Physics 389K Homework Set 12: Schrödinger & Heisenberg Equations, Time-Dependent Hamiltoni, Assignments of Quantum Mechanics

Download Physics 389K Homework Set 12: Schrödinger & Heisenberg Equations, Time-Dependent Hamiltoni and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 389K QM1, Homework Set 12 Solutions Matthias Ihl 12/09/2006 Note: I will post updated versions of the homework solutions on my home- page: http://zippy.ph.utexas.edu/~msihl/PHY389K/ We will frequently work in God-given units c = ~ = 1. The casual reader may also want to set 1 = 2 = π = −1. 1 Problem 1 The Schrödinger equation of motion is i~ d dt φ(t) = Hφ(t). (1) Now, with φ(t) = φ0e −iHt/~, i~ d dt ( φ0e −iHt/~ ) = i~ ( − iH ~ φ0e −iHt/~ ) = Hφ(t). (2) We can also show the opposite direction, namely from (1) follows, after in- tegration, that ∫ φ φ0 dφ̃ φ̃ = − iH ~ ∫ t 0 dt̃ ln φ − ln φ0 = − i ~ Ht φ(t) = φ0e −iHt/~. 1 2 Problem 2 Let A(t) = eiHt/~A0e −iHt/~. (3) We want to check whether A(t) satisfies Heisenberg’s equation of motion, i~ dA(t) dt = [A(t), H ]. (4) The calculation is straightforward: i~ dA(t) dt = i~ ( iH ~ eiHt/~A0e −iHt/~ − eiHt/~A0 iH ~ e−iHt/~ ) = −HeiHt/~A0e −iHt/~ + eiHt/~A0e −iHt/~H = [A(t), H ]. This proves the assertion. 3 Problem 3 If H = H(t), it is not possible to write U(t) = eiHt/~ (unitary time evolution). Ignoring this for the moment, we can write, i~ dA(t) dt = i~ ( i ~ (H + ∂H ∂t t)eiHt/~A0e −iHt/~ − eiHt/~A0 i ~ (H + ∂H ∂t t)e−iHt/~ ) = [A(t), H ] − ( ∂H ∂t teiHt/~A0e −iHt/~ − eiHt/~A0 ∂H ∂t te−iHt/~ ) . In case [ H, ∂H ∂t ] = 0, this can be written as i~ dA(t) dt = [A(t), H ] + t [ A(t), ∂H ∂t ] . (5) We conclude that in this case A(t) = eiHt/~A0e −iHt/~ is not a solution to Heisenberg’s equation of motion, but really, it doesn’t even make sense to work with an ansatz like that in this case.