Time Ordering for Changing Hamiltonians

A time-dependent Hamiltonian describes a system whose controls change during evolution: a rotating magnetic field, for example, changes which spin component generates the next rotation. The Schrödinger equation still applies, but the order of the successive changes can affect the final state.

Helpful background. Time-Evolution Operator explains composition; Pauli Matrices supplies the spin algebra in the exercise.

Driven Evolution and Time Ordering

The state and its propagator obey

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle=H(t)\lvert\psi(t)\rangle, \qquad i\hbar\,\partial_t U(t,t_0)=H(t)U(t,t_0),$$

with $U(t_0,t_0)=I$. These equations define a unitary propagator for a piecewise continuous Hermitian matrix $H(t)$. For unbounded Hamiltonians, domains and time regularity require further conditions; self-adjointness at each instant alone is not a general existence theorem.

In the finite-dimensional case, if $[H(t),H(s)]=0$ for all relevant times, then

$$U(t,t_0)=\exp\!\left[-\frac{i}{\hbar}\int_{t_0}^{t}H(s)\,ds\right].$$

For unbounded operators, the analogous simplification needs appropriate spectral commutation and a well-defined integrated generator. When generators do not commute, the general solution is a time-ordered exponential: later factors stand to the left. For two constant pulses of duration $\tau$,

$$U(2\tau,0)=e^{-iH_2\tau/\hbar}e^{-iH_1\tau/\hbar}.$$

The Time-Dependent Hamiltonians treatment develops the full construction and energy balance; Time Ordering derives the ordered expansion.

Exercise: The Same Pulses in Opposite Orders

Set $H_x=\hbar\Omega\sigma_x/2$ and $H_z=\hbar\Omega\sigma_z/2$, with $\Omega\tau=\pi/2$. Starting from $\lvert0\rangle$, compare the mean of $\sigma_x$ after an $x$ pulse followed by a $z$ pulse with the result for the opposite order. Here $\sigma_z\lvert0\rangle=\lvert0\rangle$.

Solution

A pulse about axis $j$ has

$$U_j=\frac{I-i\sigma_j}{\sqrt2}.$$

The first sequence gives

$$U_zU_x\lvert0\rangle =e^{-i\pi/4}\frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}.$$

This is the $+1$ eigenstate of $\sigma_x$ up to a global phase, so $\langle\sigma_x\rangle=1$. In the reversed sequence the initial $z$ pulse changes only the phase of $\lvert0\rangle$:

$$U_xU_z\lvert0\rangle =e^{-i\pi/4}\frac{\lvert0\rangle-i\lvert1\rangle}{\sqrt2}.$$

This state has $\langle\sigma_x\rangle=0$. The integrated Hamiltonian is the same for the two protocols, yet the predictions differ. This is why a single ordinary exponential of that integral cannot generally replace time ordering.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, ch. 2.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, ch. 18.