Categories: Physics, Quantum mechanics.

Amplitude rate equations

In quantum mechanics, the amplitude rate equations give the evolution of a quantum state’s superposition coefficients through time. They are known as the precursors for time-dependent perturbation theory, but by themselves they are exact and widely applicable.

Let H^0\hat{H}_0 be a “simple” time-independent part of the full Hamiltonian, and H^1\hat{H}_1 a time-varying other part, whose contribution need not be small:

H^(t)=H^0+H^1(t)\begin{aligned} \hat{H}(t) = \hat{H}_0 + \hat{H}_1(t) \end{aligned}

We assume that the time-independent problem H^0n=Enn\hat{H}_0 \Ket{n} = E_n \Ket{n} has already been solved, such that its general solution is a superposition as follows:

Ψ0(t)=ncnneiEnt/\begin{aligned} \Ket{\Psi_0(t)} = \sum_{n} c_n \Ket{n} e^{- i E_n t / \hbar} \end{aligned}

Since these n\Ket{n} form a complete basis, the full solution for H^0+H^1\hat{H}_0 + \hat{H}_1 can be written in the same form, but now with time-dependent coefficients cn(t)c_n(t):

Ψ(t)=ncn(t)neiEnt/\begin{aligned} \Ket{\Psi(t)} = \sum_{n} c_n(t) \Ket{n} e^{- i E_n t / \hbar} \end{aligned}

We put this ansatz into the full Schrödinger equation, and use the known solution for H^0\hat{H}_0:

0=H^0Ψ(t)+H^1Ψ(t)iddtΨ(t)=n(cnH^0n+cnH^1ncnEnnidcndtn)eiEnt/=n(cnH^1nidcndtn)eiEnt/\begin{aligned} 0 &= \hat{H}_0 \Ket{\Psi(t)} + \hat{H}_1 \Ket{\Psi(t)} - i \hbar \dv{}{t}\Ket{\Psi(t)} \\ &= \sum_{n} \Big( c_n \hat{H}_0 \Ket{n} + c_n \hat{H}_1 \Ket{n} - c_n E_n \Ket{n} - i \hbar \dv{c_n}{t} \Ket{n} \Big) e^{- i E_n t / \hbar} \\ &= \sum_{n} \Big( c_n \hat{H}_1 \Ket{n} - i \hbar \dv{c_n}{t} \Ket{n} \Big) e^{- i E_n t / \hbar} \end{aligned}

We then take the inner product with an arbitrary stationary basis state m\Ket{m}:

0=n(cnmH^1nidcndtmn)eiEnt/\begin{aligned} 0 &= \sum_{n} \Big( c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \dv{c_n}{t} \inprod{m}{n} \Big) e^{- i E_n t / \hbar} \end{aligned}

Thanks to orthonormality, this moves the latter term outside the summation:

idcmdteiEmt/=ncnmH^1neiEnt/\begin{aligned} i \hbar \dv{c_m}{t} e^{- i E_m t / \hbar} &= \sum_{n} c_n \matrixel{m}{\hat{H}_1}{n} e^{- i E_n t / \hbar} \end{aligned}

We divide by the left-hand exponential and define ωmn(EmEn)/\omega_{mn} \equiv (E_m - E_n) / \hbar to arrive at the desired set of amplitude rate equations, one for each basis state m\ket{m}:

idcmdt=ncn(t)mH^1(t)neiωmnt\begin{aligned} \boxed{ i \hbar \dv{c_m}{t} = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} e^{i \omega_{mn} t} } \end{aligned}

We have not made any approximations, so it is possible to exactly solve for cn(t)c_n(t) in some simple systems. This is worth pointing out, because these equations’ most famous uses are for deriving time-dependent-perturbation theory (by making a truncated power series approximation) and Rabi oscillation (by making the rotating wave approximation).

References

  1. D.J. Griffiths, D.F. Schroeter, Introduction to quantum mechanics, 3rd edition, Cambridge.

© 2023 Marcus R.A. Newman, CC BY-NC-SA 4.0.
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