Metaplectic operator approach to a time-dependent generalized harmonic oscillator

Abstract

The metaplectic operator is a unitary operator corresponding to a time-dependent classical linear canonical transformation. We present the explicit expression for the metaplectic operator. The parameter in the metaplectic operator is expressed in terms of the solution of a classical time-dependent generalized harmonic oscillator. The time evolution operator, Lewis and Riesenfeld invariant, and the wave function of the time-dependent generalized harmonic oscillator are studied using the metaplectic operator, and the results are compared with those from other studies. The metaplectic operator method is a simpler method for studying the time-dependent generalized harmonic oscillator.

Appendix

We will prove that the unitary operator ${\hat{M}}(t)$ satisfies the time-dependent Schrödinger equation, Eq. (31). To prove it, we will use the formula for the derivative of an exponential operator given by in the following form [32]:

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t} e^{ {\hat{X}}(t)}= & {} \left[ \int _0^1 e^{ s {\hat{X}}(t) } \frac{\mathrm{{d}}{\hat{X}}(t)}{\mathrm{{d}}t} e^{ -s {\hat{X}}(t)} \mathrm{{d}}s \right] e^{ {\hat{X}}(t)} \end{aligned}$$

(54)

for an time-dependent operator ${\hat{X}}(t)$. Taking ${\hat{X}}(t)$ as ${\hat{X}}(t) = -i \frac{1}{2} {\hat{z}}^\mathrm{{T}} W {\hat{z}}$ in Eq. (54) leads to

$$\begin{aligned} \frac{\mathrm{{d}}{\hat{M}}(t)}{\mathrm{{d}}t}= & {} \int _0^1 \mathrm{{d}}s \left[ e^{ -is \frac{1}{2} {\hat{z}}^\mathrm{{T}} W {\hat{z}}} \left( -i \frac{1}{2} {\hat{z}}^\mathrm{{T}} \frac{\mathrm{{d}}W}{\mathrm{{d}}t} {\hat{z}}\right) e^{ +is \frac{1}{2} {\hat{z}}^\mathrm{{T}} W {\hat{z}}} \right] {\hat{M}}(t) \nonumber \\= & {} -i \frac{1}{2} \int _0^1 \mathrm{{d}}s \Big [ e^{-sJW} {\hat{z}}\Big ]^\mathrm{{T}} \frac{\mathrm{{d}}W}{\mathrm{{d}}t} \Big [ e^{-sJW} {\hat{z}}\Big ] {\hat{M}}(t) , \end{aligned}$$

(55)

where $e^{-is \frac{1}{2} {\hat{z}}^\mathrm{{T}} W {\hat{z}}} {\hat{z}}_\mu e^{+is \frac{1}{2} {\hat{z}}^\mathrm{{T}} W {\hat{z}}} = (e^{-sJW} {\hat{z}})_\mu$ is used. Using that $[ e^{-sJW} ]^\mathrm{{T}} = e^{sJ^\mathrm{{T}}JWJ} = J^\mathrm{{T}} e^{sJW} J$ for a symmetric matrix W, we can write Eq. (55) as

$$\begin{aligned} \frac{\mathrm{{d}}{\hat{M}}(t)}{\mathrm{{d}}t} {\hat{M}}^\dagger (t)= & {} -i \frac{1}{2} {\hat{z}}^\mathrm{{T}} J^\mathrm{{T}} \int _0^1 \mathrm{{d}}s \Big [ e^{sJW} \frac{\mathrm{{d}}(JW)}{\mathrm{{d}}t} e^{-sJW} \Big ] {\hat{z}}. \end{aligned}$$

(56)

Now, we apply the relation in Eq. (54) to the derivative of $e^{JW}$ with respect to time t, and get the relation

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t} e^{JW} = \left[ \int _0^1 \Big [ e^{sJW} \frac{\mathrm{{d}}(JW)}{\mathrm{{d}}t} e^{-sJW} \Big ] \mathrm{{d}}s \right] e^{JW}, \end{aligned}$$

(57)

which is the form in Eq. (56). Therefore, using that $e^{JW(t)} = S(t)$ and $\frac{\mathrm{{d}}S(t)}{\mathrm{{d}}t} = Jh(t)S(t)$, we obtain

$$\begin{aligned} \frac{\mathrm{{d}}{\hat{M}}(t)}{\mathrm{{d}}t} {\hat{M}}^\dagger (t)= & {} -i \frac{1}{2} {\hat{z}}^\mathrm{{T}} J^\mathrm{{T}} \left( \frac{\mathrm{{d}}}{\mathrm{{d}}t} e^{JW} \right) e^{-JW} {\hat{z}}\nonumber \\= & {} -i \frac{1}{2} {\hat{z}}^\mathrm{{T}} J^\mathrm{{T}} \left( \frac{\mathrm{{d}}S}{\mathrm{{d}}t} \right) S^{-1} {\hat{z}}\nonumber \\= & {} -i \frac{1}{2} {\hat{z}}^\mathrm{{T}} h(t) {\hat{z}}. \end{aligned}$$

(58)

Thus, time derivative of ${\hat{M}}(t)$ satisfies the time-dependent Schrödinger equation (TDSE)

$$\begin{aligned} i \frac{\partial }{\partial t}{\hat{M}}(t) = {\hat{H}}(t) {\hat{M}}(t), \end{aligned}$$

(59)

where ${\hat{H}}(t) = \frac{1}{2} {\hat{z}}^\mathrm{{T}} h(t) {\hat{z}}$. For the case that the unitary operator ${\hat{M}}(t)$ is expressed in three terms as in Eq. (13), we can also show that ${\hat{M}}(t) = {\hat{U}}_l(t) {\hat{U}}_d(t) {\hat{U}}_r(t)$ satisfies the TDSE.