Module 5: Time Evolution
Nothing is Static in Quantum Mechanics
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1 The Time-Dependent Schrödinger Equation
Everything we’ve done so far found the stationary states — the energy eigenstates \(|n\rangle\) that don’t change shape over time (only their phase oscillates). But a general quantum state is a superposition, and it does change over time.
The full time-dependent Schrödinger equation (TDSE) is:
\[ i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle = \hat{H}|\Psi(t)\rangle \]
Textbook reference: Shankar Section 4.3; Griffiths Section 2.1; Cohen-Tannoudji Vol. 1, Chapter III.
2 Time Evolution of Energy Eigenstates
For an energy eigenstate \(|n\rangle\):
\[ i\hbar\frac{\partial}{\partial t}|n,t\rangle = E_n|n,t\rangle \]
This integrates immediately:
\[ |n,t\rangle = e^{-iE_nt/\hbar}|n\rangle = e^{-i\omega(n+1/2)t}|n\rangle \]
The only effect of time evolution on an energy eigenstate is a global phase \(e^{-i\omega(n+1/2)t}\). Global phases have no physical consequence: \(|\psi(t)|^2 = |\psi(0)|^2\) for all \(t\).
This is why energy eigenstates are called stationary states — all observable quantities are time-independent in these states. The energy eigenstate doesn’t “do anything.”
This doesn’t mean nothing is happening — the wavefunction is rotating in the complex plane at frequency \((n + \frac{1}{2})\omega\). But since we only measure \(|\psi|^2\), we can’t see this rotation.
3 The Time Evolution Operator
The formal solution to the TDSE is:
\[ |\Psi(t)\rangle = \hat{U}(t)|\Psi(0)\rangle, \quad \hat{U}(t) = e^{-i\hat{H}t/\hbar} \]
For the QHO, we can write this in the Fock basis:
\[ \hat{U}(t) = e^{-i\omega(\hat{N}+1/2)t} = e^{-i\omega t/2}\, e^{-i\omega t\hat{N}} \]
Since \(\hat{N}|n\rangle = n|n\rangle\):
\[ \hat{U}(t)|n\rangle = e^{-i\omega t(n+1/2)}|n\rangle \]
The key identity: \(e^{-i\omega t\hat{N}}\hat{a} = \hat{a}e^{-i\omega t(\hat{N}-1)} = e^{-i\omega t}\hat{a}e^{-i\omega t\hat{N}}\), or:
\[ \hat{U}(t)\hat{a}\hat{U}^\dagger(t) = e^{-i\omega t}\hat{a} \]
This means the ladder operators in the Heisenberg picture rotate:
\[ \hat{a}(t) = e^{-i\omega t}\hat{a}(0), \qquad \hat{a}^\dagger(t) = e^{+i\omega t}\hat{a}^\dagger(0) \]
4 Time Evolution of Superposition States
A general initial state is a superposition:
\[ |\Psi(0)\rangle = \sum_{n=0}^{\infty} c_n|n\rangle, \quad \sum_n |c_n|^2 = 1 \]
Time evolution:
\[ |\Psi(t)\rangle = \sum_{n=0}^{\infty} c_n e^{-i\omega(n+1/2)t}|n\rangle \]
In position space:
\[ \Psi(x,t) = \sum_{n=0}^{\infty} c_n e^{-i\omega(n+1/2)t}\psi_n(x) \]
4.1 Example: Superposition of \(|0\rangle\) and \(|1\rangle\)
Consider: \[ |\Psi(0)\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle + |1\rangle\right) \]
Time evolution: \[ |\Psi(t)\rangle = \frac{e^{-i\omega t/2}}{\sqrt{2}}\left(|0\rangle + e^{-i\omega t}|1\rangle\right) \]
The expectation value of position:
\[ \langle\hat{x}\rangle(t) = \frac{x_0}{\sqrt{2}}\langle\Psi(t)|(\hat{a} + \hat{a}^\dagger)|\Psi(t)\rangle \]
The cross terms survive: \(\langle 0|\hat{a}|1\rangle = 1\), \(\langle 1|\hat{a}^\dagger|0\rangle = 1\).
\[ \langle\hat{x}\rangle(t) = \frac{x_0}{\sqrt{2}}\cdot\frac{1}{2}\cdot 2\cos(\omega t) = \frac{x_0}{\sqrt{2}}\cos(\omega t) \]
The expectation value of position oscillates at frequency \(\omega\) — like a classical oscillator!
5 Ehrenfest’s Theorem
5.1 General Statement
Ehrenfest’s theorem says that expectation values obey classical equations of motion:
\[ \frac{d}{dt}\langle\hat{A}\rangle = \frac{i}{\hbar}\langle[\hat{H}, \hat{A}]\rangle + \left\langle\frac{\partial\hat{A}}{\partial t}\right\rangle \]
For time-independent operators in the QHO:
\[ \frac{d}{dt}\langle\hat{x}\rangle = \frac{i}{\hbar}\langle[\hat{H}, \hat{x}]\rangle = \frac{\langle\hat{p}\rangle}{m} \]
\[ \frac{d}{dt}\langle\hat{p}\rangle = \frac{i}{\hbar}\langle[\hat{H}, \hat{p}]\rangle = -m\omega^2\langle\hat{x}\rangle \]
Combined: \[ \frac{d^2}{dt^2}\langle\hat{x}\rangle = -\omega^2\langle\hat{x}\rangle \]
This is exactly Newton’s law for the harmonic oscillator! The center of the wavepacket \(\langle\hat{x}\rangle\) oscillates like a classical particle.
Ehrenfest’s Theorem for the QHO:
\[\frac{d^2}{dt^2}\langle\hat{x}\rangle = -\omega^2\langle\hat{x}\rangle\]
The quantum expectation value always satisfies the classical equation of motion — for any quantum state!
5.2 Ehrenfest in the Heisenberg Picture
In the Heisenberg picture, operators evolve:
\[ \hat{x}(t) = x_0\cos(\omega t)\,\hat{x}(0) + \frac{\sin(\omega t)}{m\omega}\hat{p}(0) \]
\[ \hat{p}(t) = -m\omega\sin(\omega t)\,\hat{x}(0) + \cos(\omega t)\,\hat{p}(0) \]
This is exactly the classical solution \(x(t) = x_0\cos\omega t + \frac{p_0}{m\omega}\sin\omega t\) promoted to operators. The QHO Heisenberg equations are linear, which is why they have exact classical analogs.
6 Quantum Revivals
6.1 When Superpositions Come Back
For a general superposition \(|\Psi(t)\rangle = \sum_n c_n e^{-in\omega t}|n\rangle\) (dropping the common phase \(e^{-i\omega t/2}\)):
The state returns exactly to itself when all phases complete full rotations:
\[ e^{-in\omega T} = 1 \text{ for all } n \quad \Rightarrow \quad T = \frac{2\pi}{\omega} \]
This is the classical period — the QHO is exactly periodic! Every state (not just energy eigenstates) returns to itself after time \(T = 2\pi/\omega\).
For a particle in a box, the energy levels go as \(n^2\), so different Fourier components have incommensurable periods. The wavepacket does NOT return exactly after one period.
The QHO is special: its equally spaced energy levels (spacing \(\hbar\omega\)) mean all phase factors are integer multiples of \(e^{-i\omega t}\), guaranteeing exact periodicity.
7 The Heisenberg Picture: Operators Evolve, States Don’t
In the Heisenberg picture, the time dependence is placed on operators rather than states:
\[ \hat{a}_H(t) = e^{i\hat{H}t/\hbar}\,\hat{a}\,e^{-i\hat{H}t/\hbar} = e^{-i\omega t}\hat{a} \]
\[ \hat{a}^\dagger_H(t) = e^{+i\omega t}\hat{a}^\dagger \]
The position and momentum operators: \[ \hat{x}_H(t) = \frac{x_0}{\sqrt{2}}\left(e^{-i\omega t}\hat{a} + e^{+i\omega t}\hat{a}^\dagger\right) \] \[ = x_0\cos(\omega t)\,\hat{x}(0) + \frac{\sin(\omega t)}{m\omega}\hat{p}(0) \]
This is the quantum oscillator in the Heisenberg picture — it oscillates exactly like the classical oscillator.
8 The Wigner Function: Phase-Space Quantum Mechanics
8.1 What Is the Wigner Function?
In classical mechanics, we can define a probability distribution in phase space \(f(x, p)\). In quantum mechanics, the uncertainty principle forbids a true joint probability distribution. But we can define the Wigner function:
\[ W(x, p) = \frac{1}{\pi\hbar}\int_{-\infty}^{\infty} \psi^*\!\left(x + y\right)\psi\left(x - y\right) e^{2ipy/\hbar}\,dy \]
Properties of \(W(x,p)\):
- Real: \(W(x,p)\) is always real
- Marginals: \(\int W\,dp = |\psi(x)|^2\) and \(\int W\,dx = |\phi(p)|^2\)
- Not always positive: \(W\) can be negative in some regions (a signature of quantum behavior)
- Area normalization: \(\int\!\!\int W(x,p)\,dx\,dp = 1\)
8.2 Wigner Function of the Ground State
For \(\psi_0 = (\pi x_0^2)^{-1/4}e^{-x^2/2x_0^2}\):
\[ W_0(x,p) = \frac{1}{\pi\hbar}\,e^{-x^2/x_0^2 - p^2 x_0^2/\hbar^2} \]
This is a Gaussian in phase space — entirely positive! The ground state is the most classical-looking quantum state.
8.3 Wigner Function of Excited States
For excited states \(|n\rangle\), the Wigner function is:
\[ W_n(x,p) = \frac{(-1)^n}{\pi\hbar}\,L_n\!\left(\frac{2r^2}{x_0^2}\right)\,e^{-r^2/x_0^2} \]
where \(r^2 = x^2 + p^2 x_0^2/\hbar^2\) and \(L_n\) are Laguerre polynomials.
Key feature: for \(n \geq 1\), \(W_n\) takes negative values — a purely quantum signature. The negative regions appear as rings (for \(n = 1\)) or more complex structures for higher \(n\).
The Wigner function is the closest thing quantum mechanics has to a phase-space probability distribution. In quantum optics, measuring the Wigner function (via homodyne tomography) is a standard experimental technique. Negative values of \(W\) are direct evidence of non-classical states.
9 Time Evolution of Expectation Values: Summary
For a general state \(|\Psi(t)\rangle = \sum_n c_n e^{-i\omega(n+1/2)t}|n\rangle\):
| Quantity | Time dependence |
|---|---|
| \(\langle\hat{x}\rangle(t)\) | Oscillates at \(\omega\) |
| \(\langle\hat{p}\rangle(t)\) | Oscillates at \(\omega\), \(90°\) out of phase |
| \(\langle\hat{x}^2\rangle(t)\) | Oscillates at \(2\omega\) and constant |
| \(\langle\hat{H}\rangle\) | Constant (energy conservation) |
| \(\langle\hat{N}\rangle\) | Constant |
| Period of full return | \(T = 2\pi/\omega\) |
Before moving on, make sure you can:
- Write the time evolution of an energy eigenstate \(|n\rangle\).
- Prove that \(\langle\hat{x}\rangle(t)\) oscillates classically for any state.
- State Ehrenfest’s theorem for the QHO and verify it gives Newton’s law.
- Explain why the QHO state returns exactly after \(T = 2\pi/\omega\).
- Define the Wigner function and state its key properties.
- Why can the Wigner function be negative? What does this mean physically?
Advanced: Compute the Wigner function for \(|1\rangle\) and show it has negative values.