First-Order Transition Probability

First-order time-dependent perturbation theory computes transition amplitudes produced by a weak time-dependent interaction. It is the entry point for resonance, absorption, emission, decay rates, and Fermi’s golden rule.

Split the Hamiltonian as

$$H(t)=H_0+V(t),$$

where $H_0$ is exactly solved:

$$H_0\lvert n\rangle = E_n\lvert n\rangle.$$

The goal is to estimate the amplitude for a system initially in $\lvert i\rangle$ to be found in $\lvert f\rangle$ at time $t$.

Interaction-Picture Setup

In the interaction picture, the state obeys

$$i\hbar\frac{d}{dt}\lvert\psi_I(t)\rangle = V_I(t)\lvert\psi_I(t)\rangle,$$

where

$$V_I(t) = e^{iH_0t/\hbar} V(t) e^{-iH_0t/\hbar}.$$

The first-order Dyson expansion gives

$$\lvert\psi_I(t)\rangle \approx \left[ I-\frac{i}{\hbar} \int_{t_0}^{t}dt'\,V_I(t') \right] \lvert i\rangle.$$

Thus the first-order transition amplitude is

$$c_f^{(1)}(t) = -\frac{i}{\hbar} \int_{t_0}^{t}dt'\, \langle f|V_I(t')|i\rangle.$$

This is the central formula.

Phase Factors

If $V(t)$ is written in the Schrödinger picture and $\lvert i\rangle,\lvert f\rangle$ are eigenstates of $H_0$, then

$$\langle f|V_I(t)|i\rangle = e^{i\omega_{fi}t} \langle f|V(t)|i\rangle,$$

with

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

The transition amplitude is therefore a time integral of the perturbation matrix element multiplied by a phase. Transitions are enhanced when the time dependence of $V(t)$ compensates this phase.

Constant Perturbation Switched On for a Finite Time

As a diagnostic example, suppose $V(t)=V$ is constant from $0$ to $T$ and zero outside that interval. Then

$$c_f^{(1)}(T) = -\frac{i}{\hbar}V_{fi} \int_0^Tdt\,e^{i\omega_{fi}t}.$$

The integral gives

$$c_f^{(1)}(T) = -\frac{i}{\hbar}V_{fi}\, e^{i\omega_{fi}T/2} \frac{2\sin(\omega_{fi}T/2)}{\omega_{fi}}.$$

The leading transition probability is

$$P_{i\to f}^{(1)}(T) = \frac{4\lvert V_{fi}\rvert^2}{\hbar^2} \frac{\sin^2(\omega_{fi}T/2)} {\omega_{fi}^2}.$$

This expression has a peak near $\omega_{fi}=0$, whose width is of order $1/T$.

Harmonic Driving and Resonance

If the perturbation contains a harmonic factor such as

$$V(t)=W e^{-i\Omega t}+W^\dagger e^{i\Omega t},$$

then the amplitude contains phase factors like

$$e^{i(\omega_{fi}-\Omega)t} \quad\text{and}\quad e^{i(\omega_{fi}+\Omega)t}.$$

The first term is resonant when

$$\Omega\approx\omega_{fi}.$$

This is the basic mechanism behind absorption and stimulated emission in weak-drive calculations.

Probability and Its Limits

The first-order probability is

$$P_{i\to f} \approx \lvert c_f^{(1)}(t)\rvert^2.$$

This is a leading-order probability, not an all-time exact probability. It is reliable only while the transition probability remains small and higher-order amplitudes remain negligible.

For long times near resonance in a two-level system, first-order perturbation theory eventually fails because probability would otherwise grow beyond its allowed range. Nonperturbative two-level dynamics or a rate description may then be needed.

Short-Time Behavior

For very short times, the amplitude is approximately linear in $T$:

$$c_f^{(1)}(T) \approx -\frac{i}{\hbar}V_{fi}T.$$

Thus the probability starts quadratically:

$$P_{i\to f}^{(1)}(T) \approx \frac{\lvert V_{fi}\rvert^2T^2}{\hbar^2}.$$

This short-time quadratic behavior is different from the linear-in-time behavior of a golden-rule rate, which emerges only after additional continuum and time-scale assumptions.

Discrete Final States Versus Continua

For a single discrete final state, the finite-time probability oscillates or grows only within the limits of perturbation theory. For a continuum of final states, the narrow sinc-squared peak can be integrated against a density of states. That long-time continuum limit produces Fermi’s golden rule.

The difference between a probability and a rate is therefore physical. It is not just notation.

Common Mistakes

• Forgetting the interaction-picture phase $e^{i\omega_{fi}t}$.
• Squaring the amplitude and calling the result exact.
• Using first-order theory after the transition probability is no longer small.
• Confusing short-time quadratic growth with golden-rule linear growth.
• Applying a continuum rate formula to a single isolated final state.

Cross-Links

• Time-Dependent Perturbation Theory and Transitions
• Interaction Picture for Perturbation Theory
• Dyson Expansion for Transition Amplitudes
• Resonant Driving
• Rabi Formula in the Weak-Drive Limit
• Interaction Picture
• Time-Dependent Hamiltonians
• Small Parameters and Error Estimates
• Fermi’s Golden Rule
• Transition Rates applies the finite-time and continuum distinction to spectroscopy channels, degeneracies, cross sections, and measured signals.
• Unitary Time Evolution
• Two-Level System

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

1. Derive the finite-time probability for a constant perturbation switched on from $0$ to $T$.

Solution

Start from

$$c_f^{(1)}(T) = -\frac{i}{\hbar}V_{fi} \int_0^Tdt\,e^{i\omega_{fi}t}.$$

The integral is

$$\int_0^Tdt\,e^{i\omega t} = e^{i\omega T/2} \frac{2\sin(\omega T/2)}{\omega}.$$

Taking the magnitude squared gives

$$P_{i\to f}^{(1)}(T) = \frac{4\lvert V_{fi}\rvert^2}{\hbar^2} \frac{\sin^2(\omega_{fi}T/2)} {\omega_{fi}^2}.$$

2. Why does first-order perturbation theory fail if the leading probability becomes comparable to $1$?

Solution

The first-order probability keeps only $\lvert c_f^{(1)}\rvert^2$ and ignores higher-order amplitudes, depletion of the initial state, and interference among orders. If the transition probability is order $1$, those effects are no longer negligible. The perturbative expansion is no longer self-consistent.