A spin- sits in a field of fixed strength that turns once around the axis in a time

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch Time evolution as a unitary group, and the three pictures

Problem

A spin-

\tfrac12

sits in a field of fixed strength that turns once around the

z

axis in a time

T

, giving the Hamiltonian

\hat H(t)=\tfrac12\hbar\Omega\big(\cos\Omega't\,\hat\sigma_x+\sin\Omega't\,\hat\sigma_y\big),\qquad 0\le t\le T,\quad \Omega'T=2\pi,

where

\Omega

is a constant angular frequency set by the field strength,

\Omega'=2\pi/T

is the rate at which the field turns, and

\hat\sigma_x,\hat\sigma_y

are Pauli matrices. (a) Starting from the time-sliced product

\hat U(t,0)=\lim_{N\to\infty}\prod_{k=N}^{1}e^{-i\hat H(t_k)\Delta t/\hbar}

(later slices to the left), derive the general second-order term of the Dyson series,

\hat U^{(2)}(t,0)=\left(\frac{-i}{\hbar}\right)^2\int_0^t dt_1\int_0^{t_1}dt_2\,\hat H(t_1)\hat H(t_2),

stating why the inner integral runs only up to

t_1

and not to

t

. (b) For the stated

\hat H(t)

, show that the naive second-order term

\tfrac12\big(-\tfrac{i}{\hbar}\int_0^T\hat H(t)\,dt\big)^2

vanishes, then evaluate

\hat U^{(2)}(T,0)

exactly as a

2\times2

matrix, demonstrating that the ordering is not a bookkeeping nicety.