26 Jun 2025
Schrödinger Equations and Stationary States
For a non-relativistic particle in a scalar potential,
\[E=\frac{p^2}{2m}+V(\mathbf r,t).\]The de Broglie plane wave $e^{i(\mathbf p\cdot\mathbf r-Et)/\hbar}$ is an eigenfunction of $i\hbar\partial_t$ with eigenvalue $E$ and of $-i\hbar\nabla$ with eigenvalue $\mathbf p$. This fixes the operator substitutions
\[E\longmapsto i\hbar\frac{\partial}{\partial t}, \qquad \mathbf p\longmapsto-i\hbar\nabla.\]Applying the classical energy expression as an operator relation motivates the time-dependent Schrödinger equation; the equation itself is a postulate of non-relativistic quantum dynamics:
\[\boxed{i\hbar\frac{\partial\Psi}{\partial t}=\hat H\Psi}, \qquad \boxed{\hat H=-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf r,t)}.\]The minus sign in the kinetic term is $(-i)^2=-1$. Its units are those of energy because $\hbar^2\nabla^2/m$ has dimensions
\[\frac{(\mathrm{J\,s})^2}{\mathrm{kg\,m^2}} =\mathrm J.\]The potential $V$ must have the same units. The equation is first order in time, so a specified initial wavefunction determines its later evolution; it is second order in space, so spatial boundary conditions are also required.
Conservation under dynamical evolution
Write the equation and its adjoint in state notation:
\[\lvert\dot\Psi\rangle=-\frac{i}{\hbar}\hat H\lvert\Psi\rangle, \qquad \langle\dot\Psi\rvert=\frac{i}{\hbar}\langle\Psi\rvert\hat H^\dagger.\]For a self-adjoint Hamiltonian,
\[\begin{aligned} \frac d{dt}\langle\Psi\vert\Psi\rangle &=\langle\dot\Psi\vert\Psi\rangle +\langle\Psi\vert\dot\Psi\rangle\\ &=\frac{i}{\hbar}\langle\Psi\vert\hat H\vert\Psi\rangle -\frac{i}{\hbar}\langle\Psi\vert\hat H\vert\Psi\rangle=0. \end{aligned}\]Thus normalization is preserved. If $\hat H$ is time independent, the evolution operator is
\[\boxed{U(t,t_0)=\exp\!\left[-\frac{i}{\hbar}\hat H(t-t_0)\right]},\] \[\lvert\Psi(t)\rangle=U(t,t_0)\lvert\Psi(t_0)\rangle.\]Differentiation verifies $i\hbar\,\partial_tU=\hat HU$ and $U(t_0,t_0)=I$. Since $\hat H=\hat H^\dagger$,
\[U^\dagger U =e^{i\hat H(t-t_0)/\hbar}e^{-i\hat H(t-t_0)/\hbar}=I,\]so the evolution is unitary.
Time-independent Schrödinger equation
When $V(\mathbf r)$ has no explicit time dependence, set
\[\Psi(\mathbf r,t)=\psi(\mathbf r)T(t).\]Substitution gives
\[i\hbar\psi\frac{dT}{dt} =T\left[-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf r)\right]\psi.\]Divide by $\psi T$ wherever the product is non-zero:
\[i\hbar\frac1T\frac{dT}{dt} =\frac1\psi\left[-\frac{\hbar^2}{2m}\nabla^2+V\right]\psi.\]The left side depends only on $t$ and the right side only on $\mathbf r$; both must equal a constant $E$. Hermiticity of $\hat H$ makes $E$ real. The separated equations are
\[i\hbar\frac{dT}{dt}=ET, \qquad \boxed{\hat H\psi=E\psi}.\]Integrating $dT/T=-iE\,dt/\hbar$ gives
\[T(t)=T(0)e^{-iEt/\hbar}.\]After absorbing $T(0)$ into $\psi$, an energy eigenfunction produces the stationary state
\[\boxed{\Psi_E(\mathbf r,t)=\psi_E(\mathbf r)e^{-iEt/\hbar}}.\]Its probability density is stationary,
\[\lvert\Psi_E(\mathbf r,t)\rvert^2 =\lvert\psi_E(\mathbf r)\rvert^2,\]because the time factor has unit modulus. The state vector still changes by a phase. Spatial boundary conditions and normalizability select the allowed eigenfunctions and hence the energy eigenvalues; this selection is derived for the stated potentials in Unit III.
Solved Problems
1. Plane-wave eigenstate in a constant potential
An electron moves in the constant potential $V_0=3.00\ \mathrm{eV}$ with wave number $k=+2.00\times10^{10}\ \mathrm{m^{-1}}$. Find its total energy, angular frequency, and stationary plane-wave factor.
Solution. For $\psi=Ae^{ikx}$,
\[\frac{d^2\psi}{dx^2}=-k^2\psi.\]The kinetic contribution is therefore positive despite the minus sign in the Hamiltonian:
\[K=\frac{\hbar^2k^2}{2m_e} =15.2399\ \mathrm{eV}.\]Adding the potential with its stated sign gives
\[\boxed{E=K+V_0=18.2399\ \mathrm{eV}},\]and
\[\boxed{\omega=\frac E\hbar =2.771\times10^{16}\ \mathrm{rad\,s^{-1}}}.\]Thus one stationary solution is
\[\boxed{\Psi(x,t)=A e^{\,i(2.00\times10^{10}\ \mathrm{m^{-1}})x} e^{-i(2.771\times10^{16}\ \mathrm{s^{-1}})t}}.\]The positive $k$ selects positive momentum and probability current; changing $k$ to $-k$ reverses the current but leaves the energy unchanged. The combinations $kx$ and $\omega t$ are dimensionless. In the limit $k\to0$, the kinetic contribution vanishes and $E\to V_0$.
2. Adding a constant to every potential energy
If $\hat H\psi_n=E_n\psi_n$, show what changes when $V(\mathbf r)$ is replaced by $V(\mathbf r)+C$, where $C$ is a real constant.
Solution. The shifted Hamiltonian is
\[\hat H^{\prime}=\hat H+C.\]Acting on the old eigenfunction gives
\[\hat H^{\prime}\psi_n=(\hat H+C)\psi_n =(E_n+C)\psi_n.\]Hence
\[\boxed{\psi_n^{\prime}=\psi_n,\qquad E_n^{\prime}=E_n+C}.\]The corresponding time-dependent state is
\[\Psi_n^{\prime}(\mathbf r,t) =\psi_n(\mathbf r)e^{-i(E_n+C)t/\hbar} =e^{-iCt/\hbar}\Psi_n(\mathbf r,t).\]The extra factor is a global phase, so $\lvert\Psi_n^{\prime}\rvert^2=\lvert\Psi_n\rvert^2$ and all expectation values are unchanged. The exponent is dimensionless because $Ct/\hbar$ is energy times time divided by action. Energy differences also remain unchanged: $(E_m+C)-(E_n+C)=E_m-E_n$. For $C\to0$, the original spectrum and state are recovered.
Descriptive Questions
- Why is the time-dependent Schrödinger equation a postulate rather than a derivable consequence of the classical energy equation?
- How does self-adjointness of the Hamiltonian guarantee preservation of normalization during time evolution?
- Derive the time-independent Schrödinger equation by separation of variables and explain why its separation constant is real.
- Why is an energy eigenstate called stationary even though its state vector continues to acquire phase?
Numerical Problems
- Find the kinetic energy and angular frequency of a free electron with de Broglie wavelength $0.200\ \mathrm{nm}$.
Final answer: $\boxed{E=37.60\ \mathrm{eV},\quad\omega=5.713\times10^{16}\ \mathrm{rad\,s^{-1}}}$. - An energy eigenstate has $E=2.00\ \mathrm{eV}$. Find its phase angle after $1.00\ \mathrm{fs}$ in the convention $e^{-iEt/\hbar}$.
Final answer: $\boxed{-Et/\hbar=-3.039\ \mathrm{rad}}$. - A stationary-state time factor has ordinary frequency $5.00\times10^{14}\ \mathrm{Hz}$. Find its energy.
Final answer: $\boxed{E=h\nu=2.068\ \mathrm{eV}}$.
Every added eigenvalue, phase, and constant-shift identity is checked in the MJ-11 problem-verification worksheet; every printed residual and check is zero.
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