Phase-space treatment of the driven quantum harmonic oscillator

Abstract

A recent phase-space formulation of quantum mechanics in terms of the Glauber coherent states is applied to study the interaction of a one-dimensional harmonic oscillator with an arbitrary time-dependent force. Wave functions of the simultaneous values of position q and momentum p are deduced, which in turn give the standard position and momentum wave functions, together with expressions for the ηth derivatives with respect to q and p, respectively. Afterwards, general formulae for momentum, position and energy expectation values are obtained, and the Ehrenfest theorem is verified. Subsequently, general expressions for the cross-Wigner functions are deduced. Finally, a specific example is considered to numerically and graphically illustrate some results.

Appendices

Appendix A. Some formulae used in this work

The Hermite polynomials satisfy the following relations:

• 1. [19, Section 5.6.],

  $$\begin{array}{@{}rcl@{}} &&{} H_{\eta}(y + \sigma) = {\sum}_{k=0}^{\eta} \left( \!\begin{array}{l}\eta\\ k\end{array}\!\right) H_{k}(y) (2 \sigma)^{\eta - k}\\ &&\qquad\quad =\! {\sum}_{k=0}^{\eta} \!\left( \!\begin{array}{c}\eta\\ \eta-k\end{array}\!\right) \!H_{\eta- k}(\sigma)(2 y)^{k}\!. \end{array} $$

  (A.1)

• 2. [14(b)],

  $$\begin{array}{@{}rcl@{}} {\kern-.9pc}H_{m}(y) H_{k}(y) &=& {\sum}_{r = 0}^{\min(m, k)} 2^{r} r! \left( \!\begin{array}{l}m\\ r\end{array}\!\right) \left( \!\begin{array}{l}k\\ r\end{array}\!\right) \\&& \times H_{m+k-2r}(y). \end{array} $$

  (A.2)

• 3. [20, 8.958.2],

  $$\begin{array}{@{}rcl@{}} &&{} {\sum}_{m = 0}^{n} \left( \!\begin{array}{l}n\\ m\end{array}\!\right) H_{n - m}(x ) H_{m}(y) = 2^{n/2} H_{n}\!\left( \frac{x + y}{\sqrt{2}}\right)\!.\\ \end{array} $$

  (A.3)

• 4. The Fourier transformation of the Hermite polynomials is given by

  $$\begin{array}{@{}rcl@{}} &&{} {\int}_{-\infty}^{\infty} \exp\!\left( i x y\right) \exp\!\left( - \frac{1}{2} y^{2}\right) H_{m}(y)\mathrm{d}y \\ &&\quad = (+i)^{m} \sqrt{2 \pi} \exp\!\left( - \frac{1}{2} x^{2}\right) H_{m}(x). \end{array} $$

  (A.4)

• 5. Then, given a Hermite polynomial H _n (y + c), one can write the identity

  $$\begin{array}{@{}rcl@{}} &&{} ({\kern-.6pt}y{\kern-.6pt} +{\kern-.6pt} b{\kern-.6pt})^{m} {\kern-.6pt}H_{n}{\kern-.6pt}({\kern-.6pt}y\! +\! c{\kern-.6pt})\! =\! {\sum}_{k = 0}^{[n/2]} \frac{(-1)^{k} \, n! \, 2^{n - 2k}}{k! \, (n-2k)!} \\ &&\qquad\qquad\qquad\; \times(y + b)^{m} (y + c)^{n - 2k} \\ &&{}\qquad\qquad\qquad\quad =\! {\sum}_{k = 0}^{[n/2]} {\sum}_{\ell = 0}^{m + n - 2k} \frac{(-1)^{k} \, n! \, 2^{n - 2k}}{k! \, (n-2k)!} \\ &&\qquad\qquad\qquad\; \!\times\, A_{\ell}{\kern-.5pt}({\kern-.5pt}m{\kern-.5pt},\! n\! -\! 2{\kern-.5pt} k{\kern-.5pt};\! b{\kern-.5pt},\! c{\kern-.5pt}){\kern-.5pt} y^{\ell}{\kern-.5pt}, \end{array} $$

  (A.5)

where the coefficients A _ℓ (m,n−2k;b,c) are given by expression (D.2).

Appendix B. On the polynomials π _n,k (x)

From the definition of π _n,k (x) given by (37), and formula (A.2), it follows that

$$\begin{array}{@{}rcl@{}} {\Pi}_{n, k}(x)\! & := &\! {\sum}_{\lambda = 0}^{n} {\sum}_{r = 0}^{\min(\lambda, k)}\! \left( \!\begin{array}{l}n\\ \lambda\end{array}\!\right)\! (-2)^{r} r! \! \left( \!\begin{array}{l}\lambda\\ r\end{array}\!\right)\!\! \left( \!\begin{array}{l}k\\ r\end{array}\!\right)\! H_{n - \lambda}(x)\\ &&\times H_{\lambda+k - 2r}(x)\\ & = &\! {\sum}_{\lambda = 0}^{n} {\sum}_{r = 0}^{\min(\lambda, k)} \, {\sum}_{u = 0}^{\min(n - \lambda, \lambda + k - 2r)} {\Theta}(n, k; \lambda; r, u)\!\!\\ &&\times H_{n + k - 2r -2u}(x), \end{array} $$

(B.1)

with coefficients

$$\begin{array}{@{}rcl@{}} {\Theta}(n, k; \lambda; r, u) & := & (-1)^{r} 2^{r+u} r! u! \! \left( \!\begin{array}{l}n\\ \lambda\end{array}\!\right)\!\! \left( \!\begin{array}{l}\lambda\\ r\end{array}\!\right)\! \! \left( \!\begin{array}{l}k\\ r\end{array}\!\right)\! \\ &&\times \! \left( \!\begin{array}{c}n-\lambda\\ u\end{array}\!\right)\!\! \left( \!\begin{array}{c} \lambda\! +\! k\! -\! 2r\\u\end{array}\!\right)\!. \end{array} $$

(B.2)

Equation (B.1) can be written in a more elegant way, namely

$$\begin{array}{@{}rcl@{}} &&{}{\Pi}_{n, k}(x) = {\sum}_{N = 0 }^{n+k} \theta_{N}(n, k) H_{N}(x), \qquad\\ &&{}{\Pi}_{n, k}(-x) = (-1)^{n + k}\, {\Pi}_{n, k}(x), \end{array} $$

(B.3)

where the values 𝜃 _N (n,k) are determined by equating the coefficients of x ^N in both sides of the equality

$$\begin{array}{@{}rcl@{}} {\sum}_{N = 0}^{n+k}\! \theta_{N}{\kern-.5pt}({\kern-.5pt}n{\kern-.5pt},{\kern-.5pt} k{\kern-.5pt}) x^{N}\! & = &\! {\sum}_{\lambda = 0}^{n} {\sum}_{r = 0}^{\min(\lambda, k)} \, {\sum}_{u = 0}^{\min(n - \lambda, \lambda + k - 2r)}\\ && \times {\Theta}{\kern-.5pt}({\kern-.5pt}n{\kern-.5pt},{\kern-.5pt} k{\kern-.5pt};{\kern-.5pt} \lambda{\kern-.5pt};{\kern-.5pt} r,{\kern-.5pt} u{\kern-.5pt}) x^{n + k - 2r -2u}. \end{array} $$

(B.4)

After exploring various examples, one finds that 𝜃 _N (n,k)=0, if (n + k) is even and N is odd, or if (n + k) is odd and N is even. Using this fact and the reflection formula H _N (−x) = (−1)^N H _N (x) for the Hermite polynomials, one confirms that the polynomial π _n,k (x) is even or odd depending on the value of n + k.

At this point, in (B.3) and (B.4) consider the particular situation in which k=0 and, from (38), recall that ${\Pi }_{n, 0}(x) = 2^{n/2} H_{n}(x \sqrt {2})$. Then, a formula linking the Hermite polynomials $H_{n}(x \sqrt {2})$ with H _N (x) [14(c)] allows one to write

$$\begin{array}{@{}rcl@{}} {\Pi}_{n, 0}(x) & = & 2^{n/2} H_{n}(x \sqrt{2}) = {\sum}_{j=0}^{[n/2]} \frac{n!}{j! (n - 2j)! } 2^{n- j}\\ &&\times H_{n - 2j}(x). \end{array} $$

(B.5)

Hence, by comparing this relation with (B.3), one finds the coefficients

$$\begin{array}{@{}rcl@{}} \theta_{N}(n, 0) = \theta_{j}(n) := \frac{n!}{j! (n - 2j)! }\, 2^{n- j}, \end{array} $$

(B.6)

if N = n−2j and j=0,…,[n/2], and 𝜃 _N (n,0)=0, otherwise.

Appendix C. Evaluation of the integral $F_{M N}(\mathcal {X}, a, b, c)$

For a>0, consider the integral

$$\begin{array}{@{}rcl@{}} &&{\kern-.6pc} F_{M N}(\mathcal{X}, a, b, c)\\ &&\quad := {\int}_{-\infty}^{\infty}{\kern-.6pt} \mathrm{d}y \exp(- i \mathcal{X} y {\kern-.6pt})\exp(- a y^{2})\\ &&\qquad \times H_{M}{\kern-.6pt}(y{\kern-.6pt} +{\kern-.6pt} b{\kern-.6pt}){\kern-.6pt} H_{N}{\kern-.6pt}(y{\kern-.6pt} +{\kern-.6pt} c). \end{array} $$

(C.1)

By using (A.1) and (A.2), and the integral

$$\begin{array}{@{}rcl@{}} &&{\kern-.6pc}{\int}_{-\infty}^{\infty} \exp\!\left( - i x y\right) y^{k} \exp(- a y^{2}) \mathrm{d}y \\ &&\quad= (-i)^{k} \sqrt{\frac{\pi}{a}} \left( \frac{1}{2 \sqrt{a}}\right)^{k} \exp\!\left( - \frac{x^{2}}{4 a}\right)\!\! H_{k}\!\!\left( \frac{x}{2 \sqrt{a}}\right),\\ \end{array} $$

(C.2)

one finds that

$$\begin{array}{@{}rcl@{}} &&{} F_{M N}(\mathcal{X}, a, b, c) = \sqrt{\frac{\pi}{a}}\, \exp\!\left( - \frac{\mathcal{X}^{2}}{4 a}\right) \\ &&{}\qquad\quad \times {\sum}_{k=0}^{M} {\sum}_{\ell =0}^{N}\! \left( \!\begin{array}{c}M\\ M\! -\! k\end{array}\!\right)\! \! \left( \!\begin{array}{c}N\\ N-\ell\end{array}\!\right)\! \left( \! - \frac{i}{ \sqrt{a}}\right)^{k + \ell}\\ &&{}\qquad\quad \times H_{k + \ell}\left( \frac{ \mathcal{X} }{2 \sqrt{a}}\right)\! H_{M - k}(b) H_{N - \ell}(c). \end{array} $$

(C.3)

Note that $F_{M N}(-\mathcal {X}, a, b, c) {=} F_{M N}^{\star }(\mathcal {X}, a, b, c)$, where ^⋆ denotes the complex conjugate.

Appendix D. Evaluation of the integral $\mathcal {F}(m, M, n, N | \mathcal {X}, a, b, c, {\Lambda }, \lambda )$

$$\begin{array}{@{}rcl@{}} &&{\kern-.6pc}\mathcal{F}(\mu, M, \eta, N | \mathcal{X}, a, b, c, {\Lambda}, \lambda)\\ &&\quad := {\int}_{-\infty}^{\infty} \exp\!\left( - i \mathcal{X} y \right) \exp(- a y^{2}) (y + {\Lambda})^{\mu}\\ && {\kern.4pc}\quad\; \times H_{M}(y + b) (y + \lambda)^{\eta} H_{N}(y + c) \mathrm{d}y. \end{array} $$

(D.1)

On the one hand, for μ = η=0, one has $\mathcal {F}(0, M, 0,$ $N | \mathcal {X}, a, b, c, {\Lambda }, \lambda ) \,=\, F_{M N}(\mathcal {X}, a, b, c)$. On the other hand, using the binomial theorem followed by a change of indices ℓ: = μ + ν, for μ and η integers, one gets the relation

$$\begin{array}{@{}rcl@{}} (y + {\Lambda})^{\mu} (y+\lambda)^{\eta} = {\sum}_{r = 0}^{\mu + \eta} A_{r}(\mu, \eta; {\Lambda}, \lambda) y^{r}, \end{array} $$

(D.2)

where, for Λ ≠ 0 and λ ≠ 0, the coefficients A _r (μ,η; Λ,λ) are given by

$$\begin{array}{@{}rcl@{}} A_{r}{\kern-.5pt}({\kern-.5pt}\mu{\kern-.5pt}, \eta; {\Lambda}, \lambda)\! =\! {\sum}_{j = 0}^{r}\!\left( \!\begin{array}{l}\mu\\ j\end{array}\!\right)\! \left( \!\begin{array}{c}\eta\\ r\! -\! j\end{array}\!\right)\! {\Lambda}^{\mu - j} \lambda^{\eta -r + j}. \end{array} $$

(D.3)

In addition, the following special cases should be considered: (i) for Λ=0 and λ ≠ 0,

$$\begin{array}{@{}rcl@{}} A_{r}{\kern-.5pt}({\kern-.4pt}\mu,{\kern-.5pt} \eta;{\kern-.5pt} 0,{\kern-.5pt} \lambda{\kern-.4pt}){\kern-.5pt} =\! \left( \!\begin{array}{c}\eta\\ r-\mu\end{array}\!\right)\! \lambda^{\eta -r+ \mu}, \quad \text{if}\, \mu{\kern-.5pt} \le{\kern-.5pt} r{\kern-.5pt} \le{\kern-.5pt} \mu{\kern-.5pt} +{\kern-.5pt} \eta{\kern-.4pt},\!\!\!\!\!\!\\ \end{array} $$

(D.4)

(ii) for Λ≠0 and λ=0,

$$\begin{array}{@{}rcl@{}} A_{r}{\kern-.5pt}(\mu{\kern-.5pt},{\kern-.5pt} \eta{\kern-.5pt};{\kern-.5pt} {\Lambda}{\kern-.5pt},{\kern-.5pt} 0{\kern-.5pt})\! =\!\left( \!\begin{array}{c}\mu\\ r-\eta\end{array}\!\right){\Lambda}^{\eta -r+ \mu}{\kern-.5pt}, \quad \text{if}\, \eta{\kern-.5pt} \le{\kern-.5pt} r{\kern-.5pt} \le{\kern-.5pt} \mu{\kern-.5pt}+{\kern-.5pt}\eta,\!\!\!\!\!\!\\ \end{array} $$

(D.5)

(iii) for Λ=0 and λ=0,

$$\begin{array}{@{}rcl@{}} A_{r}(\mu, \eta; 0, 0) = 1 \quad \text{if}\, r = \mu+\eta, \end{array} $$

(D.6)

where A _r (μ,η;Λ,λ)=0, otherwise.

Then, the identity $(i \partial /\partial \mathcal {X})^{r} \exp (- i \mathcal {X} y)\, {=}\, y^{r}$ $\times \exp (- i\mathcal {X} y)$ implies that

$$\begin{array}{@{}rcl@{}} &&{\kern-.6pc}\mathcal{F}(\mu, M, \eta, N | \mathcal{X}, a, b, c, {\Lambda}, \lambda)\\ &&{\kern.6pc}= {\sum}_{r = 0}^{\mu + \eta} A_{r}(\mu, \eta; {\Lambda}, \lambda) \bigg(i \frac{\partial}{\partial \mathcal{X}} \bigg)^{r} F_{M N}(\mathcal{X}, a, b, c).\\ \end{array} $$

(D.7)

Thus, because $F_{M N}(\mathcal {X}, a, b, c)$ is given by expression (C.3), one can define an auxiliary function $\mathbb {G}_{k + \ell , r}(\chi , a)$ by the relation

$$\begin{array}{@{}rcl@{}} \mathbb{G}_{k + \ell, r}{\kern-.5pt}({\kern-.5pt}\chi{\kern-.5pt},{\kern-.5pt} a{\kern-.5pt})\! & :=& \! \exp\!\left( \! + \frac{\chi^{2}}{4a}\! \right)\!\! \left( \! i \,\frac{\partial}{\partial \mathcal{X}}\! \right)^{r} F(\mathcal{X}) G_{k + \ell}(\mathcal{X}) \\ &=&\!\! \left( \frac{i}{2 \sqrt{a}} \right)^{r} {\sum}_{s=0}^{r} \left( \begin{array}{l}r\\ s\end{array}\right)\theta(k + \ell- s)\\ &&\times\, \frac{2^{s} (k+\ell)!}{(k+ \ell -s)!} (-1)^{r-s}\, H_{r -s}\bigg(\frac{\chi}{2\sqrt{a}} \bigg)\\ && \times\, H_{k+\ell-s}\bigg(\frac{\chi}{2\sqrt{a}} \bigg), \end{array} $$

(D.8)

where $F(\mathcal {X}) = \exp (- {\chi ^{2}}/{(4a)})$ and $G_{k + \ell }(\mathcal {X}) = H_{k +\ell }(\chi /(2 \sqrt {a}\,))$. For getting (D.8), one uses Leibnitz’s rule for the rth derivative of a product of functions, Rodrigues’s formula for the Hermite polynomials, and the expression

$$\begin{array}{@{}rcl@{}} \left( \!\frac{\partial}{\partial x}\!\right)^{s}\! H_{k + \ell}(x)\! =\! \theta(k{\kern-.5pt} +{\kern-.5pt} \ell{\kern-.5pt}-{\kern-.5pt} s{\kern-.5pt}) \frac{2^{s} (k{\kern-.5pt}+{\kern-.5pt}\ell)!}{(k{\kern-.5pt}+{\kern-.5pt} \ell{\kern-.5pt} -{\kern-.5pt}s{\kern-.5pt})!} H_{k+\ell-s}(x)\!\!\!\!\!\\ \end{array} $$

(D.9)

for the sth derivative of the Hermite polynomial H _{k + ℓ} (x). Here, 𝜃(y) is the unit step function: it is equal to 0 for y<0 and 1 for y≥0.

To summarize, the result of the calculation can be cast into the form

$$\begin{array}{@{}rcl@{}} &&{\kern-.6pc}\mathcal{F}(\mu, M, \eta, N | \mathcal{X}, a, b, c, {\Lambda}, \lambda)= \sqrt{\frac{\pi}{a}}\,\exp\!\left( - \frac{\chi^{2}}{4a}\right)\\ &&\quad \times {\sum}_{k=0}^{M} {\sum}_{\ell =0}^{N}\left( \!\begin{array}{c}M\\ M-k\end{array}\!\right)\left( \!\begin{array}{c}N\\ N-\ell\end{array}\!\right) \left( - \frac{i}{ \sqrt{a}}\right)^{k + \ell}\\ &&\quad \times \bigg[ {\sum}_{r = 0}^{\mu + \eta} A_{r}(\mu, \eta; {\Lambda}, \lambda) \mathbb{G}_{k + \ell, r}(\chi, a) \bigg]\\ &&\quad \times H_{M-k}(b)H_{N-\ell}(c). \end{array} $$

(D.10)

From (C.3) and (D.10) it is immediately verificable that $\mathcal {F}(0, M, 0, N | \mathcal {X}, a, b, c, {\Lambda }, \lambda ) = F_{M N}(\mathcal {X}, a, b, c)$, because (D.8) implies that $\mathbb {G}_{k + \ell ,\, 0}(\chi , a) = H_{k + \ell } \times ({ \mathcal {X} }/{(2 \sqrt {a}\,)})$.

Appendix E. A system with Hamiltonian $\hat {H}(t) = \hat {K} + {V}(t)$

Instead of the driven harmonic oscillator, consider now a system with a Hamiltonian $\hat {H}$ that splits into a time-independent unperturbed part $\hat {K}$ and a perturbation $\hat {V}(t)$, that is, $\hat {H} = \hat {K} + \hat {V}(t)$. Instead of eq. (9), one write the Schrödinger evolution operator in the form, with τ = t−t ₀,

$$\begin{array}{@{}rcl@{}} \hat{U}\! (t{\kern-.5pt},{\kern-.5pt} t_{0}{\kern-.5pt})\! =\! \exp\!\left( \!-\frac{i}{\hbar} \tau \hat{K}\!\right) \hat{U}_{\mathrm{I}}(t, t_{0}), \quad \mathcal{U}_{\mathrm{I}}({\kern-.5pt}\hat{q}{\kern-.5pt}, \hat{p}; t_{0}{\kern-.5pt}, t_{0}{\kern-.5pt})\! =\! \hat{1},\!\!\!\!\!\!\\ \end{array} $$

(E.1)

where $\hat {U}_{\mathrm {I}}(t, t_{0})$ is the evolution operator in the interaction picture and $\hat {U}_{0}(t, t_{0}) := \exp (\!- i \tau \hat {K}/ \hbar )$ is the time-evolution operator associated with the Hamiltonian $\hat {K}$. By inserting the identity $\hat {1} \,=\, \exp (+\frac {i}{\hbar } \tau \hat {K})\times \exp \!\big (\!\!-\frac {i}{\hbar } \tau \hat {K}\big )$ to the right-hand side of $\hat {U}_{\mathrm {I}}(t, t_{0})$, the generalization of (14) is given by

$$\begin{array}{@{}rcl@{}} && \hat{U}\!({\kern-.5pt}t{\kern-.5pt}, t_{0}{\kern-.5pt})\! =[ \hat{U}_{0}(t, t_{0})\hat{U}_{\mathrm{I}}(t, t_{0}) \hat{U}_{0}^{+}(t, t_{0})]\\ && \qquad\quad\;\;\;\times \exp\!\left( -\frac{i}{\hbar} \tau \hat{K}\right)\!. \end{array} $$

(E.2)

Formally, the solution of the equation of motion governing the time evolution of $\hat {U}_{\mathrm {I}}(t, t_{0})$ can be expressed as $\hat {U}_{\mathrm {I}}(t, t_{0}) := \mathcal {U}(\hat {q}, \hat {p}; t, t_{0})$, where $\mathcal {U}(\hat {q}, \hat {p}; t, t_{0})$ is a function of the position and momentum operators $\hat {q}$ and $\hat {p}$, beside the time dependence due to $\hat {V}(t)$. Now, according to the Weyl prescription, one can write

$$\begin{array}{@{}rcl@{}} \mathcal{U}(\hat{q}, \hat{p}; t, t_{0}) & = & \frac{1}{(2\pi \hbar)^{f}} \int \mathcal{U}_{\mathrm{W}}(q^{\prime}, p^{\prime}; t, t_{0})\\ &&\times \hat{D}(q^{\prime}, p^{\prime}) \mathrm{d}q^{\prime} \mathrm{d}p^{\prime}, \end{array} $$

(E.3)

where the scalar function $\mathcal {U}_{\mathrm {W}}(q^{\prime }, p^{\prime }; t, t_{0}) \! =\! \text {Tr}(\mathcal {U}(\hat {q},$ $\hat {p}; t, t_{0}) \hat {D}(-q^{\prime }, -p^{\prime }))$ is the Weyl symbol associated with $\mathcal {U}(\hat {q}, \hat {p}; t, t_{0})$ and $\hat {D}(q, p)$ is the Weyl operator defined in (1).

Consequently, the evolution operator in the Schrödinger picture can be written as

$$\begin{array}{@{}rcl@{}} \hat{U}(t, t_{0}) \!\!&=&\!\! \left[ \frac{1}{(2\pi \hbar)^{f}} \int \mathcal{U}_{\mathrm{W}}(q^{\prime}, p^{\prime}; t, t_{0})\right.\\ && \left. \!\!\!\! \times \hat{D}_{\bullet}(q^{\prime}, p^{\prime}, \tau) \mathrm{d}q^{\prime} \mathrm{d}p^{\prime} \vphantom{\frac{1}{(2\pi \hbar)^{f}}}\!\right]\! \exp\!\left( \!-\frac{i}{\hbar} \tau \hat{K}\!\right)\!, \end{array} $$

(E.4)

where $\hat {D}_{\bullet }(q^{\prime }, p^{\prime }, \tau )$ is the time-dependent Weyl operator defined by the relation

$$\begin{array}{@{}rcl@{}} \hat{D}_{\bullet}(q^{\prime}, p^{\prime}, \tau) := \exp\!\left( \frac{i}{\hbar} [p^{\prime}{} \hat{Q}(\tau) - q^{\prime}{}\hat{P}(\tau)]\right), \end{array} $$

(E.5)

and

$$\begin{array}{@{}rcl@{}} &&{}\hat{Q}(\tau) := \hat{U}_{0}(t, t_{0})\, \hat{q}\, \hat{U}_{0}^{+}(t, t_{0}) = Q(\hat{q}, \hat{p}; \tau),\\ &&{}\hat{P}(\tau) := \hat{U}_{0}(t, t_{0})\, \hat{p}\, \hat{U}_{0}^{+}(t, t_{0}) = P(\hat{q}, \hat{p}; \tau) \end{array} $$

(E.6)

are time-dependent operators, which satisfy the commutation relation $[\hat {Q}(\tau ) , \hat {P}(\tau )]\, {=}\, i \hbar$. Thus, for a system with Hamiltonian $\hat {H} {=} \hat {K} {+} \hat {V}(t)$, the state of the system at time t takes the form

$$\begin{array}{@{}rcl@{}} \left| {\Psi (t)} \right\rangle & = & \left[ \frac{1}{(2\pi \hbar)^{f}} \int \mathcal{U}_{\mathrm{W}}(q^{\prime}, p^{\prime}; t, t_{0})\hat{D}_{\bullet}(q^{\prime}, p^{\prime}, \tau)\right. \\ &&\left. \vphantom{\frac{1}{(2\pi \hbar)^{f}}} \times\! \mathrm{d}q^{\prime} \mathrm{d}p^{\prime} \right] \!\exp\!\left( -\frac{i}{\hbar} \tau \hat{K}\right)\!| {\Psi (t_{0})}\rangle\!. \end{array} $$

(E.7)

This generalization of eq. (16) is of course formal as long as explicit expressions of the operators $\hat {Q}(\tau )$ and $\hat {P}(\tau )$, the function $\mathcal {U}(\hat {q}, \hat {p}; t, t_{0})$ and the corresponding Weyl symbol $\mathcal {U}_{\mathrm {W}}(q^{\prime }, p^{\prime }; t, t_{0})$ are in general not known, and they can be difficult to obtain for a given specific system beyond the system discussed in this paper.