Time-Independent Hamiltonians

A Hamiltonian is time independent when it has no explicit dependence on the time parameter. In that case the time-evolution operator is the ordinary exponential

$$U(t,t_0)=e^{-iH(t-t_0)/\hbar}.$$

This is the simplest and most important exact solution of closed-system quantum dynamics.

Time-independent unitary evolution

The formula applies when the same self-adjoint Hamiltonian $H$, with a fixed domain, generates evolution throughout the interval from $t_0$ to $t$. It does not require $H$ to be diagonal in the basis being used, but it does require $H$ not to change with time.

For unbounded Hamiltonians, the exponential is defined through spectral theory. In finite dimensions, it is the usual matrix exponential.

Spectral Decomposition

If $H$ has a discrete spectral decomposition

$$H=\sum_n E_n P_n,$$

then

$$U(t,t_0) =\sum_n e^{-iE_n(t-t_0)/\hbar}P_n.$$

For nondegenerate eigenstates, $P_n=\lvert E_n\rangle\langle E_n\rvert$, so

$$U(t,t_0) =\sum_n e^{-iE_n(t-t_0)/\hbar} \lvert E_n\rangle\langle E_n\rvert.$$

This makes energy-basis evolution especially transparent: each energy component acquires a phase. Stationary states and phases explains why a superposition of different energies can still have time-dependent probabilities.

Degeneracy

If an energy level is degenerate, every state inside that eigenspace gets the same phase under $H$ alone. Under $H$ alone the ray of every vector in that eigenspace is fixed. A basis change does not create dynamics. An added perturbation can split or mix the subspace, and a measurement can change the state through a separate state-update operation.

Continuous Spectra

For continuous spectra, sums are replaced by spectral integrals. The free particle is the standard example: momentum components evolve by phases determined by $E=p^2/(2m)$.

The formal lesson is unchanged. Diagonalize the Hamiltonian, attach phases to spectral components, then transform back if needed.

Finite-Dimensional Matrix Exponentials

In a finite-dimensional basis, $H$ is a Hermitian matrix. If it is diagonalized as

$$H=V D V^\dagger,$$

then

$$e^{-iH\tau/\hbar} = V e^{-iD\tau/\hbar}V^\dagger, \qquad \tau=t-t_0.$$

This is often the stable way to compute exact finite-dimensional time evolution.

Examples

For a diagonal two-level Hamiltonian

$$H=\frac{\hbar\omega}{2}\sigma_z,$$

the evolution is

$$U(t,0) = \begin{pmatrix} e^{-i\omega t/2} & 0\\ 0 & e^{i\omega t/2} \end{pmatrix}.$$

For the harmonic oscillator,

$$H=\hbar\omega\left(a^\dagger a+\frac12\right),$$

number states evolve as

$$\lvert n;t\rangle =e^{-i\omega(n+1/2)t}\lvert n\rangle.$$

Common Mistakes

• Substituting an instantaneous $H(t)$ into this formula. Even if all $H(t)$ commute, the exponent uses $\int_{t_0}^tH(s)\,ds$, not generally $H(t)(t-t_0)$.
• Thinking a time-independent Hamiltonian means all observables are time independent.
• Forgetting relative phases between different energy eigenstates.
• Treating degeneracy as if it automatically fixes a preferred basis inside the degenerate subspace.
• Ignoring domain issues for unbounded Hamiltonians in rigorous arguments.

References

• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Suppose $H=\sum_n E_nP_n$. Show that $U(t,t_0)=\sum_n e^{-iE_n(t-t_0)/\hbar}P_n$ satisfies the operator Schrödinger equation.

Solution

Differentiate the spectral expression:

$$i\hbar\frac{\partial U}{\partial t} = \sum_n E_n e^{-iE_n(t-t_0)/\hbar}P_n.$$

Since $HP_n=E_nP_n$, the right-hand side is $HU(t,t_0)$.