Heisenberg Picture

The Heisenberg picture is the formulation in which states are fixed and observables carry the time dependence.

For a compact Core Formalism comparison with the Schrödinger and interaction pictures, see Pictures of Motion Overview.

For a time-evolution operator $U(t,t_0)$,

$$A_H(t) =U^\dagger(t,t_0)A_SU(t,t_0).$$

Fixed States, Evolving Operators

The Heisenberg state is usually identified with the Schrödinger state at the reference time:

$$\lvert\psi_H\rangle =\lvert\psi_S(t_0)\rangle.$$

The time dependence moves to observables:

$$A_S \longrightarrow A_H(t).$$

Expectation Values Match

The Schrödinger-picture expectation value is

$$\langle\psi_S(t)\rvert A_S\lvert\psi_S(t)\rangle.$$

Using $\lvert\psi_S(t)\rangle=U(t,t_0)\lvert\psi_H\rangle$, this becomes

$$\langle\psi_H\rvert U^\dagger(t,t_0)A_SU(t,t_0) \lvert\psi_H\rangle = \langle\psi_H\rvert A_H(t)\lvert\psi_H\rangle.$$

The pictures are equivalent when transformed consistently.

Equation of Motion

For an operator with possible explicit time dependence,

$$\frac{dA_H}{dt} = \frac{i}{\hbar}[H_H,A_H] + \left(\frac{\partial A}{\partial t}\right)_H.$$

The dedicated Heisenberg-equation page derives and applies this formula.

Examples

For a free particle,

$$x_H(t)=x_H(0)+\frac{p_H(0)}{m}t, \qquad p_H(t)=p_H(0).$$

For the harmonic oscillator,

$$a_H(t)=a_H(0)e^{-i\omega t}.$$

These examples show why the Heisenberg picture is often economical: the operator motion can be simpler than the full wavefunction.

QFT Bridge

Field theory often treats local Heisenberg operators as the fundamental objects. Learning operator time dependence here prepares the later shift from particle wavefunctions to quantum fields.

Common Mistakes

• Thinking fixed Heisenberg states mean nothing changes physically.
• Forgetting to transform operators when keeping states fixed.
• Dropping explicit time dependence in $A$.
• Mixing Schrödinger and Heisenberg operators in one expectation value without labeling them.
• Assuming the Heisenberg picture is only useful in abstract formalism.

Cross-Links

• Schrödinger Picture
• Pictures of Quantum Mechanics
• Picture Transformations
• Density Operators in Different Pictures
• Heisenberg Equations of Motion
• Translation Table of Formulations
• Commutators
• Harmonic Oscillator to Fields

References

• 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.
• S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.

Exercises

1. Show that Schrödinger and Heisenberg expectation values agree when $A_H(t)=U^\dagger A_SU$.

Solution

Using $\lvert\psi_S(t)\rangle=U\lvert\psi_H\rangle$,

$$\langle\psi_S(t)\rvert A_S\lvert\psi_S(t)\rangle = \langle\psi_H\rvert U^\dagger A_SU\lvert\psi_H\rangle = \langle\psi_H\rvert A_H(t)\lvert\psi_H\rangle.$$