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Chemistry LibreTexts

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 tex2html_wrap_inline437 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, tex2html_wrap_inline441 , which is the amplitude for obtaining the value tex2html_wrap_inline443 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 tex2html_wrap_inline443 of H, apply to each the factor tex2html_wrap_inline453 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 tex2html_wrap_inline461 , 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 tex2html_wrap_inline467 is an eigenfunction of the Hamiltonian.