a ϕb basis? Is this operator unitary? c) Verify that this basis is normalized and orthogonal. 3. Convince yourself that exp(iGλ)A exp(−iGλ) = A + iλ[G, A]+ i2λ2 [G,[G, A]]+ … 2! inλn + [G,[G,[G…[G, A]]]…]+ … n! where G is a Hermetian operator and λ is a real parameter.5.74, Problem Set #1 Page 2 (,0 )= exp [−iHt4. Just as Ut t ] is the time-evolution operator which displaces ψ(r, t )in time, (, p Dr r 0 )= exp −i ⋅(r − r0 ) is the spatial displacement operator that moves ψ in space. a) Defining D () λ=exp −ip ⋅λ , show that the transformation D†rD=r+λ where λ is a displacement vector. The relationship in Problem 3 will be useful here. b) Show that the wavefunction of the state φ= Dψ is the same as the wavefunction of the state λ . Write out φ()= x xφ explicitly if ψ , only shifted a distance φ is the ground state of the one-dimensional harmonic oscillator. 5. The Hamiltonian for a degenerate two-level system is Ho = |a〉 ε0 〈a| + |b〉 ε0 〈b| At time t = 0 a perturbation is applied: V(t) = |a〉 Vba(t) 〈b| + |b〉 Vab(t) 〈a| where Vab(t) = Vba(t)* = V(1 – exp(−γt)) . a) Does the Hamiltonian commute at all times? b) If the system is initially in prepared in state |b〉 (t ≤ 0), what is the state of the system for t > 0? c) What is the probability of finding the system in |a〉 for t > 0? d) Describe the behavior of this system in the limits γ → 0 and γ → ∞.5.74, Problem Set #1 Page 3 6. Time-Development of the Density Matrix (a) Using the time-dependent Schrödinger equation, show that the time-dependence of the density matrix ρ = |ψ〉〈ψ| is given by the Liouville-Von Neumann equation: ∂ρ−i =[H, ρ]∂t (b) Show that the time dependence of ρ obtained by directly integrating the Liouville-Von Neumann equation from 0 to t is the same as ρ(t)= Uρ(0)U†
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