9.7: Time Evolution of the State Vector

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

\[i\hbar {\partial \over \partial t} \vert\Psi(t)\rangle = H\vert\Psi(t)\rangle \nonumber \]

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

\[\vert\Psi(t)\rangle = e^{-iHt/\hbar}\vert\Psi(0)\rangle \nonumber \]

where \(\vert\Psi(0)\rangle \) is the initial state vector. The operator

\[ U(t) = e^{-iHt\hbar} \nonumber \]

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

\[H\vert E_i\rangle = E_i \vert E_i\rangle \nonumber \]

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

\[\vert\Psi(t)\rangle = \sum_i c_i(t) \vert E_i\rangle \nonumber \]

where, of course, \(c_i(t) = \langle E_i\vert\Psi(t)\rangle \), which is the amplitude for obtaining the value \(E_i\) 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:

\[ \begin{align*} \vert\Psi(t)\rangle &= \displaystyle e^{-iHt\hbar}\vert\Psi(0)\rangle \\[4pt] &= e^{-iHt/\hbar}\sum_i\vert E_i\rangle \langle E_i\vert\Psi(0)\rangle \\[4pt] &=\sum_i e^{-iE_i t/\hbar}\vert E_i\rangle \langle E_i\vert\Psi(0)\rangle \end{align*} \]

Thus, we need to compute all the initial amplitudes for obtaining the different eigenvalues \(E_i\) of \(H\), apply to each the factor \(\exp(-iE_it/\hbar)\vert E_i\rangle\) 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 \(\langle x | \), we obtain

\[ \begin{align*} \langle x\vert H(X,P)\vert\Psi(t)\rangle &= i\hbar {\partial \over \partial t}\langle x\vert\Psi(t)\rangle \\[4pt]  H\left(x,{\hbar \over i}{\partial \over \partial x}\right)\Psi(x,t) &= i\hbar {\partial \over \partial t}\Psi(x,t) \end{align*} \]

If the classical Hamiltonian takes the form

\[H(x,p) = {p^2 \over 2m} + U(x) \nonumber \]

then the Schrödinger equation becomes

\[\left[-{\hbar^2 \over 2m}{\partial^2 \over \partial x^2} + U(x)\right]\Psi(x,t)= i\hbar {\partial \over \partial t}\Psi(x,t) \nonumber \]

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:

\[ \begin{align*} \langle x\vert H\vert E_i\rangle \nonumber &= E_i \langle x\vert E_i\rangle \\[4pt]  H\left(x,{\hbar \over i}{\partial \over \partial x}\right)\psi_i(x)  &=  E_i \psi_i(x) \\[4pt]  \left[-{\hbar^2 \over 2m}{\partial^2 \over \partial x^2} + U(x)\right]\psi_i(x)  &= E_i \psi_i(x) \end{align*}\]

where \(\psi_i(x) = \langle x\vert E_i\rangle \) is an eigenfunction of the Hamiltonian.