Quantum corrections to the Weyl quantization of the classical time of arrival

Abstract

A time of arrival (TOA) operator that is conjugate with the system Hamiltonian was constructed by Galapon without canonical quantization in Galapon (J. Math. Phys. 45:3180–3215, 2004). The constructed operator was expressed as an infinite series but only the leading term was investigated which was shown to be equal to the Weyl-quantized TOA-operator for entire analytic potentials. In this paper, we give a full account of the said operator by explicitly solving all the terms in the expansion. We interpret the terms beyond the leading term as the quantum corrections to the Weyl quantization of the classical arrival time. These quantum corrections are expressed as some integrals of the interaction potential and their properties are investigated in detail. In particular, the quantum corrections always vanish for linear systems but nonvanishing for nonlinear systems. Finally, we consider the case of an anharmonic oscillator potential as an example.

Appendix

Derivation of the leading quantum correction

We consider the leading quantum correction $T_1(u,v)$ given by (40) where the coefficients $\alpha _{m,j}^{(1)}$ satisfy the recurrence relation given by (41). Direct substitution of (41) into (40) leads to

$$\begin{aligned} T_{1} (u,v) = & \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{j = 0}}^{\infty } {u^{m} } } v^{{2j}} \left( {\frac{\mu }{{2\hbar ^{2} }}} \right)^{{j - 1}} \frac{1}{{m \cdot 2j}}\sum\limits_{{l = 1}}^{{m - 1}} {\frac{{la_{l} }}{{2^{{l - 1}} }}} \alpha _{{m - l,j - 1}}^{{(1)}} \\ & + \sum\limits_{{m = 0}}^{\infty } {\sum\limits_{{j = 0}}^{\infty } {u^{m} } } v^{{2j}} \left( {\frac{\mu }{{2\hbar ^{2} }}} \right)^{{j - 1}} \frac{1}{{m \cdot 2j}}\sum\limits_{{l = 3}}^{{m + 1}} {\frac{{a_{l} }}{{2^{{l - 1}} }}} \left( {\begin{array}{*{20}c} l \\ 3 \\ \end{array} } \right)\alpha _{{m - l + 2,j - 2}}^{{(0)}} . \\ \end{aligned}$$

(134)

Performing series rearrangements, shifting of indices, and using the following relations

$$\begin{aligned} \int _{0}^{u} \textrm{d}s \, s^{m+l+1}=\frac{u^{m+l+2}}{m+l+2}, \qquad \int _{0}^{v} \textrm{d}w \, w^{2j+1}=\frac{v^{2j+2}}{2j+2}, \end{aligned}$$

(135)

Equation (134) simplifies to

$$\begin{aligned} \begin{aligned} T_1(u,v)=&\left( \frac{\mu }{2\hbar ^2}\right) \int _{0}^{u} \textrm{d}s \, V'\left( \frac{s}{2}\right) \int _{0}^{v} \textrm{d}w \, w \, T_1(s,w)+\frac{1}{24}\left( \frac{\mu }{2\hbar ^2}\right) \int _{0}^{u} ds \, V'''\left( \frac{s}{2}\right) \int _{0}^{v} \textrm{d}w \, w^3 \, T_0(s,w). \end{aligned} \end{aligned}$$

(136)

Taking $\partial ^2/\partial u\partial v$ and using again the Leibniz integral rule given by (55), we get the following partial differential equation for $T_1(u,v)$

$$\begin{aligned} \frac{\partial ^2 T_1(u,v)}{\partial v \partial u}=\left( \frac{\mu }{2\hbar ^2}\right) V'\left( \frac{u}{2}\right) v\,T_1(u,v)+\frac{1}{24}\left( \frac{\mu }{2\hbar ^2}\right) v^3 \, V'''\left( \frac{u}{2}\right) T_0(u,v). \end{aligned}$$

(137)

It is straightforward to show the uniqueness of the solution $T_1(u,v)$ with boundary conditions $T_1(u,0)=T_1(0,v)=0$. Suppose that $T_{1,a}(u,v)$ and $T_{1,b}(u,v)$ both satisfy (137). Since the leading kernel factor $T_0(u,v)$ is unique, it can be shown using the triangle inequality that $|T_{1,a}(u,v)$ - $T_{1,b}(u,v)|\rightarrow 0$ so that $T_{1,a}(u,v)$ = $T_{1,b}(u,v)$. Hence, $T_1(u,v)$ is also unique. In fact, the uniqueness of $T_1(u,v)$ is also guaranteed by the use of the method of successive approximations later.

Notice that (137) is dependent on $T_1(s,w)$ and $T_0(s,w)$ but the latter is just the leading kernel factor which is already known at this point. To solve for $T_1(u,v)$, we apply again the method of successive approximations used in Sect. (3.1). Since we are solving for $T_1(u,v)$, our zeroth-order approximation is the second term of (136), that is,

$$\begin{aligned} T_{1,0}(u,v)=\frac{1}{24}\left( \frac{\mu }{2\hbar ^2}\right) \int _{0}^{u} ds \, V'''\left( \frac{s}{2}\right) \int _{0}^{v} dw \, w^3 \, T_0(s,w). \end{aligned}$$

(138)

The nth-order approximation can then be determined from the following equation,

$$\begin{aligned} T_{1,n}(u,v)=T_{1,0}(u,v)+\left( \frac{\mu }{2\hbar ^2}\right) \int _{0}^{u} ds \, V'\left( \frac{s}{2}\right) \int _{0}^{v} dw \, w \, T_{1,n-1}(s,w). \end{aligned}$$

(139)

The solution $T_1(u,v)$ of the integral equation in (136) is derived by taking the limit, $T_1(u,v)=\lim _{n\rightarrow \infty }T_{1,n}(u,v)$.

From (139), we determine the first few iterates and also infer the general form for arbitrary n. For $n=1$, we have

$$\begin{aligned} T_{1,1}(u,v)&=\frac{1}{24}\left( \frac{\mu }{2\hbar ^2}\right) \int _{0}^{u} \textrm{d}s \, V'''\left( \frac{s}{2}\right) \int _{0}^{v} \textrm{d}w \, w^3 \, T_0(s,w)\\&+\frac{1}{24}\left( \frac{\mu }{2\hbar ^2}\right) ^2\int _{0}^{u} \textrm{d}s \, V'''\left( \frac{s}{2}\right) \left[ V \left( \frac{u}{2}\right) -V \left( \frac{s}{2}\right) \right] \int _{0}^{v} \textrm{d}w \, w^3 (v^2-w^2)\, T_0(s,w). \end{aligned}$$

(140)

For $n=2$, we have

$$\begin{aligned} T_{1,2}(u,v)&=\frac{1}{24}\left( \frac{\mu }{2\hbar ^2}\right) \int _{0}^{u} \textrm{d}s \, V'''\left( \frac{s}{2}\right) \int _{0}^{v} \textrm{d}w \, w^3 \, T_0(s,w)\\&+\frac{1}{24}\left( \frac{\mu }{2\hbar ^2}\right) ^2\int _{0}^{u} \textrm{d}s \, V'''\left( \frac{s}{2}\right) \left[ V \left( \frac{u}{2}\right) -V \left( \frac{s}{2}\right) \right] \int _{0}^{v} \textrm{d}w \, w^3 (v^2-w^2)\, T_0(s,w)\\&+\frac{1}{96}\left( \frac{\mu }{2\hbar ^2}\right) ^3\int _{0}^{u} \textrm{d}s V'''\left( \frac{s}{2}\right) \left[ V \left( \frac{u}{2}\right) -V \left( \frac{s}{2}\right) \right] ^2 \int _{0}^{v} \textrm{d}w \, w^3 (v^2-w^2)^2\, T_0(s,w). \end{aligned}$$

(141)

Doing the same calculations for $n \ge 3$, we infer the following form of $T_{1,n}(u,v)$ for arbitrary n

$$\begin{aligned} T_{1,n}(u,v)=&\left( \frac{\mu }{48\hbar ^2}\right) \int _{0}^{u} \textrm{d}s \, V'''\left( \frac{s}{2}\right) \int _{0}^{v} \textrm{d}w \, w^3 \, T_0(s,w) \\&\times \sum _{k=0}^{n}\frac{1}{(1)_k k!}\left( \frac{\mu }{2\hbar ^2}\right) ^k(v^2-w^2)^k \left[ V \left( \frac{u}{2}\right) -V \left( \frac{s}{2}\right) \right] ^k. \end{aligned}$$

(142)

Equation (142) is also proven formally via mathematical induction. Taking the limit $n \rightarrow \infty$ and using the definition of the hypergeometric function given by (68), we find the leading kernel factor correction to be

$$\begin{aligned} \begin{aligned} T_1(u,v)&=\left( \frac{\mu }{48\hbar ^2}\right) \int _{0}^{u} \textrm{d}s \, V'''\left( \frac{s}{2}\right) \int _{0}^{v} \textrm{d}w \, w^3 \, T_0(s,w)\,\,{}_0F_1 \left( ;1;\left( \frac{\mu }{2\hbar ^2}\right) (v^2-w^2)\left[ V \left( \frac{u}{2}\right) -V \left( \frac{s}{2}\right) \right] \right) \end{aligned} \end{aligned}$$

(143)

in its integral form. Equation (143) clearly shows the dependence of the leading kernel correction $T_1(u,v)$ on the potential V(q) and the leading kernel factor $T_0(u,v)$ which is also dependent on the potential. It is straightforward to show that (91) satisfies the partial differential equation for the leading correction given by (137) subject to the boundary conditions $T_1(u,0)=T_1(0,v)=0$.