0.4: Time evolution of the state vector

The time evolution of the state vector is prescribed by the Schrödinger equation

where H is the Hamiltonian operator. This equation can be solved, in principle, yielding

where is the initial state vector. The operator

is the time evolution operator or quantum propagator. Let us introduce the eigenvalues and eigenvectors of the Hamiltonian H that satisfy

The eigenvectors for an orthonormal basis on the Hilbert space and therefore, the state vector can be expanded in them according to

where, of course, , which is the amplitude for obtaining the value at time t if a measurement of H is performed. Using this expansion, it is straightforward to show that the time evolution of the state vector can be written as an expansion:

Thus, we need to compute all the initial amplitudes for obtaining the different eigenvalues of H, apply to each the factor and then sum over all the eigenstates to obtain the state vector at time t.

If the Hamiltonian is obtained from a classical Hamiltonian H(x,p), then, using the formula from the previous section for the action of an arbitrary operator A(X,P) on the state vector in the coordinate basis, we can recast the Schrödiner equation as a partial differential equation. By multiplying both sides of the Schrödinger equation by , we obtain

If the classical Hamiltonian takes the form

then the Schrödinger equation becomes

which is known as the Schrödinger wave equation or the time-dependent Schrödinger equation.

In a similar manner, the eigenvalue equation for H can be expressed as a differential equation by projecting it into the X basis:

where is an eigenfunction of the Hamiltonian.