Categories: Perturbation, Physics, Quantum mechanics.

Time-dependent perturbation theory

In quantum mechanics, time-dependent perturbation theory exists to deal with time-varying perturbations to the Schrödinger equation. This is in contrast to time-independent perturbation theory, where the perturbation is stationary.

Let H^0\hat{H}_0 be the base time-independent Hamiltonian, and H^1\hat{H}_1 be a time-varying perturbation, with “bookkeeping” parameter λ\lambda:

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

We assume that the unperturbed time-independent problem H^0∣n⟩=En∣n⟩\hat{H}_0 \Ket{n} = E_n \Ket{n} has already been solved, such that the general solution for the full H^\hat{H} can be written as:

∣Ψ(t)⟩=∑ncn(t)∣n⟩exp⁡(−iEnt/ℏ)\begin{aligned} \Ket{\Psi(t)} = \sum_{n} c_n(t) \Ket{n} \exp(- i E_n t / \hbar) \end{aligned}

These time-dependent coefficients are then governed by the amplitude rate equations:

iℏdcmdt=∑ncn(t)⟨m∣λH^1(t)∣n⟩exp⁡(iωmnt)\begin{aligned} i \hbar \dv{c_m}{t} = \sum_{n} c_n(t) \matrixel{m}{\lambda \hat{H}_1(t)}{n} \exp(i \omega_{mn} t) \end{aligned}

So far, we have not made any approximations at all. We rewrite this in integral form:

cm(t)=cm(0)−λiℏ∑n∫0tcn(τ)⟨m∣H^1(τ)∣n⟩exp⁡(iωmnτ)dτ\begin{aligned} c_m(t) = c_m(0) - \lambda \frac{i}{\hbar} \sum_{n} \int_0^t c_n(\tau) \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau} \end{aligned}

If this cannot be solved exactly, we must approximate it. We expand cm(t)c_m(t) as a power series, with the initial condition cm(j)(0)=0c_m^{(j)}(0) = 0 for j>0j > 0:

cm(t)=cm(0)+λcm(1)(t)+λ2cm(2)(t)+...\begin{aligned} c_m(t) = c_m^{(0)} + \lambda c_m^{(1)}(t) + \lambda^2 c_m^{(2)}(t) + ... \end{aligned}

We then insert this into the integral and collect the nonzero orders of λ\lambda:

cm(1)(t)=−iℏ∑n∫0tcn(0)⟨m∣H^1(τ)∣n⟩exp⁡(iωmnτ)dτcm(2)(t)=−iℏ∑n∫0tcn(1)(τ)⟨m∣H^1(τ)∣n⟩exp⁡(iωmnτ)dτcm(3)(t)=−iℏ∑n∫0tcn(2)(τ)⟨m∣H^1(τ)∣n⟩exp⁡(iωmnτ)dτ\begin{aligned} c_m^{(1)}(t) &= - \frac{i}{\hbar} \sum_{n} \int_0^t c_n^{(0)} \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau} \\ c_m^{(2)}(t) &= - \frac{i}{\hbar} \sum_{n} \int_0^t c_n^{(1)}(\tau) \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau} \\ c_m^{(3)}(t) &= - \frac{i}{\hbar} \sum_{n} \int_0^t c_n^{(2)}(\tau) \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau} \end{aligned}

And so forth. The pattern here is clear: we can calculate the (j ⁣+ ⁣1)(j\!+\!1)th correction using our result for the jjth correction. The only purpose of λ\lambda was to help us collect its orders; in the end we simply set λ=1\lambda = 1 or absorb it into H^1\hat{H}_1. Now we have the essence of time-dependent perturbation theory, and we cannot go any further without considering a specific H^1\hat{H}_1.

References

  1. D.J. Griffiths, D.F. Schroeter, Introduction to quantum mechanics, 3rd edition, Cambridge.
  2. R. Shankar, Principles of quantum mechanics, 2nd edition, Springer.