Heisenberg Picture: Observables in Motion

Table of Contents

1. Introduction
2. Schrödinger vs Heisenberg Picture
3. Core Idea of the Heisenberg Picture
4. Time Evolution of Operators
5. Heisenberg Equation of Motion
6. Constant State Vectors
7. Time Evolution of Expectation Values
8. Comparison with Classical Mechanics
9. Example: Free Particle
10. Example: Quantum Harmonic Oscillator
11. Commutation Relations in Heisenberg Picture
12. Advantages of the Heisenberg Picture
13. Time-Dependent and Time-Independent Hamiltonians
14. Connection to Quantum Field Theory
15. Applications in Quantum Optics and Control
16. Conclusion

1. Introduction

In quantum mechanics, the Heisenberg picture offers an alternative way to understand time evolution: instead of evolving state vectors, we evolve operators over time. This approach, introduced by Werner Heisenberg, is particularly useful in quantum field theory and high-energy physics.

2. Schrödinger vs Heisenberg Picture

Both pictures are physically equivalent and give the same predictions.

3. Core Idea of the Heisenberg Picture

We start with a state \( |\psi\rangle \) at time \( t = 0 \) and keep it fixed. Time evolution is absorbed into the operator:

\[
\hat{A}_H(t) = \hat{U}^\dagger(t)\hat{A}_S \hat{U}(t)
\]

Where:

• \( \hat{A}_H(t) \) is the Heisenberg picture operator
• \( \hat{A}_S \) is the Schrödinger picture operator
• \( \hat{U}(t) = e^{-i\hat{H}t/\hbar} \)

4. Time Evolution of Operators

Using:

\[
\hat{A}_H(t) = e^{i\hat{H}t/\hbar} \hat{A}_S e^{-i\hat{H}t/\hbar}
\]

The operator \( \hat{A}_H(t) \) evolves, encapsulating the system’s dynamics.

5. Heisenberg Equation of Motion

Differentiating \( \hat{A}_H(t) \):

\[
\frac{d\hat{A}_H(t)}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}_H(t)] + \left(\frac{\partial \hat{A}}{\partial t}\right)_H
\]

This is the Heisenberg equation of motion — the quantum analogue of Hamilton’s equations.

6. Constant State Vectors

In this picture:

\[
|\psi(t)\rangle_H = |\psi(0)\rangle = |\psi\rangle
\]

All dynamics are carried by operators, making the picture ideal for observables over time.

7. Time Evolution of Expectation Values

\[
\langle \hat{A}(t) \rangle = \langle \psi | \hat{A}_H(t) | \psi \rangle
\]

This shows that the Heisenberg picture yields the same expectation values as the Schrödinger picture — they are experimentally indistinguishable.

8. Comparison with Classical Mechanics

The Heisenberg equation resembles classical Poisson brackets:

\[
\frac{df}{dt} = {f, H} \quad \longleftrightarrow \quad \frac{d\hat{A}}{dt} = \frac{i}{\hbar} [\hat{H}, \hat{A}]
\]

This forms the basis of quantization rules and classical-quantum correspondence.

9. Example: Free Particle

Let \( \hat{H} = \frac{\hat{p}^2}{2m} \), then:

• \( \frac{d\hat{x}}{dt} = \frac{\hat{p}}{m} \)
• \( \frac{d\hat{p}}{dt} = 0 \)

Solving gives:

\[
\hat{x}(t) = \hat{x}(0) + \frac{\hat{p}}{m} t, \quad \hat{p}(t) = \hat{p}(0)
\]

10. Example: Quantum Harmonic Oscillator

For \( \hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{1}{2} m \omega^2 \hat{x}^2 \):

• \( \frac{d\hat{x}}{dt} = \frac{\hat{p}}{m} \)
• \( \frac{d\hat{p}}{dt} = -m\omega^2 \hat{x} \)

Solutions:

\[
\hat{x}(t) = \hat{x}(0)\cos\omega t + \frac{\hat{p}(0)}{m\omega} \sin\omega t
\]
\[
\hat{p}(t) = \hat{p}(0)\cos\omega t – m\omega \hat{x}(0)\sin\omega t
\]

11. Commutation Relations in Heisenberg Picture

The canonical commutation relations are preserved:

\[
[\hat{x}_H(t), \hat{p}_H(t)] = i\hbar
\]

This guarantees consistency with the uncertainty principle and quantum postulates.

12. Advantages of the Heisenberg Picture

• More aligned with observable evolution
• Simpler in quantum field theory and interaction pictures
• Ideal when Hamiltonian is time-independent
• Operators evolve under algebraic rules — useful in symmetries and conserved quantities

13. Time-Dependent and Time-Independent Hamiltonians

• If \( \hat{H}(t) \) depends on time, the evolution operator becomes:

\[
\hat{U}(t) = \mathcal{T} \exp\left(-\frac{i}{\hbar} \int_0^t \hat{H}(t’) dt’\right)
\]

Where \( \mathcal{T} \) is the time-ordering operator

This leads to Dyson series in perturbation theory.

14. Connection to Quantum Field Theory

In quantum field theory:

• Fields are operator-valued functions evolving in the Heisenberg picture
• States define particle content
• Interaction picture mixes Schrödinger and Heisenberg pictures for scattering calculations

15. Applications in Quantum Optics and Control

• Laser-matter interaction
• Quantum feedback systems
• Time-resolved spectroscopy
• Optical cavities modeled with Heisenberg equations for creation/annihilation operators

16. Conclusion

The Heisenberg picture provides a powerful framework for quantum dynamics where observables evolve and states remain fixed. Especially useful in systems with strong symmetry or time-invariant Hamiltonians, it gives deep insight into conservation laws and bridges quantum mechanics with classical analogs. Mastery of this picture is essential in advanced quantum mechanics, quantum field theory, and quantum optics.