1.2. Schrödinger and Heisenberg pictures¶
In quantum mechanics, experimental results are expressed in terms of matrix elements of operators. In general, for an operator (which we assume to be timeindependent), such a matrix element is These matrix elements change in time as the wave functions are timedependent: where on the right hand side is the wave function at .
Therefore, we can write From this formulation, it is immediately clear that we can take two viewpoints:

We take independent of time and let and evolve in time according to the timedependent Schrödinger equation, or

We take the wave functions and fixed and introduce a timedependent operator : In that case, the matrix element is written as
The first viewpoint is called the Schrödinger picture and the second the Heisenberg picture. For systems where the Hamiltonian can be split into an 'easy' and a 'difficult' part, it makes sense to use a third picture, called interaction picture. This will be covered later in these notes.