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 only our previous 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, we cannot go any further without considering a specific H^1\hat{H}_1.

Sinusoidal perturbation

Arguably the most important perturbation is a sinusoidally-varying potential, which represents e.g. incoming electromagnetic waves, or an AC voltage being applied to the system. In this case, H^1\hat{H}_1 has the following form:

H^1(r⃗,t)≡V(r⃗)sin⁡(ωt)=12iV(r⃗) (exp⁡(iωt)−exp⁡(−iωt))\begin{aligned} \hat{H}_1(\vec{r}, t) \equiv V(\vec{r}) \sin(\omega t) = \frac{1}{2 i} V(\vec{r}) \: \big( \exp(i \omega t) - \exp(-i \omega t) \big) \end{aligned}

We abbreviate Vmn=⟨m∣V∣n⟩V_{mn} = \matrixel{m}{V}{n}, and take the first-order correction formula:

cm(1)(t)=−12ℏ∑nVmncn(0)∫0texp⁡ ⁣(iτ(ωmn ⁣+ ⁣ω))−exp⁡ ⁣(iτ(ωmn ⁣− ⁣ω))dτ=i2ℏ∑nVmncn(0)(exp⁡ ⁣(it(ωmn ⁣+ ⁣ω))−1ωmn+ω+exp⁡ ⁣(it(ωmn ⁣− ⁣ω))−1ωmn−ω)\begin{aligned} c_m^{(1)}(t) &= - \frac{1}{2 \hbar} \sum_{n} V_{mn} c_n^{(0)} \int_0^t \exp\!\big(i \tau (\omega_{mn} \!+\! \omega)\big) - \exp\!\big(i \tau (\omega_{mn} \!-\! \omega)\big) \dd{\tau} \\ &= \frac{i}{2 \hbar} \sum_{n} V_{mn} c_n^{(0)} \bigg( \frac{\exp\!\big(i t (\omega_{mn} \!+\! \omega) \big) - 1}{\omega_{mn} + \omega} + \frac{\exp\!\big(i t (\omega_{mn} \!-\! \omega) \big) - 1}{\omega_{mn} - \omega} \bigg) \end{aligned}

For simplicity, we let the system start in a known state ∣a⟩\Ket{a}, such that cn(0)=δnac_n^{(0)} = \delta_{na}, and we assume that the driving frequency is close to resonance ω≈ωma\omega \approx \omega_{ma}, such that the second term dominates the first, which can then be neglected. We thus get:

cm(1)(t)=iVma2ℏexp⁡ ⁣(it(ωma ⁣− ⁣ω))−1ωma−ω=iVma2ℏexp⁡ ⁣(it(ωma ⁣− ⁣ω)/2)−exp⁡ ⁣( ⁣− ⁣it(ωma ⁣− ⁣ω)/2)ωma−ω exp⁡ ⁣(it(ωma ⁣− ⁣ω)/2)=−Vmaℏsin⁡ ⁣(t(ωma ⁣− ⁣ω)/2)ωma−ω exp⁡ ⁣(it(ωma ⁣− ⁣ω)/2)\begin{aligned} c_m^{(1)}(t) &= i \frac{V_{ma}}{2 \hbar} \frac{\exp\!\big(i t (\omega_{ma} \!-\! \omega) \big) - 1}{\omega_{ma} - \omega} \\ &= i \frac{V_{ma}}{2 \hbar} \frac{\exp\!\big(i t (\omega_{ma} \!-\! \omega) / 2 \big) - \exp\!\big(\!-\! i t (\omega_{ma} \!-\! \omega) / 2 \big)}{\omega_{ma} - \omega} \: \exp\!\big(i t (\omega_{ma} \!-\! \omega) / 2 \big) \\ &= - \frac{V_{ma}}{\hbar} \frac{\sin\!\big( t (\omega_{ma} \!-\! \omega) / 2 \big)}{\omega_{ma} - \omega} \: \exp\!\big(i t (\omega_{ma} \!-\! \omega) / 2 \big) \end{aligned}

Taking the norm squared yields the transition probability: the probability that a particle that started in state ∣a⟩\Ket{a} will be found in ∣m⟩\Ket{m} at time tt:

Pa→m=∣cm(1)(t)∣2=∣Vma∣2ℏ2sin⁡2 ⁣((ωma−ω)t/2)(ωma−ω)2\begin{aligned} \boxed{ P_{a \to m} = |c_m^{(1)}(t)|^2 = \frac{|V_{ma}|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_{ma} - \omega) t / 2 \big)}{(\omega_{ma} - \omega)^2} } \end{aligned}

The result would be the same if H^1≡Vcos⁡(ωt)\hat{H}_1 \equiv V \cos(\omega t). However, if instead H^1≡Vexp⁡(−iωt)\hat{H}_1 \equiv V \exp(- i \omega t), the result is larger by a factor of 44, which can cause confusion when comparing literature.

In any case, the probability oscillates as a function of tt with period T=2π/(ωma ⁣− ⁣ω)T = 2 \pi / (\omega_{ma} \!-\! \omega), so after one period the particle is back in ∣a⟩\Ket{a}, and after T/2T/2 the particle is in ∣b⟩\Ket{b}. See Rabi oscillation for a more accurate treatment of this “flopping” behaviour.

However, when regarded as a function of ω\omega, the probability takes the form of a sinc-function centred around (ωma ⁣− ⁣ω)(\omega_{ma} \!-\! \omega), so it is highest for transitions with energy ℏω=Em ⁣− ⁣Ea\hbar \omega = E_m \!-\! E_a.

Also note that the sinc-distribution becomes narrower over time, which roughly means that it takes some time for the system to “notice” that it is being driven periodically. In other words, there is some “inertia” to it.

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.