27 Jun 2025
Energy-Eigenfunction Expansion and Time Evolution
Let a time-independent Hamiltonian have a complete orthonormal set of discrete energy eigenfunctions,
\[\hat H\phi_n(\mathbf r)=E_n\phi_n(\mathbf r), \qquad \int\phi_m^{\ast}(\mathbf r)\phi_n(\mathbf r)d^3r=\delta_{mn}.\]Completeness means that an arbitrary square-integrable initial state can be expanded as
\[\Psi(\mathbf r,0)=\sum_n c_n\phi_n(\mathbf r).\]Multiply by $\phi_m^{\ast}$ and integrate:
\[\int\phi_m^{\ast}\Psi(\mathbf r,0)d^3r =\sum_n c_n\int\phi_m^{\ast}\phi_n d^3r =\sum_n c_n\delta_{mn}=c_m.\]Therefore
\[\boxed{c_n=\int\phi_n^{\ast}(\mathbf r)\Psi(\mathbf r,0)d^3r}.\]If the initial state is normalized, orthonormality gives
\[\begin{aligned} 1&=\int\lvert\Psi(\mathbf r,0)\rvert^2d^3r\\ &=\sum_{m,n}c_m^{\ast}c_n \int\phi_m^{\ast}\phi_n d^3r =\sum_n\lvert c_n\rvert^2. \end{aligned}\]Thus $\lvert c_n\rvert^2$ is the probability of obtaining $E_n$ in an energy measurement.
General time-dependent solution
Each eigenfunction supplies a stationary solution $\phi_n e^{-iE_nt/\hbar}$. Linearity then gives
\[\boxed{ \Psi(\mathbf r,t)= \sum_n c_n\phi_n(\mathbf r)e^{-iE_nt/\hbar} }.\]The sign in the phase is fixed directly by substitution:
\[i\hbar\frac{\partial}{\partial t} e^{-iE_nt/\hbar}=E_ne^{-iE_nt/\hbar}.\]Consequently,
\[i\hbar\partial_t\Psi =\sum_nE_nc_n\phi_ne^{-iE_nt/\hbar} =\sum_nc_n(\hat H\phi_n)e^{-iE_nt/\hbar} =\hat H\Psi.\]At $t=0$ the expression returns the prescribed initial state, so it is the required solution. Only the phases change; hence the energy probabilities $\lvert c_n\rvert^2$ remain constant. The expectation value is
\[\boxed{\langle H\rangle=\sum_n\lvert c_n\rvert^2E_n},\]and is time independent for a time-independent Hamiltonian.
Density of a superposition
Although a single energy eigenstate has a time-independent density, a superposition need not. For two components,
\[\Psi=c_1\phi_1e^{-iE_1t/\hbar} +c_2\phi_2e^{-iE_2t/\hbar},\]so
\[\begin{aligned} \lvert\Psi\rvert^2={}&\lvert c_1\phi_1\rvert^2+\lvert c_2\phi_2\rvert^2\\ &+2\operatorname{Re}\!\left[ c_1c_2^{\ast}\phi_1\phi_2^{\ast} e^{-i(E_1-E_2)t/\hbar} \right]. \end{aligned}\]The interference term oscillates at angular frequency
\[\omega_{12}=\frac{\lvert E_1-E_2\rvert}{\hbar}.\]If $E_1=E_2$, the relative phase is constant. If the Hamiltonian also has a continuous spectrum, the corresponding part of the expansion is an integral,
\[\Psi(\mathbf r,t) =\sum_n c_n\phi_n e^{-iE_nt/\hbar} +\int c(E)\phi_E(\mathbf r)e^{-iEt/\hbar}dE,\]with Dirac-delta rather than Kronecker-delta normalization. This is the same spectral principle for both bound and free components.
Discussion