Comments
Description
Transcript
Hints and answers
QUANTUM OPERATORS 19.4 Hints and answers 19.1 19.3 19.5 19.7 19.9 Show that the commutator is anti-Hermitian. Use the Hermitian conjugate of the given equation to obtain the time dependence of ψ|. The rate of change of ψ|A|ψ is iψ| [ H, A ] |ψ. Note that [ H, px ] = [ V , px ] and [ H, x ] = p2x , x /2m. Show that C2 = 2I. √ √ sin 2 1 0 1 1 , sin C = √ cos C = cos 2 . 0 1 1 −1 2 Express the total Hamiltonian in terms of B = A + gI and determine the value of B, B † . Show that, if F (n) is the nth derivative of F(λ), then F (n+1) = A, F (n) . Use a Taylor series in λ to evaluate F(1), using derivatives evaluated at λ = 0. Successively reduce the level of nesting of each multiple commutator by using the result of evaluating the previous term. The given expression reduces to cos θ Ly − sin θ Lz . 674