← Part VI/Time-Dependent Perturbation

4. Time-Dependent Perturbation Theory

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Transitions between quantum states induced by time-varying perturbations.

The Setup

Time-dependent Hamiltonian:

$$\hat{H}(t) = \hat{H}_0 + \hat{H}'(t)$$

where $\hat{H}_0$ has known eigenstates $|n\rangle$ with energies $E_n$

Interaction Picture

Expand state in basis of unperturbed eigenstates:

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

Schrödinger equation gives:

$$i\hbar\dot{c}_n(t) = \sum_m H'_{nm}(t)e^{i\omega_{nm}t}c_m(t)$$

where $\omega_{nm} = (E_n - E_m)/\hbar$ and $H'_{nm} = \langle n|\hat{H}'|m\rangle$

First-Order Perturbation Theory

Assume initially in state $|i\rangle$: $c_i(0) = 1$, $c_{n\neq i}(0) = 0$

To first order:

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

Transition amplitude from state $|i\rangle$ to state $|f\rangle$

Transition Probability

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

Probability of finding system in state $|f\rangle$ at time $t$

Sinusoidal Perturbation

Consider harmonic perturbation:

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

Transition amplitude:

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

Resonance Condition

Maximum transition rate when:

$$\omega \approx \omega_{fi} = \frac{E_f - E_i}{\hbar}$$

Energy conservation: photon energy matches transition energy

Near resonance $(\omega \approx \omega_{fi})$:

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

Absorption and Emission

Absorption: $E_f > E_i$, $\omega > 0$ - system gains energy
Stimulated emission: $E_f < E_i$, $\omega < 0$ - system loses energy
Selection rules: $V_{fi} = 0$ for forbidden transitions

Electric Dipole Approximation

For electromagnetic radiation:

$$\hat{H}'(t) = -\vec{d}\cdot\vec{E}(t) = -e\vec{r}\cdot\vec{E}_0\cos(\omega t)$$

Matrix element:

$$V_{fi} = -e\langle f|\vec{r}|i\rangle\cdot\vec{E}_0$$

$\langle f|\vec{r}|i\rangle$ is the transition dipole moment

Selection Rules for Hydrogen

Electric dipole transitions require:

$\Delta \ell = \pm 1$ (angular momentum conservation)
$\Delta m = 0, \pm 1$ (polarization dependent)
$\Delta n$ = arbitrary (but favors small values)

Examples:

Allowed: $2p \to 1s$, $3d \to 2p$
Forbidden: $2s \to 1s$, $3d \to 1s$

Sudden Perturbation

Perturbation applied instantaneously at $t = 0$:

$$\hat{H} = \begin{cases}\hat{H}_0 & t < 0\\\hat{H}_0 + \hat{V} & t \geq 0\end{cases}$$

Wave function unchanged at $t = 0$ (sudden approximation)

Transition probability:

$$P_{i\to f} = |\langle\psi_f^{new}|\psi_i^{old}\rangle|^2$$

Adiabatic Perturbation

Perturbation applied very slowly:

Adiabatic theorem: If $\hat{H}(t)$ changes slowly enough, system remains in instantaneous eigenstate

$$|\psi(t)\rangle = |n(t)\rangle e^{i\theta_n(t)}$$

where $|n(t)\rangle$ is instantaneous eigenstate and $\theta_n(t)$ is dynamic phase

Example: Atom in Laser Field

Laser drives transitions between levels $|1\rangle$ and $|2\rangle$

On resonance $(\omega = \omega_{21})$, probability oscillates:

$$P_{1\to 2}(t) = \sin^2(\Omega_R t/2)$$

where $\Omega_R = |V_{21}|/\hbar$ is the Rabi frequency

Complete population transfer at $t = \pi/\Omega_R$

Fermi's Golden Rule Preview

For transitions to continuum of final states:

$$\Gamma_{i\to f} = \frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f)$$

where $\rho(E_f)$ is density of final states

Transition rate (per unit time) - covered in detail next section

← Fine Structure Fermi's Golden Rule →

Quantum Mechanics Course