Table of Contents

1. Introduction
2. Historical Background and Significance
3. Motivation for the Rule
4. Derivation from Time-Dependent Perturbation Theory
5. Mathematical Expression
6. Transition Rate and Density of States
7. Physical Interpretation
8. Validity Conditions
9. Example: Spontaneous Emission of a Photon
10. Example: Particle Scattering in a Potential
11. Comparison with Classical Rates
12. Applications in Atomic, Nuclear, and Condensed Matter Physics
13. Limitations and Cautions
14. Variants and Extensions
15. Conclusion

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1. Introduction

Fermi’s Golden Rule provides a formula for computing the transition rate between quantum states under the influence of a weak time-dependent perturbation. It is one of the most widely used results in quantum physics, particularly in atomic transitions, scattering theory, and decay processes.

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2. Historical Background and Significance

Enrico Fermi developed this rule in the 1930s while analyzing beta decay. It helped bridge quantum theory with measurable decay rates and laid the foundation for the quantum theory of radioactive processes. Today, it is indispensable in both theoretical derivations and experimental calculations.

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3. Motivation for the Rule

When a system is exposed to a weak perturbation, it may transition from an initial quantum state \( |i\rangle \) to a final state \( |f\rangle \). The goal is to compute the transition probability per unit time (i.e., rate) for this process, particularly when the final states form a continuum.

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4. Derivation from Time-Dependent Perturbation Theory

Assume a perturbing Hamiltonian of the form:

\[
\hat{H}(t) = \hat{H}_0 + \lambda \hat{H}'(t)
\]

Let \( \hat{H}'(t) = \hat{V} e^{-i\omega t} + \hat{V}^\dagger e^{i\omega t} \), and suppose \( \hat{H}'(t) \) couples the initial and final states. From first-order time-dependent perturbation theory, the transition amplitude is:

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

The probability is then:

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

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5. Mathematical Expression

Fermi’s Golden Rule gives the transition rate \( \Gamma \) as:

\[
\Gamma_{i \to f} = \frac{2\pi}{\hbar} |\langle f | \hat{H}’ | i \rangle|^2 \rho(E_f)
\]

Where:

• \( \Gamma_{i \to f} \): Transition probability per unit time.
• \( \langle f | \hat{H}’ | i \rangle \): Matrix element of the perturbation.
• \( \rho(E_f) \): Density of final states at energy \( E_f \).

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6. Transition Rate and Density of States

The density of states \( \rho(E) \) measures how many quantum states exist per unit energy near a given energy level. In systems where the final states form a continuum, this density is essential to compute the transition rate:

\[
\rho(E) = \frac{dn}{dE}
\]

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7. Physical Interpretation

• The transition rate is proportional to the square of the coupling strength between initial and final states.
• It is also proportional to the number of available final states.
• Intuitively, stronger interactions and more available final states result in higher transition probabilities.

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8. Validity Conditions

Fermi’s Golden Rule applies under the following conditions:

• The perturbation is weak.
• The perturbation is time-independent or slowly varying.
• The final state is part of a quasi-continuum.
• The interaction time is long enough for energy conservation to emerge via the delta function \( \delta(E_f – E_i – \hbar\omega) \).

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9. Example: Spontaneous Emission of a Photon

An excited atom decays to a lower energy level and emits a photon. Fermi’s Golden Rule yields the spontaneous emission rate:

\[
\Gamma = \frac{2\pi}{\hbar} |\langle f | \hat{d} \cdot \vec{E} | i \rangle|^2 \rho(\omega)
\]

Where \( \hat{d} \) is the dipole operator and \( \vec{E} \) is the electric field mode.

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10. Example: Particle Scattering in a Potential

A particle incident on a scattering center may be deflected. The potential acts as a perturbation \( \hat{H}’ = V(\vec{r}) \). Fermi’s Golden Rule computes the differential cross-section by integrating over all final momenta.

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11. Comparison with Classical Rates

Unlike classical physics, which may assume deterministic transitions, quantum rates depend on probability amplitudes and the interference of different quantum paths. Fermi’s Golden Rule quantifies these inherently probabilistic effects.

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12. Applications in Atomic, Nuclear, and Condensed Matter Physics

• Atomic transitions (optical and microwave frequencies).
• Nuclear decay rates.
• Semiconductor band transitions.
• Phonon emission and absorption in solids.
• Quantum tunneling rates and ionization processes.

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13. Limitations and Cautions

• Not valid for short times or strong perturbations.
• Assumes non-degenerate initial states unless modified.
• Energy conservation is enforced statistically over time, not instantaneously.
• Corrections needed for resonant or divergent behaviors.

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14. Variants and Extensions

• Golden Rule for degenerate states.
• Time-dependent generalizations for pulsed interactions.
• Extensions in many-body and field-theoretic contexts (e.g., Fermi’s Golden Rule in QED).

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15. Conclusion

Fermi’s Golden Rule is a cornerstone of quantum transition rate calculations. By linking microscopic interaction strengths with macroscopic transition probabilities, it serves as a bridge between quantum dynamics and observable decay and scattering phenomena. Mastery of this concept is essential for physicists across all specializations of quantum theory.

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Table of Contents

1. Introduction
2. Motivation and Relevance
3. General Setup of Time-Dependent Perturbation
4. The Interaction Picture
5. First-Order Time-Dependent Perturbation Theory
6. Transition Probabilities
7. Fermi’s Golden Rule
8. Periodic Perturbations and Resonance
9. Example: Two-Level Atom in an Oscillating Field
10. Sudden and Adiabatic Perturbations
11. Transition Amplitudes and Matrix Elements
12. Harmonic Perturbations and Selection Rules
13. Applications in Quantum Optics
14. Limitations and Cautions
15. Conclusion

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1. Introduction

Time-dependent perturbation theory is used in quantum mechanics to study how a quantum system evolves when its Hamiltonian changes with time. While time-independent perturbation theory handles stationary systems, many important physical phenomena involve time-varying interactions, such as electromagnetic fields or sudden kicks.

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2. Motivation and Relevance

This theory is crucial for understanding:

• Absorption and emission of light.
• Atomic transitions and decay processes.
• Response to external fields in quantum optics.
• Particle scattering and time-dependent forces.

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3. General Setup of Time-Dependent Perturbation

We split the Hamiltonian as:

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

where:

• \( \hat{H}_0 \): Time-independent part with known solutions.
• \( \hat{H}'(t) \): Time-dependent perturbation, assumed small.

We expand the wavefunction in the eigenbasis of \( \hat{H}_0 \):

\[
|\psi(t)\rangle = \sum_n c_n(t) e^{-i E_n t/\hbar} |n\rangle
\]

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4. The Interaction Picture

To simplify calculations, we use the interaction picture, where operators evolve like in the Heisenberg picture but states evolve under the perturbation:

\[
i\hbar \frac{d}{dt} |\psi_I(t)\rangle = \hat{H}’_I(t) |\psi_I(t)\rangle
\]

with:

\[
\hat{H}’_I(t) = e^{i\hat{H}_0 t/\hbar} \hat{H}'(t) e^{-i\hat{H}_0 t/\hbar}
\]

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5. First-Order Time-Dependent Perturbation Theory

We solve the differential equation perturbatively. The first-order transition amplitude from state \( |i\rangle \) to \( |f\rangle \) is:

\[
c_f^{(1)}(t) = \frac{1}{i\hbar} \int_{0}^{t} \langle f | \hat{H}'(t’) | i \rangle e^{i\omega_{fi} t’} dt’
\]

Where \( \omega_{fi} = (E_f – E_i)/\hbar \).

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6. Transition Probabilities

The probability of transitioning from \( |i\rangle \) to \( |f\rangle \) is:

\[
P_{i \rightarrow f}(t) = |c_f^{(1)}(t)|^2
\]

This gives us measurable quantities like decay rates and absorption strengths.

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7. Fermi’s Golden Rule

For perturbations that are constant in time or turned on suddenly, and observed over long times:

\[
P_{i \rightarrow f}(t) \approx \frac{2\pi}{\hbar} |\langle f | \hat{H}’ | i \rangle|^2 \rho(E_f)
\]

Here \( \rho(E_f) \) is the density of final states. This rule is crucial in particle and nuclear physics.

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8. Periodic Perturbations and Resonance

For oscillatory perturbations:

\[
\hat{H}'(t) = \hat{V} \cos(\omega t)
\]

Transitions are enhanced (resonant) when:

\[
\hbar \omega = E_f – E_i
\]

This underpins spectroscopy, lasers, and resonance phenomena in quantum systems.

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9. Example: Two-Level Atom in an Oscillating Field

A system with two states \( |g\rangle \) and \( |e\rangle \), interacting with an external electric field:

\[
\hat{H}'(t) = \hbar \Omega \cos(\omega t)(|g\rangle \langle e| + |e\rangle \langle g|)
\]

Time-dependent perturbation theory predicts Rabi oscillations and transition probabilities dependent on detuning and field strength.

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10. Sudden and Adiabatic Perturbations

• Sudden: Perturbation changes faster than the system’s natural time scale. The wavefunction remains unchanged but expands in the new eigenbasis.
• Adiabatic: Perturbation changes slowly, and the system adapts continuously, staying in the instantaneous eigenstate.

These are limits used in both analytic calculations and quantum control.

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11. Transition Amplitudes and Matrix Elements

The core quantity is:

\[
\langle f | \hat{H}'(t) | i \rangle
\]

which determines how “connected” the initial and final states are. Strong matrix elements lead to high transition rates.

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12. Harmonic Perturbations and Selection Rules

Perturbations like:

\[
\hat{H}'(t) = q E_0 x \cos(\omega t)
\]

lead to dipole transitions. The selection rules are derived from symmetry and parity considerations. E.g., \( \Delta \ell = \pm 1 \), \( \Delta m = 0, \pm 1 \) in atoms.

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13. Applications in Quantum Optics

• Stimulated emission and absorption.
• Laser physics and coherent control.
• Photon-atom interaction in cavity QED.
• Nuclear magnetic resonance (NMR) and electron spin resonance (ESR).

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14. Limitations and Cautions

• Assumes perturbation is small compared to level spacing.
• Higher-order effects can be significant in strong fields.
• Breakdown near level crossings and avoided crossings.
• Doesn’t account for dissipation or decoherence unless extended.

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15. Conclusion

Time-dependent perturbation theory is a foundational tool in quantum mechanics that allows physicists to analyze transitions between quantum states under external influences. From understanding atomic spectra to designing quantum control protocols, it plays a central role in both theoretical and experimental quantum science.

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