Statement of the problem

The golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation H' of the Hamiltonian. The total Hamiltonian is the sum of an “original” Hamiltonian H₀ and a perturbation: ${\displaystyle H=H_{0}+H'(t)}$. In the interaction picture, we can expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system ${\displaystyle |n\rangle }$, with ${\displaystyle H_{0}|n\rangle =E_{n}|n\rangle }$.

Discrete spectrum of final states

We first consider the case where the final states are discrete. The expansion of a state in the perturbed system at a time t is ${\textstyle |\psi (t)\rangle =\sum _{n}a_{n}(t)e^{-iE_{n}t/\hbar }|n\rangle }$. The coefficients a_n(t) are yet unknown functions of time yielding the probability amplitudes in the Dirac picture. This state obeys the time-dependent Schrödinger equation:

$${\displaystyle H|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle .}$$

Expanding the Hamiltonian and the state, we see that, to first order,

$${\displaystyle \left(H_{0}+H'-\mathrm {i} \hbar {\frac {\partial }{\partial t}}\right)\sum _{n}a_{n}(t)|n\rangle e^{-\mathrm {i} tE_{n}/\hbar }=0,}$$

where E_n and |n⟩ are the stationary eigenvalues and eigenfunctions of H₀.

This equation can be rewritten as a system of differential equations specifying the time evolution of the coefficients ${\displaystyle a_{n}(t)}$:

$${\displaystyle \mathrm {i} \hbar {\frac {da_{k}(t)}{dt}}=\sum _{n}\langle k|H'|n\rangle a_{n}(t)e^{\mathrm {i} t(E_{k}-E_{n})/\hbar }.}$$

This equation is exact, but normally cannot be solved in practice.

For a weak constant perturbation H' that turns on at t = 0, we can use perturbation theory. Namely, if ${\displaystyle H'=0}$, it is evident that ${\displaystyle a_{n}(t)=\delta _{n,i}}$, which simply says that the system stays in the initial state ${\displaystyle i}$.

For states ${\displaystyle k\neq i}$, ${\displaystyle a_{k}(t)}$ becomes non-zero due to ${\displaystyle H'\neq 0}$, and these are assumed to be small due to the weak perturbation. The coefficient ${\displaystyle a_{i}(t)}$ which is unity in the unperturbed state, will have a weak contribution from ${\displaystyle H'}$. Hence, one can plug in the zeroth-order form ${\displaystyle a_{n}(t)=\delta _{n,i}}$ into the above equation to get the first correction for the amplitudes ${\displaystyle a_{k}(t)}$:

$${\displaystyle \mathrm {i} \hbar {\frac {da_{k}(t)}{dt}}=\langle k|H'|i\rangle e^{\mathrm {i} t(E_{k}-E_{i})/\hbar },}$$

whose integral can be expressed as

$${\displaystyle \mathrm {i} \hbar a_{k}(t)=\int _{0}^{t}\langle k|H'(t')|i\rangle e^{\mathrm {i} \omega _{ki}t'}dt'}$$

with ${\displaystyle \omega _{ki}\equiv (E_{k}-E_{i})/\hbar }$, for a state with a_i(0) = 1, a_k(0) = 0, transitioning to a state with a_k(t).

The probability of transition from the initial state (ith) to the final state (fth) is given by

$${\displaystyle w_{fi}=|a_{f}(t)|^{2}={\frac {1}{\hbar ^{2}}}\left|\int _{0}^{t}\langle f|H'(t')|i\rangle e^{\mathrm {i} \omega _{fi}t'}dt'\right|^{2}}$$

It is important to study a periodic perturbation with a given frequency ${\displaystyle \omega }$ since arbitrary perturbations can be constructed from periodic perturbations of different frequencies. Since ${\displaystyle H'(t)}$ must be Hermitian, we must assume ${\displaystyle H'(t)=Fe^{-\mathrm {i} \omega t}+F^{\dagger }e^{\mathrm {i} \omega t}}$, where ${\displaystyle F}$ is a time independent operator. The solution for this case is^[7]

$${\displaystyle a_{f}(t)=-\langle f|F|i\rangle {\frac {e^{\mathrm {i} (\omega _{fi}-\omega )t}}{\hbar (\omega _{fi}-\omega )}}-\langle f|F^{\dagger }|i\rangle {\frac {e^{\mathrm {i} (\omega _{fi}+\omega )t}}{\hbar (\omega _{fi}+\omega )}}.}$$

This expression is valid only when the denominators in the above expression is non-zero, i.e., for a given initial state with energy ${\displaystyle E_{i}}$, the final state energy must be such that ${\displaystyle E_{f}-E_{i}\neq \pm \hbar \omega .}$ Not only the denominators must be non-zero, but also must not be small since ${\displaystyle a_{f}}$ is supposed to be small.

Consider now the case where the perturbation frequency is such that ${\displaystyle E_{k}-E_{n}=\hbar (\omega +\varepsilon )}$ where ${\displaystyle \varepsilon }$ is a small quantity. Unlike the previous case, not all terms in the sum over ${\displaystyle n}$ in the above exact equation for ${\displaystyle a_{k}(t)}$ matters, but depends only on ${\displaystyle a_{n}(t)}$ and vice versa. Thus, omitting all other terms, we can write

$${\displaystyle i\hbar {\frac {da_{k}}{dt}}=\langle k|F|n\rangle e^{i\varepsilon t}a_{n},\quad i\hbar {\frac {da_{n}}{dt}}=\langle n|F^{\dagger }|k\rangle e^{-i\varepsilon t}a_{k}.}$$

The two independent solutions are

$${\displaystyle a_{n}=Ae^{i\alpha _{1}t},\,a_{k}=-A\hbar \alpha _{1}e^{i\alpha _{1}t}/\langle n|F^{\dagger }|k\rangle }$$

$${\displaystyle a_{n}=Be^{-i\alpha _{2}t},\,a_{k}=B\hbar \alpha _{2}e^{-i\alpha _{2}t}/\langle n|F^{\dagger }|k\rangle }$$

where

$${\displaystyle \alpha _{1}=-{\frac {1}{2}}\varepsilon +\Omega ,\quad \alpha _{2}={\frac {1}{2}}\varepsilon +\Omega ,\quad \Omega ={\sqrt {{\frac {1}{4}}\varepsilon ^{2}+|\eta |^{2}}},\quad \eta ={\frac {1}{\hbar }}\langle k|F|n\rangle }$$

and the constants ${\displaystyle A}$ and ${\displaystyle B}$ are fixed by the normalization condition.

If the system at ${\displaystyle t=0}$ is in the ${\displaystyle |\psi _{k}\rangle }$ state, then the probability of finding the system in the ${\displaystyle |\psi _{n}\rangle }$ state is given by

$${\displaystyle w_{kn}={\frac {|\eta |^{2}}{2\Omega ^{2}}}(1-\cos 2\Omega t)}$$

which is a periodic function with frequency ${\displaystyle 2\Omega }$; this function varies between ${\displaystyle 0}$ and ${\displaystyle |\eta |^{2}/\Omega ^{2}}$. At the exact resonance, i.e., ${\displaystyle \varepsilon =0}$, the above formula reduces to

$${\displaystyle w_{kn}={\frac {1}{2}}(1-\cos 2|\eta |t)}$$

which varies periodically between ${\displaystyle 0}$ and ${\displaystyle 1}$, that is to say, the system periodically switches from one state to the other. The situation is different if the final states are in the continuous spectrum.

Continuous spectrum of final states

Since the continuous spectrum lies above the discrete spectrum, ${\displaystyle E_{f}-E_{i}>0}$ and it is clear from the previous section, major role is played by the energies ${\displaystyle E_{f}}$ lying near the resonance energy ${\displaystyle E_{i}+\hbar \omega }$, i.e., when ${\displaystyle \omega _{fi}\approx \omega }$. In this case, it is sufficient to keep only the first term for ${\displaystyle a_{f}(t)}$. Assuming that perturbations are turned on from time ${\displaystyle t=0}$, we have then

$${\displaystyle a_{f}(t)=-{\frac {\mathrm {i} }{\hbar }}\int _{0}^{t}\langle f|H'(t')|i\rangle e^{\mathrm {i} \omega _{fi}t'}dt'=-\langle f|F|i\rangle {\frac {e^{\mathrm {i} (\omega _{fi}-\omega )t}-1}{\hbar (\omega _{fi}-\omega )}}}$$

The squared modulus of ${\displaystyle a_{f}}$ is

$${\displaystyle |a_{f}|^{2}=4|\langle f|F|i\rangle |^{2}{\frac {\sin ^{2}((\omega _{fi}-\omega )t/2)}{\hbar ^{2}(\omega _{fi}-\omega )^{2}}}}$$

Therefore, the transition probability per unit time, for large t, is given by

$${\displaystyle {\frac {dw_{fi}}{dt}}={\frac {d}{dt}}|a_{f}|^{2}={\frac {2\pi }{\hbar }}|\langle f|F|i\rangle |^{2}\delta (E_{f}-E_{i}-\hbar \omega )}$$

Note that the delta function in the expression above arises due to the following argument. Defining ${\displaystyle \Delta =\omega _{fi}-\omega }$ the time derivative of ${\displaystyle \sin ^{2}(\Delta t/2)/\Delta ^{2}}$ is ${\displaystyle \sin(\Delta t)/(2\Delta )}$, which behaves like a delta function at large t (for more information, please see Sinc function#Relationship to the Dirac delta distribution).

The constant decay rate of the golden rule follows.^[8] As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the a_k(t) terms invalidates lowest-order perturbation theory, which requires a_k ≪ a_i.)