5.2.3 Applications#

Prompts

How does a harmonic perturbation \(V(t) = V_0 \cos(\omega t)\) create resonant transitions between quantum states? What determines the Rabi frequency?
What are the electric dipole selection rules for hydrogen, and why do they follow from the symmetry of the dipole operator?
How does Fermi’s golden rule connect the decay rate \(\Gamma\) to the lifetime \(\tau\) and the natural linewidth \(\Delta E\) of a spectral line?
What is linear response theory, and how does the Kubo formula express conductivity in terms of current-current correlations?
Why is the Hall conductance \(\sigma_{xy} = \nu e^2/h\) exactly quantized for filled Landau levels?

Lecture Notes#

Overview#

This lesson applies first-order time-dependent perturbation theory to four key physical systems: harmonic driving that reveals resonance and Rabi oscillations, electric dipole transitions with selection rules, decay and linewidth from finite lifetime, and linear response theory leading to the quantum Hall conductance—a remarkable exact result from perturbation theory.

Harmonic Perturbation#

The simplest time-dependent perturbation is a harmonic (oscillating) drive:

\[ V(t) = V_0 \cos(\omega t) \]

We can rewrite this as a sum of positive and negative frequency components:

\[ V(t) = \frac{V_0}{2} \left( \mathrm{e}^{\mathrm{i}\omega t} + \mathrm{e}^{-\mathrm{i}\omega t} \right) \]

The transition amplitude from state \(\vert i \rangle\) to \(\vert f \rangle\) picks up resonance when the driving frequency matches the energy gap:

\[ c_f^{(1)}(t) \propto \langle f | V_0 | i \rangle \left[ \frac{\mathrm{e}^{\mathrm{i}(\omega - \omega_{fi})t} - 1}{\omega - \omega_{fi}} + \frac{\mathrm{e}^{-\mathrm{i}(\omega + \omega_{fi})t} - 1}{\omega + \omega_{fi}} \right] \]

where \(\omega_{fi} = (E_f - E_i)/\hbar\). The first term dominates when \(\omega \approx \omega_{fi}\) (absorption), and the second when \(\omega \approx -\omega_{fi}\) (stimulated emission). This is the rotating wave approximation: we keep only the resonant term and drop the rapidly oscillating counter-rotating term.

Near resonance, the transition probability oscillates at the Rabi frequency:

Rabi Oscillations

In the rotating frame near resonance, the transition probability is

\[ P(t) = \sin^2\left(\frac{\Omega_R t}{2}\right) \quad \text{where} \quad \Omega_R = \frac{|V_{fi}|}{\hbar} \]

is the Rabi frequency. Population coherently oscillates between the two states at rate \(\Omega_R\). A \(\pi\)-pulse (\(\Omega_R t = \pi\)) completely flips the state.

A key application: scanning the driving frequency \(\omega\) probes the energy level spacing. Each resonance reveals \(E_f - E_i\), forming the basis of spectroscopy.

Electric Dipole Transitions#

When light couples to matter, the interaction is dominated by the dipole term:

\[ V(t) = -\boldsymbol{d} \cdot \boldsymbol{E}(t) = -(-e\boldsymbol{r}) \cdot \boldsymbol{E}(t) = e \boldsymbol{r} \cdot \boldsymbol{E}(t) \]

where \(\boldsymbol{d} = -e\boldsymbol{r}\) is the electric dipole moment. The transition amplitude depends on the dipole matrix element \(\langle f | \boldsymbol{r} | i \rangle\).

Not all transitions are allowed. The parity and angular momentum structure of wavefunctions impose selection rules. For hydrogen (or any single-electron atom):

Selection Rules for Electric Dipole Transitions

For a transition \(n\ell m_\ell \to n'\ell' m_{\ell}'\):

\(\Delta \ell = \pm 1\) (change parity)
\(\Delta m_\ell = 0, \pm 1\) (depends on light polarization)
\(\Delta S = 0\) (spin unchanged in non-relativistic theory)

Transitions violating these are forbidden and have suppressed amplitude. States that can only decay via forbidden transitions become metastable (long-lived).

For example, the 2s state of hydrogen cannot decay to 1s by electric dipole radiation (both \(\ell = 0\), so \(\Delta \ell = 0\)), making 2s metastable. The 2p state, however, can decay resonantly.

Lifetime and Linewidth#

An excited state coupled to a continuum of final states (e.g., radiating photons) decays exponentially. From Fermi’s golden rule applied to the emission process, the decay rate is

\[ \Gamma = \frac{2\pi}{\hbar} \sum_f |\langle f | V | i \rangle|^2 \delta(E_i - E_f - \hbar\omega_{\text{photon}}) \]

The inverse is the lifetime:

Lifetime and Natural Linewidth

\[ \tau = \frac{1}{\Gamma} \]

The energy-time uncertainty principle implies that a state with finite lifetime \(\tau\) has an intrinsic energy uncertainty:

\[ \Delta E \sim \hbar \Gamma = \frac{\hbar}{\tau} \]

This gives a natural linewidth in the emission spectrum. The lineshape is Lorentzian, centered at the transition energy \(E_i - E_f\) with width \(\Gamma\).

For instance, the 2p state of hydrogen has \(\tau \sim 1.6 \times 10^{-9}\) s (lifetime), corresponding to \(\Gamma \sim 0.1\) eV (linewidth).

Linear Response and Hall Conductance#

An electric field \(\boldsymbol{E}(t)\) drives currents in the material. For weak fields, the response is linear:

\[ j_\alpha(t) = \sum_\beta \sigma_{\alpha\beta} E_\beta(t) \]

where \(\sigma_{\alpha\beta}\) is the conductivity tensor. Computing \(\sigma\) from first-order perturbation theory requires summing the current-current correlation:

Kubo Formula for Conductivity

\[ \sigma_{xy} = \frac{2\pi e^2}{\hbar} \sum_{n \neq m} \frac{f(E_n) - f(E_m)}{(E_n - E_m)^2} \cdot 2\operatorname{Im}\langle n | \hat{j}_x | m \rangle \langle m | \hat{j}_y | n \rangle \]

where \(f(E)\) is the Fermi-Dirac distribution and \(\hat{j}\) is the current operator. For \(T = 0\), the sum is over occupied (\(n\)) and empty (\(m\)) states.

Quantum Hall Conductance from Filled Landau Levels

When a Landau level is completely filled (all \(N_\phi\) states occupied, where \(N_\phi = B A / \phi_0\) is the number of states), the Kubo formula yields an exact, quantized result:

\[ \sigma_{xy} = \frac{\nu e^2}{h} \]

where \(\nu\) is the filling factor (number of filled Landau levels). For a single filled level, \(\sigma_{xy} = e^2/h\)—exact to all orders in perturbation theory, independent of disorder or interactions.

This connects back to §4.3 (Landau levels and cyclotron motion): the quantum Hall effect is not just a band structure curiosity—it is a fundamental linear response result. Perturbation theory reveals its quantization.

Discussion: Why Is the Hall Conductance Exactly Quantized?

The Kubo formula gives \(\sigma_{xy} = \nu e^2/h\) for filled Landau levels—an integer times a fundamental constant.

Fermi’s golden rule is perturbative and approximate. Yet \(\sigma_{xy}\) is exact to all orders. What makes this quantity topologically protected?
The conductance is independent of disorder, sample geometry, and interaction strength. Why does perturbation theory capture an exact result at all?
The integer \(\nu\) counts filled Landau levels. What is the physical meaning of this number from the perspective of topology and winding numbers?
If we gradually deform the Hamiltonian without closing a gap, can \(\sigma_{xy}\) change? What happens if we allow the gap to close?

Summary#

Harmonic driving at frequency \(\omega\) induces resonant transitions when \(\omega = \omega_{fi}\); Rabi oscillations probe the coupling strength.
Dipole selection rules (\(\Delta \ell = \pm 1\), etc.) determine which transitions are allowed; metastable states bypass these rules.
Finite lifetime \(\tau = 1/\Gamma\) produces natural linewidth \(\Delta E \sim \hbar/\tau\); spectral lines have intrinsic width.
Linear response and the Kubo formula predict exact Hall quantization for filled Landau levels—a triumph of first-order perturbation theory.

Homework#

1. Rabi Frequency and \(\pi\)-Pulses

A two-level system with transition frequency \(\omega_{01} = 1.0 \times 10^{15}\) rad/s is driven by a resonant oscillating field \(V(t) = V_0 \cos(\omega_{01} t)\) with \(V_0 = 0.1\) meV.

(a) Calculate the Rabi frequency \(\Omega_R = |V_{01}|/\hbar\).

(b) What drive duration is needed for a \(\pi\)-pulse (complete inversion)?

2. Off-Resonance Driving

The same system is now driven at frequency \(\omega = \omega_{01} + \delta\), where \(\delta = 0.1 \Omega_R\) is a small detuning.

(a) In the rotating frame, write the effective Hamiltonian (include the detuning term).

(b) What is the modified oscillation frequency?

3. Selection Rules for Hydrogen

Which of the following electric dipole transitions in hydrogen are allowed? For each, state whether it satisfies all three selection rules (\(\Delta \ell = \pm 1\), \(\Delta m_\ell = 0, \pm 1\), \(\Delta S = 0\)).

(a) \(1s \to 2s\)

(b) \(2p \to 1s\)

(d) \(3d \to 2p\)

(e) \(3d \to 2s\)

4. Lifetime of the 2p State

The 2p state of hydrogen decays to 1s by emitting a Lyman-\(\alpha\) photon. The dipole matrix element is \(|\langle 1s | r | 2p \rangle| = 2 a_0 / \sqrt{3}\) (where \(a_0\) is the Bohr radius), and the transition frequency is \(\omega = (E_2 - E_1)/\hbar = 1.55 \times 10^{16}\) rad/s.

(a) Use Fermi’s golden rule to estimate the decay rate:

\[ \Gamma = \frac{\omega^3}{3\pi \epsilon_0 \hbar c^3} |\langle 1s | e r | 2p \rangle|^2 \]

(b) Calculate the lifetime \(\tau = 1/\Gamma\) in seconds and nanoseconds.

5. Energy-Time Uncertainty

A metastable state has a linewidth of \(\Delta E = 10^{-5}\) eV.

(a) What is the lifetime?

(b) Over what time scale can you measure the energy to better than \(\Delta E\)?

6. Kubo Formula for a Two-Level System

Consider a two-level system in a magnetic field with Hamiltonian \(\hat{H}_0 = -\frac{\hbar\omega_0}{2}\hat{\sigma}^z\). A weak time-dependent field couples via \(\hat{H}_1(t) = -\hat{\sigma}^x E_x(t)\) (proportional to dipole coupling).

(a) Identify the current operators \(\hat{j}_x\) and \(\hat{j}_y\) in this language. (Hint: in the quantum Hall context, current is proportional to \(\hat{\sigma}^x\) and \(\hat{\sigma}^y\).)

(b) Compute \(\langle 0 | \hat{\sigma}^x | 1 \rangle\) and \(\langle 1 | \hat{\sigma}^y | 0 \rangle\).

(d) At \(T = 0\), does \(\sigma_{xy}\) vanish or not? Explain.

7. Hall Conductance for a Single Landau Level

For a 2D electron gas in a perpendicular magnetic field with one filled Landau level containing \(N_\phi\) electrons:

(a) What is the total charge? (Express in terms of \(N_\phi\) and \(e\).)

(b) Show that the Hall conductance \(\sigma_{xy} = (e \times \text{velocity}) / E\) equals \(e^2/h\) per filled level. (Hint: use dimensional analysis and Landau level degeneracy.)

(d) If the system has two partially filled Landau levels with fillings \(\nu_1 = 1\) and \(\nu_2 = 1/3\), what is the net Hall conductance?