Heisenberg picture in Quantum Mechanics

The Heisenberg picture is one of two commonly used formulations in quantum mechanics, the other being the Schrödinger picture. In the Heisenberg picture, the observables (operators) evolve with time while the states remain fixed, which is in contrast to the Schrödinger picture where the states evolve with time and the operators remain fixed.

To explain the Heisenberg picture more simply:

Imagine you have a movie where the background remains the same, but the characters and objects change and move around. In the Heisenberg picture, the "background" is the state of the system, which stays constant, and the "characters and objects" are the observables (like position, momentum, etc.), which change with time.

Here are a few key points about the Heisenberg picture:

  1. Operators Evolve: In the Heisenberg picture, the operators (like position, momentum, energy) change with time according to the equations of motion. The equations that govern this evolution are derived from the Heisenberg equations of motion, which are similar to the classical equations of motion but adapted for quantum operators.

  2. States Stay Fixed: Unlike the Schrödinger picture where the states evolve over time, in the Heisenberg picture, the states remain fixed. This means that if you know the state of the system at a particular time, you can use the evolving operators to calculate the expected values of observables at different times.

  3. Uncertainty Principle: The Heisenberg picture naturally preserves the Heisenberg Uncertainty Principle. This principle states that there is an inherent limit to the precision with which certain pairs of observables (like position and momentum) can be simultaneously known. This uncertainty is a fundamental feature of quantum mechanics.

  4. Useful for Time-Dependent Systems: The Heisenberg picture is particularly useful when dealing with systems where the interactions or external forces change with time. It simplifies the calculations and analysis of how observables evolve under such conditions.

In Heisenberg picture, state vector do not change in time and operators change in time.

The state vector in Schrodinger picture at particular moment t = to is taken as state vector l 𝜓_H> in Heisenberg picture

Thus the operator

changes the state vector in Schrodinger picture to the state vector in Heisenberg picture. Also the operator is Heisenberg picture and that in Schrodinger picture are related by

This relation indicates that the operator in Heisenberg picture depends upon time even when As doesn't depend upon time explicitly.

If As doesn't depend on time explicitly and As commutes with A_H, then

If As doesn't depend upon time explicitly ∂As/∂t = 0

In summary, the Heisenberg picture provides an alternative way of describing the behaviour of quantum systems, focusing on how observables change with time while keeping the states fixed. It's a powerful tool for understanding the dynamics of quantum systems, especially in scenarios where time-dependent changes play a significant role.

ELI5: The expression dA_H/dt = (1/iℏ) [A_H, H_H] in the Heisenberg picture of quantum mechanics in simpler terms:

Imagine you have a special toy box that makes toys do different things. Inside the box, you have two toys: Toy A and Toy H. These toys can spin around and change with time.

Now, you're curious about how Toy A changes over time. You want to know how fast it spins or wiggles. But here's the tricky part: whenever Toy A changes, it also affects Toy H, and when Toy H changes, it affects Toy A too!

In the Heisenberg picture, we use a secret code to understand how these toys change together. The code looks like this: dA_H/dt = (1/iℏ) [A_H, H_H].

  • "dA_H/dt" means how fast Toy A changes over time in this special way we're looking at it.

  • "[A_H, H_H]" is like a secret handshake between the toys. It shows us how much Toy A and Toy H mess with each other when they change.

Here's the fun part: if you want to know how Toy A changes, you just need to look at how it dances with Toy H using this secret code. It's like watching their dance moves to figure out how Toy A is spinning and wiggling over time.

And that "1/iℏ" part? Well, that's like a magic ingredient in the code that helps us talk the special language of quantum mechanics. It's a bit like a secret ingredient in a recipe that makes the whole thing work!

So, in the Heisenberg picture, we use this secret code to figure out how special quantum toys like Toy A change over time while they're dancing with each other. It helps us understand how things move and wiggle in the quantum world!

This note is a part of the Physics Repository.