Introduction to Quantum Mechanics · Problem 11.23

Quantum Dynamics — David J. Griffiths

Physics — Quantum Mechanics · dyson-formula-time-ordered-product

Problem

In Problem 11.1 you showed that the solution to dfdt=k(t) f(t)\frac{df}{dt} = k(t)\,f(t) (where k(t)k(t) is a function of tt) is f(t)=eK(t)f(0),whereK(t)≡∫0tk(t′) dt′.f(t) = e^{K(t)}f(0), \quad \text{where} \quad K(t) \equiv \int_0^t k(t')\,dt'. This suggests that the solution to the Schrödinger equation (11.1) might be Ψ(t)=eG^(t)Ψ(0),whereG^(t)≡−iℏ∫0tH^(t′) dt′.(11.108)\Psi(t) = e^{\hat{G}(t)}\Psi(0), \quad \text{where} \quad \hat{G}(t) \equiv -\frac{i}{\hbar}\int_0^t \hat{H}(t')\,dt'. \tag{11.108} It doesn't work, because H^(t)\hat{H}(t) is an operator, not a function, and H^(t1)\hat{H}(t_1) does not (in general) commute with H^(t2)\hat{H}(t_2). (a) Try calculating iℏ ∂Ψ/∂ti\hbar\,\partial\Psi/\partial t, using Equation 11.108. *Note:* as always, the exponentiated operator is to be interpreted as a power series: eG^≡1+G^+12G^G^+13!G^G^G^+⋯ .e^{\hat{G}} \equiv 1 + \hat{G} + \frac{1}{2}\hat{G}\hat{G} + \frac{1}{3!}\hat{G}\hat{G}\hat{G} + \cdots. Show that if [G^,H^]=0\left[\hat{G}, \hat{H}\right] = 0, then Ψ\Psi satisfies the Schrödinger equation. (b) Check that the correct solution in the general case ([G^,H^]≠0)\left(\left[\hat{G}, \hat{H}\right] \neq 0\right) is Ψ(t)={1+(−iℏ)∫0tH^(t1) dt1+(−iℏ)2∫0tH^(t1)[∫0t1H^(t2) dt2]dt1+(−iℏ)3∫0tH^(t1)[∫0t1H^(t2)(∫0t2H^(t3) dt3)dt2]dt1+⋯}Ψ(0).(11.109)\begin{aligned} \Psi(t) = \Bigg\{1 + \left(-\frac{i}{\hbar}\right)\int_0^t \hat{H}(t_1)\,dt_1 &+ \left(-\frac{i}{\hbar}\right)^2 \int_0^t \hat{H}(t_1)\left[\int_0^{t_1} \hat{H}(t_2)\,dt_2\right]dt_1 \tag{11.109}\\ &+ \left(-\frac{i}{\hbar}\right)^3 \int_0^t \hat{H}(t_1)\left[\int_0^{t_1} \hat{H}(t_2)\left(\int_0^{t_2} \hat{H}(t_3)\,dt_3\right)dt_2\right]dt_1 + \cdots\Bigg\}\Psi(0). \end{aligned} (c) Show that T[G^G^]=−2ℏ2∫0tH^(t1)[∫0t1H^(t2) dt2]dt1,\mathbf{T}\left[\hat{G}\hat{G}\right] = -\frac{2}{\hbar^2}\int_0^t \hat{H}(t_1)\left[\int_0^{t_1} \hat{H}(t_2)\,dt_2\right]dt_1, and generalize to higher powers of G^\hat{G}. In place of G^n\hat{G}^n, in equation 11.108, we really want T[G^n]\mathbf{T}\left[\hat{G}^n\right]: Ψ(t)=T[e−iℏ∫0tH^(t′) dt′]Ψ(0).(11.112)\Psi(t) = \mathbf{T}\left[e^{-\frac{i}{\hbar}\int_0^t \hat{H}(t')\,dt'}\right]\Psi(0). \tag{11.112} This is Dyson's formula; it's a compact way of writing Equation 11.109, the formal solution to Schrödinger's equation. Dyson's formula plays a fundamental role in quantum field theory.

Browse the problem library