23.4 The harmonic oscillator propagator

23.4 The harmonic oscillator propagator

In section 12.5 we saw that for the free particle quantum system, energy eigenstates were momentum eigenstates, and in the momentum space representation time evolution by a time interval $T$ was given by a kernel (see equation 12.6)

$$\tilde{U}(T,k)=\frac{1}{\sqrt{2\pi }}{e}^{-i\frac{1}{2m}{k}^{2}T}$$

The position space propagator was found by computing the Fourier transform of this. For the harmonic oscillator, energy eigenstates are no longer momentum eigenstates and diferent methods are needed to compute the action of the time evolution operator $e^{-iHT}$

23.4.1 The propagator in the Bargmann-Fock representation

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue

In the Bargmann-Fock representation the Hamiltonian is the operator

$$H=\omega \left(a^{†}a+\frac{1}{2}\right)=\omega \left(z\frac{d}{dz}+\frac{1}{2}\right)$$

(here we choose $\hslash =1$ and $m=1$ , but no longer fix $\omega =1)$ and energy eigenstates are the states n

$$\frac{z^n}{\sqrt{n!}}=|n⟩$$

with energy eigenvalues

$$\omega \left(n+\frac{1}{2}\right)$$

$e^{-iHT}$ will be diagonal in this basis, with

$$e^{-iHT}|n⟩=e^{-i\omega (n+\frac{1}{2})T}|n⟩$$

Instead of the Schr¨odinger picture in which states evolve and operators are constant, one can instead go to the Heisenberg picture (see section 7.3) where states are constant and operators $\mathcal{O}$ evolve in time according to

$$\frac{d}{dt}\mathcal{O}(t)=i[H,\mathcal{O}(t)]$$

with solution

$$\mathcal{O}(t)=e^{itH}\mathcal{O}(0)e^{-itH}$$

In the harmonic oscillator problem we can express other operators in terms of the annihilation and creation operators, which evolve according to

$$\frac{d}{dt}a(t)=i[H,a(t)]=-i\omega a,\frac{d}{dt}{a}^{†}(t)=i[H,a^{†}(t)]=i\omega a^{†}$$

with solutions

$$a(t)=e^{-i\omega t}a(0),a^{†}(t)=e^{i\omega t}{a}^{†}(0)$$

The Hamiltonian operator is time invariant.

Questions about time evolution now become questions about various products of annihilation and creation operators taken at various times, applied to various Heisenberg picture states. Since an arbitrary state is given as a linear combination of states produced by repeatedly applying $a^{†}(0)$ to |0⟩, such problems can be reduced to evaluating expressions involving just the state |0⟩, with various creation and annihilation operators applied at diferent times. Non-zero results will come from terms involving

$$⟨0|a(T)a^{†}(0)|0⟩=e^{-i\omega T}$$

which for $T>0$ has an interpretation as an amplitude for the process of adding one quantum to the lowest energy state at time $t=0$ , then removing it at time $t=T$

23.4.2 The coherent state propagator

One possible reason these states are given the name “coherent” is that they remain coherent states as they evolve in time (for the harmonic oscillator Hamiltonian), with $\alpha$ evolving in time along a classical phase space trajectory. If the state at $t=0$ is a coherent state labeled by ${\alpha }_0\left(|\psi (0)⟩=|{\alpha }_0⟩\right)$ , by 23.2, at later times one has

$$\begin{array}{rl}|\psi (t)⟩& =e^{-iHt}|{\alpha }_0⟩\\ & =e^{-iHt}{e}^{-\frac{|{\alpha }_0{|}^2}{2}}\sum _{n=0}^{\infty }\frac{{\alpha }_0^n}{\sqrt{n!}}|n⟩\\ & =e^{-i\frac{1}{2}\omega t}{e}^{-\frac{|{\alpha }_0{|}^2}{2}}\sum _{n=0}^{\infty }\frac{e^{-i\omega nt}{\alpha }_0^n}{\sqrt{n!}}|n⟩\\ & =e^{-i\frac{1}{2}\omega t}{e}^{-\frac{|e^{-i\omega t}{\alpha }_0{|}^2}{2}}\sum _{n=0}^{\infty }\frac{(e^{-i\omega t}{\alpha }_0{)}^n}{\sqrt{n!}}|n⟩\\ & =e^{-i\frac{1}{2}\omega t}|e^{-i\omega t}{\alpha }_0⟩\end{array}\text{\hspace{1em}(23.9)}$$

Up to the phase factor $e^{-i\frac{1}{2}\omega t}$ , this remains a coherent state, with time dependence of the label $\alpha$ given by the classical time-dependence of the complex coordinate $\begin{array}{r}\overline{z}(t)=\frac{1}{\sqrt{2}}(\sqrt{\omega }q(t)+\frac{i}{\sqrt{\omega }}p(t))\end{array}$ for the harmonic oscillator (see 22.1) with $\overline{z}(0)={\alpha }_0$

Equations 23.3 and 23.9 can be used to calculate a propagator function in terms of coherent states, with the result

$$⟨{\alpha }_T|e^{-iHT}|{\alpha }_0⟩=\exp (-\frac{1}{2}(|{\alpha }_0{|}^2+|{\alpha }_T{|}^2)+{\overline{\alpha }}_T{\alpha }_0{e}^{-i\omega T}-\frac{i}{2}\omega T)\text{\hspace{1em}(23.10)}$$

23.4.3 The position space propagator

Coherent states can be expressed in the position space representation by calculating

$$\begin{array}{rl}⟨q|\alpha ⟩& =⟨q|e^{-\frac{|\alpha {|}^2}{2}}{e}^{\alpha a^{†}}|0⟩\\ & =e^{-\frac{|\alpha {|}^2}{2}}{e}^{\frac{\alpha }{\sqrt{2}}(\sqrt{\omega }q-\frac{1}{\sqrt{\omega }}\frac{d}{dq})}(\frac{\omega }{\pi }{)}^{\frac{1}{4}}{e}^{-\frac{\omega }{2}{q}^2}\\ & =(\frac{\omega }{\pi }{)}^{\frac{1}{4}}{e}^{-\frac{|\alpha {|}^2}{2}}{e}^{\alpha \sqrt{\frac{\omega }{2}}q}{e}^{-\frac{\alpha }{\sqrt{2\omega }}\frac{d}{dq}}{e}^{-\frac{{\alpha }^2}{4}}{e}^{-\frac{\omega }{2}{q}^2}\\ & =(\frac{\omega }{\pi }{)}^{\frac{1}{4}}{e}^{-\frac{|\alpha {|}^2}{2}}{e}^{-\frac{{\alpha }^2}{4}}{e}^{\alpha \sqrt{\frac{\omega }{2}}q}{e}^{-\frac{\omega }{2}(q-\frac{\alpha }{\sqrt{2\omega }}{)}^2}\\ & =(\frac{\omega }{\pi }{)}^{\frac{1}{4}}{e}^{-\frac{|\alpha {|}^2}{2}}{e}^{-\frac{{\alpha }^2}{2}}{e}^{-\frac{\omega }{2}{q}^2}{e}^{\sqrt{2\omega }\alpha q}\end{array}\text{\hspace{1em}(23.11)}$$

This expression gives the transformation between the position space basis and coherent state basis. The propagator in the position space basis can then be calculated as

$$⟨q_T|e^{-iHT}|q_0⟩=\frac{1}{{\pi }^2}{\int }_{{\mathbf{C}}^2}⟨q_T|{\alpha }_T⟩⟨{\alpha }_T|e^{-iHT}|{\alpha }_0⟩⟨{\alpha }_0|q_0⟩d^2{\alpha }_T{d}^2{\alpha }_0$$

using equations 23.10, 23.11 (and its complex conjugate), as well as equation 23.7.

We will not perform this (rather dificult) calculation here, but just quote the result, which is

$$⟨q_T|e^{-iHT}|q_0⟩=\sqrt{\frac{\omega }{i2\pi \sin (\omega T)}}\exp \left(\frac{i\omega }{2\sin (\omega T)}((q_0^2+q_T^2)\cos (\omega T)-2q_0{q}_T)\right)\text{\hspace{1em}(23.12)}$$

One can easily see that as $T0$ this will approach the free particle propagator (equation 12.9, with $m=1)$

$$⟨q_T|e^{-iHT}|q_0⟩\approx \sqrt{\frac{1}{i2\pi T}}{e}^{\frac{i}{2T}(q_T-q_0{)}^2}$$

and as in that case becomes the distribution $\delta (q_0-q_T)$ as $T0$ . Without too much dificulty, one can check that 23.12 satisfies the harmonic oscillator Schr¨odinger equation (in $q=q_T$ and $t=T$ , for any initial $\psi (q_0,0))$ .

As in the free particle case, the harmonic oscillator propagator can be defined first as a function of a complex variable $s=\tau +iT$ , holomorphic for $\tau >0$ , then taking the boundary value as $\tau 0.$ This fixes the branch of the square root in 23.12 and one finds (see for instance section 7.6.7 of [108]) that the square root factor needs to be taken to be

$$\sqrt{\frac{\omega }{i2\pi \sin (\omega T)}}=e^{-i\frac{\pi }{4}}{e}^{-in\frac{\pi }{2}}\sqrt{\frac{\omega }{2\pi |\sin (\omega T)|}}$$

for $\omega T\in [n\pi ,(n+1)\pi ]$

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