Part VI, Chapter 4 | Page 1 of 3

Time-Dependent Perturbation Theory

Transitions between quantum states induced by time-varying perturbations

While time-independent perturbation theory corrects energy levels and stationary states, time-dependent perturbation theory addresses a fundamentally different question: given a system initially in a known eigenstate, what is the probability of finding it in a different state after a time-varying perturbation is applied? This framework describes all quantum transitions: absorption, emission, scattering, and decay.

The Setup

The Hamiltonian has a time-independent part with known solutions and a time-dependent perturbation switched on at $t = 0$:

$$\hat{H}(t) = \hat{H}_0 + \hat{H}'(t), \quad \hat{H}'(t) = 0 \text{ for } t < 0$$

The unperturbed Hamiltonian has known eigenstates: $\hat{H}_0|n\rangle = E_n|n\rangle$. At $t = 0$, the system is in a definite eigenstate $|i\rangle$ of $\hat{H}_0$.

The Interaction Picture

We expand the time-evolving state in the basis of unperturbed eigenstates, factoring out the known time evolution due to $\hat{H}_0$:

$$|\psi(t)\rangle = \sum_n c_n(t)\, e^{-iE_n t/\hbar}\, |n\rangle$$

Substituting into the Schrodinger equation $i\hbar\partial_t|\psi\rangle = \hat{H}|\psi\rangle$, we obtain the exact equation for the coefficients:

$$i\hbar\,\dot{c}_n(t) = \sum_m \langle n|\hat{H}'(t)|m\rangle\, e^{i\omega_{nm}t}\, c_m(t)$$

where $\omega_{nm} = (E_n - E_m)/\hbar$ is the Bohr frequency connecting states $|n\rangle$ and $|m\rangle$. This is still exact -- no approximation has been made.

First-Order Perturbation Theory

The perturbative approximation: replace $c_m(t)$ on the right side by its initial value. Since the system starts in state $|i\rangle$:

$$c_m(0) = \delta_{mi} \quad \Rightarrow \quad c_m^{(0)}(t) = \delta_{mi}$$

Substituting into the equation of motion, the first-order transition amplitude from $|i\rangle$ to a different final state $|f\rangle$ ($f \neq i$) is:

$$\boxed{c_f^{(1)}(t) = \frac{1}{i\hbar}\int_0^t \langle f|\hat{H}'(t')|i\rangle\, e^{i\omega_{fi}t'}\, dt'}$$

This is the central formula of first-order time-dependent perturbation theory. Everything else follows from evaluating this integral for specific forms of $\hat{H}'(t)$.

Transition Probability

The probability of finding the system in state $|f\rangle$ at time $t$ is:

$$\boxed{P_{i \to f}(t) = |c_f(t)|^2}$$

For this probability to be meaningful, we need $P_{i\to f} \ll 1$ (so the perturbation hasn't significantly depleted the initial state). When $P_{i\to f}$ approaches unity, higher-order corrections become essential.

Constant Perturbation Turned On at t=0

The simplest case: $\hat{H}'(t) = \hat{V}\,\theta(t)$ (constant perturbation, suddenly switched on). The integral evaluates to:

$$c_f^{(1)}(t) = \frac{V_{fi}}{i\hbar}\int_0^t e^{i\omega_{fi}t'}\, dt' = \frac{V_{fi}}{i\hbar}\frac{e^{i\omega_{fi}t} - 1}{i\omega_{fi}}$$

The transition probability is:

$$P_{i\to f}(t) = \frac{|V_{fi}|^2}{\hbar^2}\frac{4\sin^2(\omega_{fi}t/2)}{\omega_{fi}^2} = \frac{|V_{fi}|^2}{\hbar^2}\, t^2\, \text{sinc}^2\!\left(\frac{\omega_{fi}t}{2}\right)$$

This function is sharply peaked near $\omega_{fi} = 0$ (energy conservation: $E_f \approx E_i$). The peak height grows as $t^2$ and the width shrinks as $1/t$, with the area under the peak growing linearly in $t$. In the long-time limit, this leads to Fermi's Golden Rule (covered on the next page).

Sinusoidal Perturbation

For a harmonic perturbation (e.g., electromagnetic radiation):

$$\hat{H}'(t) = \hat{V}\cos(\omega t) = \frac{\hat{V}}{2}(e^{i\omega t} + e^{-i\omega t})$$

The transition amplitude becomes a sum of two terms:

$$c_f^{(1)}(t) = \frac{V_{fi}}{2i\hbar}\left[\frac{e^{i(\omega_{fi}+\omega)t}-1}{\omega_{fi}+\omega} + \frac{e^{i(\omega_{fi}-\omega)t}-1}{\omega_{fi}-\omega}\right]$$

Each term is peaked at a different resonance condition, corresponding to absorption and stimulated emission.

Resonance: Absorption and Stimulated Emission

The two terms in the transition amplitude dominate in different regimes:

Absorption ($E_f > E_i$, so $\omega_{fi} > 0$):

The second term dominates when $\omega \approx \omega_{fi}$. The system absorbs energy $\hbar\omega$ from the perturbation and transitions upward.

Stimulated emission ($E_f < E_i$, so $\omega_{fi} < 0$):

The first term dominates when $\omega \approx |\omega_{fi}|$. The system emits energy and transitions downward, stimulated by the perturbation.

Near resonance ($\omega \approx |\omega_{fi}|$), keeping only the dominant term:

$$P_{i\to f}(t) \approx \frac{|V_{fi}|^2}{4\hbar^2}\frac{\sin^2[(\omega_{fi}-\omega)t/2]}{[(\omega_{fi}-\omega)/2]^2}$$

Maximum transition probability at exact resonance: $P_{\max} = |V_{fi}|^2 t^2/(4\hbar^2)$, growing quadratically with time (until perturbation theory breaks down).

The Energy-Time Uncertainty Principle

The sinc-squared function in the transition probability has a width $\Delta\omega \sim 2\pi/t$ in frequency space. This means:

$$\Delta E \cdot t \sim 2\pi\hbar = h$$

This is a manifestation of the energy-time uncertainty principle:

Short-time perturbations ($t$ small) allow transitions to a wide range of final state energies (energy conservation is fuzzy)
Long-time perturbations ($t$ large) restrict transitions to a narrow energy window (energy conservation becomes sharp)
In the limit $t \to \infty$, perfect energy conservation is enforced: $E_f = E_i \pm \hbar\omega$

Looking Ahead

On the next page, we derive Fermi's Golden Rule -- the long-time limit of the transition probability that gives a constant transition rate. This is perhaps the single most useful result in all of quantum mechanics, with applications from atomic spectroscopy to nuclear decay to particle physics.

← Variational Method Fermi's Golden Rule →

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