JUPITER SCIENCE
Where Exploration Meets Excellence
Explore
Jupiter Wiki›Reference Entry
Jupiter Wiki · Reference Note

Heisenberg Picture

Updated Jun 13, 2026

The Heisenberg Picture is a formulation of quantum mechanics in which the state vectors are constant in time, while the operators (observables) evolve according to the Heisenberg equation of motion. This approach, developed by Werner Heisenberg in 1925, stands in contrast to the Schrödinger picture, where the state vectors evolve and the operators remain stationary. Despite these conceptual differences, both pictures are mathematically equivalent and yield the same physical predictions for expectation values and measurement probabilities.

On This Page

  1. Short Definition
  2. Detailed Explanation
  3. Key Formula / Syntax / Principle
  4. Important Characteristics
  5. Practical Example
  6. Common Confusions or Misconceptions
  7. Related Terms
  8. Why It Matters

Short Definition

The Heisenberg Picture is a representation of quantum dynamics where the time dependence is shifted from the state of the system (the wave function) to the observables (the operators). In this framework, a quantum state ##|\psi\rangle## is fixed at its initial value, while an operator ##A## evolves through time via a unitary transformation involving the system's Hamiltonian.

Detailed Explanation

In the standard Schrödinger picture, the state of a system evolves according to the time-dependent Schrödinger equation. If we consider a system with a Hamiltonian ##H##, the state at time ##t## is given by ## \psi_S(t)\rangle = U(t, t_0) \psi_S(t_0)\rangle##, where ##U## is the unitary time-evolution operator. In this picture, an observable ##A_S## is typically time-independent unless it has an explicit external time dependence.

The Heisenberg Picture arises from the realization that physical predictions in quantum mechanics depend on the "sandwich" or expectation value:

### \langle A \rangle = \langle \psi(t) A_S \psi(t) \rangle ###

By substituting the evolution of the state vector into this expression, we get:

### \langle A \rangle = \langle \psi(t_0) U^\dagger(t, t_0) A_S U(t, t_0) \psi(t_0) \rangle ###

We can define a new operator in the Heisenberg Picture, ##A_H(t)##, such that:

### A_H(t) = U^\dagger(t, t_0) A_S U(t, t_0) ###

In this picture, the state vector ##|\psi_H\rangle## is defined as the state at the reference time ##t_0##:

### \psi_H\rangle = \psi_S(t_0)\rangle ###

Because the state vector does not change with time, the entire dynamical information of the system is contained within the operators. This formulation is particularly useful because it highlights the correspondence between quantum mechanics and classical mechanics, where observables like position and momentum are functions of time.

The Time-Evolution Operator

For a time-independent Hamiltonian ##H##, the unitary evolution operator is expressed as:

### U(t, t_0) = e^{-\dfrac{iH(t-t_0)}{\hbar}} ###

Consequently, the Heisenberg operator evolves as:

### A_H(t) = e^{\dfrac{iH(t-t_0)}{\hbar}} A_S e^{-\dfrac{iH(t-t_0)}{\hbar}} ###

Mathematical Equivalence

The transition between the Schrödinger and Heisenberg pictures is a unitary transformation. Since unitary transformations preserve the inner product and the eigenvalues of operators, all physical observables—such as the probability of finding a particle in a certain region or the expected energy of a state—remain identical regardless of which picture is used for the calculation.

Key Formula / Syntax / Principle

The fundamental equation governing the evolution of operators in the Heisenberg Picture is the Heisenberg Equation of Motion. It is derived by differentiating the definition of ##A_H(t)## with respect to time.

The Heisenberg Equation of Motion

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

Where:

  • ##A_H(t)## is the operator in the Heisenberg picture.
  • ##H## is the Hamiltonian of the system.
  • ##[H, A_H(t)] = H A_H(t) - A_H(t) H## is the commutator of the Hamiltonian and the operator.
  • ##\left( \dfrac{\partial A}{\partial t} \right)_H## represents the explicit time dependence of the operator (e.g., if the operator depends on an external time-varying field).

Comparison Table: Schrödinger vs. Heisenberg

Feature Schrödinger Picture Heisenberg Picture
State Vector ##|\psi\rangle##
Time-dependent: ##i\hbar \dfrac{\partial}{\partial t} \psi_S\rangle = H \psi_S\rangle##
Constant: ## ## \psi_H\rangle = \psi_S(0)\rangle## ##

Operators ##A##

Constant (usually): ##\dfrac{dA_S}{dt} = 0##

Time-dependent: ##\dfrac{dA_H}{dt} = \dfrac{i}{\hbar}[H, A_H]##

Basis Vectors

Constant

Time-dependent

Important Characteristics

Practical Example

The Quantum Harmonic Oscillator

Consider a simple harmonic oscillator with the Hamiltonian:

### H = \dfrac{p^2}{2m} + \dfrac{1}{2}m\omega^2 x^2 ###

In the Heisenberg picture, we want to find the time dependence of the position operator ##x_H(t)## and momentum operator ##p_H(t)##. Using the Heisenberg equation of motion:

### \dfrac{d}{dt} x_H = \dfrac{i}{\hbar} [H, x_H] = \dfrac{p_H}{m} ###
### \dfrac{d}{dt} p_H = \dfrac{i}{\hbar} [H, p_H] = -m\omega^2 x_H ###

These are exactly the same forms as the classical equations of motion for a harmonic oscillator. Solving these coupled differential equations yields:

### x_H(t) = x_H(0) \cos(\omega t) + \dfrac{p_H(0)}{m\omega} \sin(\omega t) ###
### p_H(t) = p_H(0) \cos(\omega t) - m\omega x_H(0) \sin(\omega t) ###

This example demonstrates how the Heisenberg picture allows us to treat quantum operators in a manner very similar to classical variables.

Common Confusions or Misconceptions

Term Description
Schrödinger Picture A representation where states evolve and operators are stationary.
Interaction Picture Also known as the Dirac Picture; both states and operators evolve, typically used in perturbation theory.
Hamiltonian The operator corresponding to the total energy of the system, acting as the generator of time evolution.
Unitary Operator An operator that preserves the inner product, ensuring the total probability remains 1.
Commutator A mathematical operation ##[A, B] = AB - BA## that measures the degree to which two operators fail to commute.

Why It Matters

The Heisenberg Picture is indispensable in several advanced areas of physics:

By shifting the focus from the wave-like evolution of states to the algebraic evolution of observables, the Heisenberg Picture offers a robust alternative perspective that simplifies complex time-dependent problems and clarifies the relationship between quantum and classical dynamics.

RESOURCES