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 $\hat{A}$ (which we assume to be time-independent), such a matrix element is $$\left\langle\phi \right| \hat{A} \left| \psi \right\rangle.$$ These matrix elements change in time as the wave functions are time-dependent: $$\left| \psi(t) \right\rangle = e^{-{\rm i}H t /\hslash} \left| \psi \right\rangle$$ where $\left| \psi \right\rangle$ on the right hand side is the wave function at $t=0$.

Therefore, we can write $$\left\langle\phi(t) \right| \hat{A} \left| \psi(t) \right\rangle= \left\langle\phi \right| e^{{\rm i}H t /\hslash} \hat{A} e^{-{\rm i}H t /\hslash} \left| \psi \right\rangle.$$ From this formulation, it is immediately clear that we can take two viewpoints:

• We take $\hat{A}$ independent of time and let $\left| \psi \right\rangle$ and $\left| \phi \right\rangle$ evolve in time according to the time-dependent Schrödinger equation, or

• We take the wave functions $\left| \psi \right\rangle$ and $\left| \phi \right\rangle$ fixed and introduce a time-dependent operator $\hat{A}(t)$: $$\hat{A}(t) = e^{{\rm i}H t /\hslash} \hat{A} e^{-{\rm i}H t /\hslash}.$$ In that case, the matrix element is written as $$\left\langle\phi \right| \hat{A}(t) \left| \psi \right\rangle.$$

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.