Schrodinger picture

The Schrödinger picture is one of the two fundamental ways to describe the behaviour of quantum systems in quantum mechanics, the other being the Heisenberg picture. These two pictures provide different perspectives on how quantum states and observables evolve over time.

In the Schrödinger picture:

1. Quantum States: The focus is on the quantum states of a system. A quantum state describes the probability distribution of different possible outcomes for measurements of various observables.

2. Time Evolution: In the Schrödinger picture, the quantum states of a system evolve over time according to the Schrödinger equation, a fundamental equation in quantum mechanics. This equation describes how the state of a system changes as time passes.

3. Operators: The operators that represent observables (like position, momentum, energy, etc.) do not change with time in the Schrödinger picture. They remain fixed.

4. Equations of Motion: The equations of motion are applied to the quantum states themselves. These equations describe how the state evolves over time.

5. Observables: Observables, which are represented by operators, have fixed values for a given quantum state. These values are used to calculate probabilities of measurement outcomes.

To summarize, in the Schrödinger picture, the focus is on how the quantum states of a system evolve over time according to the Schrödinger equation, while the operators representing observables remain constant. It's a way of understanding quantum mechanics that emphasizes the changing probabilities of measurement outcomes as the system's state evolves.

If the eigen values of an operator do not change with time then the mathematical form of the operator, generally doesn't depend upon time. The time evolution of the system is entirely contained in state vector, l 𝜓 > which is the solution of Schrodinger equation. Such representation in which the operator does not depend upon the time and state vector l 𝜓(t) > changes with time is known as Schrodinger picture (or Schrodinger representation). Let l 𝜓_s(t_o) > and l 𝜓_s(t) > represent the state vector in Schrodinger picture at the moments to and t respectively, t> to

Let u(t, to) changes the ket l 𝜓_s(t_o) > to l 𝜓_s(t) > represent the state vector in Schrodinger picture at the moments to and t respectively. t> to.

Let u(t, to) change the ket l 𝜓_s(t_o) > to l 𝜓_s(t) >

Since the state vector is normalised at each instant.

U+ must be unitary operator.

We have the Schrodinger equation

From this we can write

Solution of this equation is

Let l u_n> be the set of eigen value kets of it.

Matrix element of operator As in Schrodinger picture

If As does not depend upon time explicitly ∂As\∂t = 0

This is time derivative involving the expectation value of the commutator of operators in the Schrodinger picture.

It describes how the time derivative of the expectation value of the commutator of operators "As" and [A{S, H}] between two different states 𝜓{s}(t) and 𝜙{s}(t) influences the dynamics of a quantum subsystem in the Schrodinger picture. This type of equation is used to study how observables change over time and is a fundamental aspect of quantum mechanics.

This note is a part of the Physics Repository.