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,

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

Here $H_H(t)=U^\dagger H_S(t)U$ is the energy observable in the Heisenberg picture. For time-dependent $H_S$, it need not equal $H_S(t)$. The Heisenberg equation follows by differentiating all three factors, with $A_S(t)$ in the middle when the observable is explicit in time.

The fixed Heisenberg state has zero generator in its state equation. Its energy is still measured by $H_H$, which is generally nonzero; picture transformations explains this distinction.

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 operator solutions apply to every initial state for which the moments exist. For a free packet, for example,

$$(\Delta x)^2_t=(\Delta x)^2_0+\frac{2t}{m}C_{xp}(0) +\frac{t^2}{m^2}(\Delta p)^2_0,$$

where $C_{xp}=\langle\{x-\langle x\rangle,p-\langle p\rangle\}\rangle/2$. Thus fixed states in this picture describe spreading as well as center motion.

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.

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

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