Categories: Physics, Quantum mechanics.

Amplitude rate equations

In quantum mechanics, the amplitude rate equations give the evolution of a quantum state in a time-varying potential. Although best known as the precursors of time-dependent perturbation theory, by themselves they are exact and widely applicable.

Let H^0\hat{H}_0 be the time-independent part of the total Hamiltonian, and H^1\hat{H}_1 the time-varying part (whose contribution need not be small), so H^(t)=H^0+H^1(t)\hat{H}(t) = \hat{H}_0 + \hat{H}_1(t).

Suppose that the time-independent problem H^0∣n⟩=En∣n⟩\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)⟩=∑ncn∣n⟩e−iEnt/ℏ\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)∣n⟩e−iEnt/ℏ\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^0∣Ψ(t)⟩+H^1∣Ψ(t)⟩−iℏddt∣Ψ(t)⟩=∑n(cnH^0∣n⟩+cnH^1∣n⟩−cnEn∣n⟩−iℏdcndt∣n⟩)e−iEnt/ℏ=∑n(cnH^1∣n⟩−iℏdcndt∣n⟩)e−iEnt/ℏ\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(cn⟨m∣H^1∣n⟩−iℏdcndt⟨m∣n⟩)e−iEnt/ℏ\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:

iℏdcmdte−iEmt/ℏ=∑ncn⟨m∣H^1∣n⟩e−iEnt/ℏ\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≡(Em−En)/ℏ\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}:

iℏdcmdt=∑ncn(t)⟨m∣H^1(t)∣n⟩eiωmnt\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.