Schrödinger picture ↩

shorthands: {}
aliases: [Schrodinger picture]
created: 2021-11-07 20:57:56
modified: 2022-01-16 18:07:01

Overview

The time evolution operator drives the time evolution from a state vector at time to a state vector at time :

This is only true for a time-independent Hamiltonian operator, . The exponent is evaluated via its Taylor series.

Introduction

This is the most used picture to describe time evolution in QM.
For simplicity let's consider a time independent Hamiltonian operator:

In this picture, the time dependent Schrödinger equation describes the time evolution of the physical states:

And the initial condition is fixed:

The time evolution operator

Now we can introduce the time evolution operator as:

Where the initial condition is obvious:

We can substitute this new form of into the Schrödinger equation:

This is a simple differential equation so after integration we get this:

This can be solved using successive approximation:

Here we can realize that this is the Taylor expansion of the exponential function, so the operator takes this form:

This also satisfies the commutation relation:

The time evolution operator is unitary

Statement: the time evolution operator is unitary.

Proof

Take the adjoint of both sides of the substituted Schrödinger equation:

From this we can see that the time derivative of is zero:

And because of the initial condition

We can see that it remains the identity operator all along:

The inverse of

The inverse of the time evolution operator describes reversed time evolution:

For a time dependent hamiltonian

1. If the Hamiltonian is dependent on time and the Hamiltonians at different times commute, then the time evolution operator can be written as:

2. If the Hamiltonian is dependent on time, but the Hamiltonians at different times do not commute, then the time evolution operator can be written as:
   Where is the time-ordering operator