# 0.4: Time evolution of the state vector

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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.