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:

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

We calculate transitions between distinct normalized states $f\ne i$ in an orthonormal eigenbasis of $H_0$. The perturbation $V$ includes its physical coupling strength; a “first-order probability” below means the square of a first-order amplitude, hence second order in that strength.

Required background. Interaction Picture fixes the phase convention, and Dyson Transition Amplitudes explains amplitude versus probability order.

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

$$V_I(t) = e^{iH_0(t-t_0)/\hbar} V(t) e^{-iH_0(t-t_0)/\hbar}.$$

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

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

For $f=i$, this is only a correction to the zeroth-order survival amplitude $1$; squaring the correction alone does not give survival probability.

Phase Factors

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

$$\langle f|V_I(t)|i\rangle = e^{i\omega_{fi}(t-t_0)} \langle f|V(t)|i\rangle,$$

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. A necessary depletion check is $\sum_{f\ne i}\lvert c_f^{(1)}\rvert^2\ll1$, with continuum integrals when needed. Small probabilities alone are not sufficient: accumulated energy-shift phases and higher-order paths must also be negligible for the observable and time interval being predicted.

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. An isolated coherent pair then requires Rabi dynamics. A rate description additionally requires a continuum or a specified loss of coherence; it does not follow just from failure of first order.

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. When the spectral weight is smooth over the finite-time peak and depletion remains small, this produces Fermi’s Golden Rule. The required interval is long compared with the reservoir correlation time but short compared with the depletion time.

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

Exercises

Many individually weak channels

At short times a constant perturbation connects the initial state to $N$ orthogonal states with equal matrix-element magnitude $v$. Find the leading total transition probability. Why can checking one channel fail?

Solution

Each probability is $v^2T^2/\hbar^2$, so their sum is $Nv^2T^2/\hbar^2$. The necessary short-time criterion is $\sqrt N\,vT/\hbar\ll1$. Completeness gives the general expression

$$\sum_{f\ne i}P_{i\to f}(T) =\frac{T^2}{\hbar^2} \left(\langle i|V^2|i\rangle-\langle i|V|i\rangle^2\right) +o(T^2).$$

This assumes $|i\rangle$ has finite variance of $V$. Since $|i\rangle$ is an $H_0$ eigenstate, this is also the variance of the full constant Hamiltonian in the initial state. It agrees with the exact short-time survival expansion.

A finite resonance limit

Take $\omega_{fi}\to0$ in the rectangular-pulse formula. At what time scale must the leading prediction for this channel cease to be reliable?

Solution

Using $\sin x/x\to1$ gives $P_{i\to f}\simeq |V_{fi}|^2T^2/\hbar^2$, with no divergence at fixed $T$. At $T\sim\hbar/|V_{fi}|$ depletion and repeated transitions matter. The exact two-state resonant result is $\sin^2(|V_{fi}|T/\hbar)$ after subtracting the common diagonal energy.

References

• 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.
• 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.