Time-Dependent Perturbation Theory and Fermi’s Golden Rule
Author
Daniel Fischer
Time-Dependent Perturbation Theory: Overview and Plan
In time-independent perturbation theory, we ask how a weak perturbation changes the stationary energies and eigenstates of a quantum system. A time-dependent perturbation raises a different question:
How can an interaction drive a quantum system from one stationary state into another?
This is the situation encountered whenever an atom interacts with an oscillating electromagnetic field, but the same formalism applies much more generally.
We write the Hamiltonian as
\[
\hat H(t)=\hat H_0+\hat V(t),
\]
where the eigenstates and eigenenergies of the unperturbed Hamiltonian are assumed to be known,
\[
\hat H_0\left\lvert n \right\rangle=E_n\left\lvert n \right\rangle.
\]
The purpose of time-dependent perturbation theory is to determine how a weak perturbation \(\hat V(t)\) changes the probability amplitudes of these states.
The specific goals for this chapter are:
Expand a time-dependent quantum state in the stationary eigenstates of an unperturbed Hamiltonian.
Derive the equations that describe how a time-dependent perturbation changes the expansion coefficients.
Obtain the first-order transition amplitude between two states.
Understand why an oscillating perturbation produces resonant transitions.
Derive Fermi’s Golden Rule and interpret the roles of the transition matrix element, energy conservation, and the density of final states.
Establish the framework that will later be used to describe radiative transitions and electric-dipole matrix elements.
ImportantCore Knowledge — Know
For
\[
\hat H(t)=\hat H_0+\hat V(t),
\qquad
\hat H_0\left\lvert n \right\rangle=E_n\left\lvert n \right\rangle,
\]
the state can be expanded as
\[
\left\lvert \Psi(t) \right\rangle=
\sum_n c_n(t)e^{-iE_n t/\hbar}\left\lvert n \right\rangle.
\]
If the system starts in state \(\left\lvert i \right\rangle\), the first-order amplitude for finding it in a different state \(\left\lvert f \right\rangle\) is
\[
\boxed{
c_f^{(1)}(t)
=
\frac{1}{i\hbar}
\int_0^t
\left\langle f \middle| \hat V(t') \middle| i \right\rangle\,
e^{i\omega_{fi}t'}\,dt'
}
\]
where
\[
\omega_{fi}
=
\frac{E_f-E_i}{\hbar}.
\]
For a weak perturbation that drives transitions into a dense set of final states, the long-time transition rate is given by Fermi’s Golden Rule,
\[
\boxed{
\Gamma_{i\rightarrow f}
=
\frac{2\pi}{\hbar}
\left|
\left\langle f \middle| \hat W \middle| i \right\rangle\right|^2
\rho(E_f),
}
\]
evaluated at the energy allowed by the perturbation. For absorption from a harmonic perturbation of angular frequency \(\omega\),
\[
\boxed{
E_f=E_i+\hbar\omega.
}
\]
NoteWorking Knowledge — Understand
You should understand that
a time-dependent perturbation changes the probability amplitudes of the stationary states of the unperturbed Hamiltonian;
the matrix element \(\left\langle f \middle| \hat V \middle| i \right\rangle\) measures how strongly the perturbation couples the initial and final states;
an oscillating perturbation produces a finite-width resonance at finite interaction time, which becomes sharply concentrated around energy conservation for long interaction times;
Fermi’s Golden Rule describes a transition rate, not merely a transition probability;
the transition rate depends both on the strength of the coupling and on the number of final states available near the energy-conserving value;
for a single isolated pair of discrete states under coherent continuous driving, the Golden Rule is not the complete long-time description; coherent dynamics such as Rabi oscillations can occur.
The detailed algebra leading from the time-dependent Schrödinger equation to the coefficient equations and from the finite-time transition probability to Fermi’s Golden Rule is included below to show where the results come from, but the intermediate steps are not intended as material to memorize.
1. The Basic Problem
Suppose that the Hamiltonian consists of a solvable time-independent part and a weak time-dependent perturbation,
\[
\boxed{
\hat H(t)
=
\hat H_0+\hat V(t).
}
\]
The unperturbed stationary states satisfy
\[
\boxed{
\hat H_0\left\lvert n \right\rangle=
E_n\left\lvert n \right\rangle.
}
\]
If the perturbation were absent, a state that starts in \(\left\lvert n \right\rangle\) would evolve only by a phase factor,
\[
\left\lvert \Psi_n(t) \right\rangle=
e^{-iE_n t/\hbar}\left\lvert n \right\rangle.
\]
A time-dependent perturbation can instead transfer amplitude between different unperturbed states. The central question is therefore:
If the system begins in a state \(\left\lvert i \right\rangle\), what is the probability that the perturbation drives it into another state \(\left\lvert f \right\rangle\)?
This is qualitatively different from time-independent perturbation theory. There we determine how stationary eigenstates and energies are modified. Here we are interested primarily in transitions between states.
2. Expansion in the Unperturbed Basis
Because the eigenstates of \(\hat H_0\) form a basis, we write the general time-dependent state as
Two ingredients determine whether a transition is likely:
the matrix element\[
\left\langle f \middle| \hat V(t) \middle| i \right\rangle,
\] which tells us how strongly the perturbation couples the two states;
the oscillating phase \[
e^{i\omega_{fi}t},
\] which determines how contributions at different times add together.
4. An Oscillating Perturbation and Resonance
Many important perturbations are periodic. For example, interaction with monochromatic light produces terms oscillating with the field frequency.
This quantity is often called the Bohr frequency for the transition \(i\rightarrow f\).
The perturbation itself oscillates with angular frequency \(\omega\). What matters for the transition is how closely this frequency matches the transition frequency. We therefore define the detuning
\[
\boxed{
\Delta
=
\omega_{fi}-\omega.
}
\]
In terms of the detuning, the first-order transition probability is
For a finite interaction time, the resonance is not infinitely narrow. The function
\[
\frac{\sin^2(\Delta t/2)}
{(\Delta/2)^2}
\]
has a central peak around \(\Delta=0\) whose characteristic width decreases as the interaction time increases.
Schematically,
\[
\boxed{
\Delta\omega
\sim
\frac{1}{t}.
}
\]
A shorter interaction therefore permits a broader range of transition frequencies, while a long interaction produces a narrow resonance.
This finite-time result is important: exact energy conservation emerges as the long-time limit of the transition probability rather than being imposed by hand.
The dependence on detuning can be explored in the interactive figure below. Move the slider to change the interaction time \(t\).
Figure 1: Finite-time resonance factor as a function of detuning. Increasing the interaction time makes the resonance taller and narrower.
As the interaction time increases,
the central peak at \(\Delta=0\) becomes taller;
the resonance becomes narrower, with a characteristic width of order
\[
\Delta\omega\sim\frac{1}{t}.
\]
Thus a short interaction allows a relatively broad range of transition frequencies, whereas a long interaction selects frequencies increasingly close to the resonance condition.
Notice that the peak height at exact resonance grows as \(t^2\). At the same time, its width decreases as \(1/t\). Consequently, the area under the resonance grows proportionally to \(t\). This is the behavior that leads to a constant transition rate when we sum over a dense set of final states.
NoteThe second oscillating term
The second term in the perturbation,
\[
\hat W^\dagger e^{+i\omega t},
\]
produces an integral containing
\[
e^{i(\omega_{fi}+\omega)t}.
\]
Its resonance condition is therefore
\[
\omega_{fi}+\omega=0,
\]
or
\[
\boxed{
E_f=E_i-\hbar\omega.
}
\]
In the atom–radiation problem, the two signs correspond to processes in which the atomic energy increases or decreases by one photon energy \(\hbar\omega\).
5. From Transition Probability to Fermi’s Golden Rule
For a single discrete final state, the finite-time probability above is the appropriate first-order result.
In many applications, however, the system can make a transition into a large number of closely spaced final states. Examples include ionization into a continuum, scattering processes, and emission into the many available modes of the electromagnetic field.
Let
\[
\rho(E_f)
\]
be the density of final states, defined so that
\[
\rho(E_f)\,dE_f
\]
is the number of available final states in an energy interval \(dE_f\).
The total transition probability is obtained by summing over the possible final states. For a dense spectrum, the sum becomes an integral,
The transition probability then grows linearly with time,
\[
P(t)\simeq \Gamma t,
\]
and the coefficient of \(t\) defines a transition rate.
5.2 Fermi’s Golden Rule
Carrying out the integral over final states gives
TipFermi’s Golden Rule
\[
\boxed{
\Gamma_{i\rightarrow f}
=
\frac{2\pi}{\hbar}
\left|
\left\langle f \middle| \hat W \middle| i \right\rangle\right|^2
\rho(E_f),
}
\]
with the final-state energy fixed by energy conservation. For absorption from a harmonic perturbation,
\[
\boxed{
E_f=E_i+\hbar\omega.
}
\]
Equivalently, before performing the final-state energy integral, the rate can be written as
\[
\boxed{
d\Gamma
=
\frac{2\pi}{\hbar}
\left|
\left\langle f \middle| \hat W \middle| i \right\rangle\right|^2
\delta(E_f-E_i-\hbar\omega)
\,dN_f,
}
\]
where \(dN_f\) represents the available final states.
The Golden Rule therefore contains three pieces of physics:
Coupling strength\[
\left|
\left\langle f \middle| \hat W \middle| i \right\rangle\right|^2
\] tells us how strongly the perturbation connects the two states.
Energy conservation\[
E_f=E_i+\hbar\omega
\] selects the resonant final states.
Availability of final states\[
\rho(E_f)
\] tells us how many states are available at the required energy.
A large matrix element does not by itself guarantee a large rate: there must also be accessible final states at the appropriate energy.
For sufficiently long times, the resonance is sharply localized near
\[
E_f=E_i+\hbar\omega.
\]
If \(\rho(E_f)\) and \(|W_{fi}|^2\) vary slowly across this narrow resonance, they can be evaluated at the resonant energy and taken outside the integral.
Fermi’s Golden Rule is extremely useful, but it is an approximation.
The assumptions made above include:
the perturbation is sufficiently weak that first-order perturbation theory remains valid;
the system remains predominantly in the initial state during the time interval considered;
the final states form a dense spectrum or continuum, so that a sum over states can be replaced by an integral;
the matrix element and density of states do not vary strongly across the narrow resonance;
the interaction time is long enough that a transition rate is meaningful.
ImportantA discrete two-level system is different
For one isolated final state, the delta-function form of Fermi’s Golden Rule should not be interpreted as a literal infinite transition rate at exact resonance.
which eventually becomes inconsistent with the requirement that a probability cannot exceed one.
The problem is not quantum mechanics; it is the breakdown of first-order perturbation theory.
A coherently driven isolated two-level system instead exhibits Rabi oscillations, which will be discussed later in the section on laser–atom interactions.
7. Connection to Atomic Transitions
The formalism developed here is general. To apply it to absorption or emission of radiation, we must identify the perturbation that couples an atom to the electromagnetic field.
For electric-dipole transitions, this interaction can be expressed in terms of the atomic dipole operator and the electric field. The transition rate then contains a matrix element of the form
\[
\left\langle f \middle|
\hat{\vec d}\cdot\vec E
\middle| i \right\rangle,
\]
or, equivalently, a dipole matrix element involving the position operator,
\[
\left\langle f \middle| \hat{\vec r} \middle| i \right\rangle.
\]
Fermi’s Golden Rule then tells us that the transition strength is proportional to the absolute square of this matrix element.
The detailed connection between the atom–field Hamiltonian, the electric-dipole approximation, and the Einstein coefficients is developed later in Dipole Matrix Elements.
Summary
Time-dependent perturbation theory describes transitions caused by a weak time-dependent interaction.
The main first-order result is
\[
\boxed{
c_f^{(1)}(t)
=
\frac{1}{i\hbar}
\int_0^t
\left\langle f \middle| \hat V(t') \middle| i \right\rangle\,
e^{i\omega_{fi}t'}\,dt'.
}
\]
For an oscillating perturbation, the transition probability is resonantly enhanced when the perturbation frequency matches the energy separation between the states.
When transitions occur into a dense set of final states, the long-time limit leads to Fermi’s Golden Rule,
\[
\boxed{
\Gamma
=
\frac{2\pi}{\hbar}
\left|
\left\langle f \middle| \hat W \middle| i \right\rangle\right|^2
\rho(E_f),
}
\]