Time Dependence
Chem 3240 · Lecture 3.8
How states evolve in time
The time-dependent Schrödinger equation governs all quantum dynamics:
\[ i\hbar\frac{\partial}{\partial t}\mid \psi \rangle =\hat{H}\mid\psi(t)\rangle \]
So far we have studied states that do not appear to move. Why?
Stationary states carry a phase
A pure eigenstate of \(\hat{H}\) evolves only by a complex phase factor:
\[\psi_n(x,t) = \psi_n(x)\, e^{-\frac{i}{\hbar}E_n t}\]
- Spatial shape \(\psi_n(x)\) is fixed
- Only the phase \(e^{-iE_n t/\hbar}\) turns
Why the probability is stationary
The phase has unit magnitude, so it cancels in \(|\psi|^2\):
\[\mid \psi_n(x,t) \mid^2 = \psi_n^*(x)\psi_n(x)\, e^{-\frac{i}{\hbar}E_n t} e^{+\frac{i}{\hbar}E_n t} = \mid \psi_n(x) \mid^2\]
Definite energy \(\Rightarrow\) time-independent probability distribution.
Expectation values freeze too
For any operator \(\hat{A}\) the phases also cancel:
\[\langle A \rangle = \int \psi_n^*(x) e^{+\frac{i}{\hbar}E_n t} \hat{A}\, \psi_n(x) e^{-\frac{i}{\hbar}E_n t}\, dx = \int \psi_n^*(x) \hat{A}\, \psi_n(x)\, dx\]
Nothing observable changes in a single eigenstate.
Superpositions bring time back
Start with a superposition at \(t=0\):
\[\mid \psi(0) \rangle = c_1\mid 1 \rangle + c_2 \mid 2 \rangle\]
Each piece picks up its own phase:
\[\mid \psi(t) \rangle = c_1 e^{-\frac{i}{\hbar}E_1 t}\mid 1 \rangle + c_2 e^{-\frac{i}{\hbar}E_2 t}\mid 2 \rangle\]
Coefficients move, basis stays fixed
\[\mid \psi(t) \rangle = c_1(t)\mid 1\rangle+c_2(t) \mid 2 \rangle\]
- The eigenstates are fixed, orthogonal directions
- The coefficients \(c_n(t)\) rotate the state vector through them
General case:
\[\mid \psi(t)\rangle = \sum_n c_n e^{-\frac{i}{\hbar}E_n t} \mid n\rangle\]
Interference makes probabilities oscillate
Different energies \(\Rightarrow\) phases turn at different rates.
- Relative phase \(e^{-\frac{i}{\hbar}(E_k - E_n)t}\) drives interference
- Drives oscillations between states in two-level systems
- The engine of all quantum dynamics
What stays constant: normalization
Orthogonality kills the cross terms, so the norm is conserved:
\[\langle \psi(t) \mid \psi(t)\rangle = \sum_n \sum_k c^*_n c_k\, e^{-\frac{i}{\hbar}(E_k - E_n)t} \delta_{kn} = \sum_n \mid c_n \mid^2 = 1\]
The particle is always somewhere.
Constants of motion
Time derivative of any expectation value:
\[\frac{\partial}{\partial t}\langle A \rangle = \frac{1}{i\hbar} \langle \psi \mid [\hat{A}, \hat{H}] \mid \psi \rangle + \langle \psi \mid \frac{\partial \hat{A}}{\partial t} \mid \psi \rangle\]
- Commutes with \(\hat{H}\) and no explicit \(t\) \(\Rightarrow\) conserved
- Energy commutes with itself \(\Rightarrow\) \(\dfrac{\partial}{\partial t}\langle E \rangle = 0\)
Quantum dynamics, visualized
Superposition of energies \(\Rightarrow\) motion you can watch.
Takeaway
A single eigenstate only spins its phase, so nothing observable moves; superpose different energies and their phases beat against each other, and quantum dynamics comes alive.