Higher-order phase-space moments for Morse oscillators and their harmonic limit

Abstract

An explicit and general formula for (m, l) diagonal higher-order phase-space moments (i.e. matrix elements) $\langle {\hat{x}}^m{\hat{p}}^l\rangle$ for the Morse oscillator potential is presented. We show that this formula is more extended than those currently available in the literature and can be carried out using a phase-space Wigner distribution function in conjunction with its characteristic function. In the case it is wanted, the obtained formula gives the explicit form of the expected values of position and momentum in terms of the potential parameters. The validity of these expressions is tested by giving a numerical comparison with those obtained by other approximations. Moreover, it makes it possible to approach straightforwardly the harmonic limit of the Morse oscillator potential using the Heisenberg uncertainty product. We show indeed that when changing the potential depth, a common pattern emerges as a linear behavior on n in the form $(\Delta x)_n(\Delta p)_n\sim n+1/2$, applied to all levels $n=0,1,2,\ldots$, indicating the signature of the so-called harmonic limit of the Morse oscillator potential.

Appendices

Appendix A: Evaluation of some summations

In this appendix, we will evaluate some summations involving products of $\Theta$’s with $l_1^kl_2^m$, i.e.

$$\begin{aligned} \Sigma _{n,N}^{k,m} = \sum _{l_1=0}^n\sum _{l_2=0}^n l_1^kl_2^m \Theta _{l_1,l_2}^{n,N}, \end{aligned}$$

for $k,m=0,1,2,\ldots$ It is obvious that the case $k=m=0$ concerns the proof of the normalized-summation given by Eq. (6), while other cases lead to evaluate, among others, Eqs. (39) and (40).

Case $k=m=0$. In this case, let us substitute the expression of $\Theta _{l_1,l_2}^{n,N}$ into Eq. (6), we get:

$$\begin{aligned} \Sigma _{n,N}^{0,0}& = {} \frac{2(N-n)\Gamma (2N-n+1)}{n!}\sum _{l_1=0}^n\frac{(-1)^{l_1}}{\Gamma (2N-2n+l_1+1)}\left( {\begin{array}{c}n\\ l_1\end{array}}\right) \nonumber \\{} & {}\quad \times \left[ \sum _{l_2=0}^n (-1)^{l_2}\left( {\begin{array}{c}n\\ l_2\end{array}}\right) \frac{\Gamma (l_2+2N-2n+l_1)}{\Gamma (l_2+2N-2n+1)}\right] \nonumber \\& = {} \frac{2(N-n)}{n!}\sum _{l_1=0}^n\frac{(-1)^{l_1}}{2N-2n+l_1}\displaystyle \left( {\begin{array}{c}n\\ l_1\end{array}}\right) \frac{\Gamma (n-l_1+1)}{\Gamma (1-l_1)}, \end{aligned}$$

(A1)

where we have used the entry 0.160(2) of Ref. [66], i.e.

$$\begin{aligned} \sum _{r=0}^{n}(-1)^r \left( {\begin{array}{c}n\\ r\end{array}}\right) \frac{\Gamma (r+b)}{\Gamma (r+a)}=\frac{B(n+a-b,b)}{\Gamma (a-b)}, \end{aligned}$$

(A2)

to evaluate the $l_2$-summation and B denotes the Eulerian beta-function. Let us decompose Eq. (A1) into its components as follows:

$$\begin{aligned} \Sigma _{n,N}^{0,0}& = {} \frac{2N-2n}{n!}\bigg [\frac{\Gamma (n+1)}{2N-2n}-\frac{1}{2N-2n+1}\frac{\Gamma (n+1)}{\Gamma (0)} \nonumber \\{} & {} \qquad+\frac{1}{\pi }\sum _{l_1=2}^n (-1)^{l_1}\left( {\begin{array}{c}n\\ l_1\end{array}}\right) \frac{\Gamma (l_1)\Gamma (n-l_1+1)}{2N-2n+l_1}\,\sin \pi l_1\bigg ] \nonumber \\& = {} \frac{2N-2n}{n!}\left[ \frac{\Gamma (n+1)}{2N-2n}-\frac{1}{2N-2n+1}\frac{\Gamma (n+1)}{\Gamma (0)}\right] , \end{aligned}$$

(A3)

where we have used the reflection formula $\Gamma (k)\Gamma (1-k)=\pi /\sin k\pi$ in the last summation which vanishes because $l_1$ was restricted to integer-values. Noting that $(-n)_n\equiv \Gamma (0)/\Gamma (-n)=(-1)^n\Gamma (n+1)$ and using once more the reflection formula, i.e. $\Gamma (-n)\Gamma (n+1)=-\pi /\sin n\pi$, then Eq. (A3) is reduced to

$$\begin{aligned} \sum _{l_1=0}^n\sum _{l_2=0}^n\Theta _{l_1,l_2}^{n,N}& = {} \frac{2(N-n)}{n!}\left[ \frac{\Gamma (n+1)}{2N-2n}+\frac{(-1)^n}{\pi }\frac{\Gamma (n+1)}{2N-2n+1}\,\sin n\pi \right] , \nonumber \\& = {} 1, \end{aligned}$$

(A4)

since $\sin n\pi =0$, for all positive-integers n. This complete the demonstration of the normalized-summation (6) from which aim the normalization of Wigner’s distribution function.

Cases $k\ne 0,m=0$ or $k=0,m\ne 0$. These particular cases lead to evaluate the terms that arise in Eqs. (39) and (40). Both cases are equivalent since all $\Theta$’s are invariant under permutation of $l_1$ and $l_2$. For this purpose, the use of Eq. (A3) and the identity $\Gamma (0)=(-1)^n\Gamma (-n)\Gamma (n+1)=-\pi (-1)^n/\sin n\pi$ lead to express $\Sigma _{n,N}^{k,0}$ as follows:

$$\begin{aligned} \Sigma _{n,N}^{k,0}& = \; \frac{2(N-n)}{n!}\bigg [\frac{\Gamma (n+1)}{2N-2n}\,l_1^k\bigg \vert _{l_1=0} -\frac{l_1^k}{2N-2n+1}\frac{\Gamma (n+1)}{\Gamma (0)}\bigg \vert _{l_1=1} \nonumber \\ &\; \qquad+\frac{1}{\pi }\sum _{l_1=2}^n (-1)^{l_1}l_1^k\left( {\begin{array}{c}n\\ l_1\end{array}}\right) \frac{\Gamma (l_1)\Gamma (n-l_1+1)}{2N-2n+l_1}\,\sin \pi l_1\bigg ] \nonumber \\ & = \; \frac{2(N-n)}{2N-2n+1}\,(-1)^n\frac{\sin n\pi }{\pi }\nonumber \\& = \; 0, \end{aligned}$$

(A5)

applied for all $n\in {\mathbb {N}}$ and for all $k\ne 0$. The same value is obtained for $\Sigma _{n,N}^{0,m}$.

Case $k=m=1$. This case makes it possible to evaluate the remaining-mixed term $l_1l_2$ involved in Eq. (40). By reducing the product $l_1l_2$ with both binomials and using Eq. (A2), the decomposition on its components gives

$$\begin{aligned} \Sigma _{n,N}^{1,1}& = \; \frac{2(N-n)\,n^2}{n!}\sum _{l_1=0}^{n-1}(-1)^{l_1}\left( {\begin{array}{c}n-1\\ l_1\end{array}}\right) \frac{\Gamma (n-l_1-1)}{\Gamma (-l_1)}\nonumber \\ & = \; \frac{2(N-n)\,n^2}{n!}\bigg [\frac{\Gamma (n-1)}{\Gamma (0)}+\sum _{l_1=1}^{n-2}(-1)^{l_1}\left( {\begin{array}{c}n-1\\ l_1\end{array}}\right) \frac{\Gamma (n-l_1-1)}{\Gamma (-l_1)} \nonumber \\ &\qquad \; +(-1)^{n-1}\frac{\Gamma (0)}{\Gamma (1-n)}\bigg ], \end{aligned}$$

(A6)

and according to the relation $\Gamma (0)=(-1)^n\Gamma (-n)\Gamma (n+1)=-\pi (-1)^n/\sin n\pi$ used hereinabove, the first-term and the summation in Eq. (A6) vanish, while the last-term gives

$$\begin{aligned}\Sigma_{n,N}^{1,1} = \frac{2(N-n)\,n^2}{n!}(-1)^{n-1}(-1)^{n-1}\Gamma (n) \equiv 2n(N-n). \end{aligned}$$

(A7)

Both Eqs. (A5) and (A7) give the expected value established in Eq. (40).

Appendix B: Derivation of Eqs. (11) and (12)

In this appendix, we will provide the derivation of Eqs. (11) and (12). To this end, let us recall (as given in the text) that $f_{\beta ',\lambda }(y)=\Gamma (\beta )\,e^{i \lambda y}$, with $\beta (y)=\beta '-i y$, and choosing to define $f_{\beta ',\lambda }(y)$ as a composite-function

$$\begin{aligned} f_{\beta ',\lambda }(y):=f_{\beta ',\lambda }\circ h(\beta )\equiv e^{h(\beta )}=\Gamma (\beta )\,e^{i \lambda y}, \end{aligned}$$

(B8)

where $h(\beta )=\ln \Gamma (\beta )-\lambda \beta +C$ and $C=\lambda \beta '$. Because the involved $\beta$-function is imaginary-linear in y, then the m-fold differentiation of $f_{\beta ',\lambda }(y)$ with respect to the variable y can be expressed as

$$\begin{aligned} \frac{\partial ^m}{\partial y^m}\,f_{\beta ',\lambda }(y) \equiv \frac{\partial ^m\beta }{\partial y^m}\frac{\partial ^m}{\partial \beta ^m} \,f_{\beta ',\lambda }\circ h(\beta ) = (-i)^m \,\frac{\partial ^m}{\partial \beta ^m} \,f_{\beta ',\lambda }\circ h(\beta ). \end{aligned}$$

(B9)

Now in order to evaluate properly Eq. (B9), we will use the Faà di Bruno formula (see, for instance, Ref. [67]) which has been applied suitably to a composite-function $f_{\beta ',\lambda }\circ h(\beta )=f_{\beta ',\lambda }[h(\beta )]$ in order to compute its mth-derivative in terms of derivatives of $f_{\beta ',\lambda }$ and h. It is written in the form

$$\begin{aligned} \frac{\partial ^m}{\partial \beta ^m}\,f_{\beta ',\lambda }\circ h(\beta )= \sum _{L\in \tau }\frac{m!}{\mu _1!\mu _2!\cdots \mu _m!}\left( \frac{\partial ^Lf_{\beta ',\lambda }}{\partial h^L(\beta )}\,\circ h(\beta )\right) \prod _{s=1}^m\left( \frac{h^{(s)}(\beta )}{s!}\right) ^{\mu _s}, \end{aligned}$$

(B10)

where the summation ranges over the set

$$\begin{aligned} \tau =\left\{ \mu _s\in {\mathbb {N}},\ \sum _{s=1}^ms\mu _s=m\quad \textrm{and}\quad \sum _{s=1}^m\mu _s=L\right\} , \end{aligned}$$

with $L=1,2,\ldots ,m$. Then from Eq. (B8), it follows straightforwardly that

$$\begin{aligned} \frac{\partial ^Lf_{\beta ',\lambda }}{\partial h^L}\,\circ h(\beta ) \equiv \frac{d^L}{d h^L}\,e^{h(\beta )} = \frac{d^{L-1}}{d h^{L-1}}\,e^{h(\beta )} =\cdots = \frac{d^2}{d h^2}\,e^{h(\beta )} = \frac{d}{d h}\,e^{h(\beta )} = e^{h(\beta )}, \end{aligned}$$

(B11)

where it is easy to verify that successive derivations of $h(\beta )$ give the mth-derivative of $h(\beta )$, i.e.

$$\begin{aligned} h'(\beta ) & = \; \frac{d}{d\beta }[\ln \Gamma (\beta )-\lambda \beta +C]=\psi (\beta )-\lambda \equiv \psi _\lambda (\beta ), \\ h''(\beta ) & = \; \psi '(\beta ),\qquad \ldots , \qquad h^{(m)}(\beta )=\psi ^{(m-1)}(\beta ), \end{aligned}$$

where $\psi ^{(m)}(\beta )$ refer to polygamma functions of the differentiation-order m and $\psi (\beta )$ denotes the digamma function. These considerations lead us to recast Eq. (B9), using Eq. (B10), as follows:

$$\begin{aligned} \frac{\partial ^m}{\partial y^m}\,f_{\beta ',\lambda }(y) & = \; (-i)^m \sum _{L\in \tau }\frac{m!}{\mu _1!}\,\psi _\lambda ^{\mu _1}(\beta )\prod _{s=2}^m\frac{1}{\mu _s!}\left[ \frac{\psi ^{(s-1)}(\beta )}{s!}\right] ^{\mu _s} \Gamma (\beta )\,e^{i \lambda y} \nonumber \\ & = \; (-i)^m\,\Xi ^{(m)}[\beta (y)]\,f_{\beta ',\lambda }(y), \end{aligned}$$

(B12)

which is Eq. (10) and making further identifications in Eq. (B12), the functional $\Xi ^{(m)}[\beta (y)]$ is expressed by

$$\begin{aligned} \Xi ^{(m)}[\beta (y)]=\sum _{L\in \tau } \mathcal C_{\mu _1,\mu _2,\ldots ,\mu _m}^{(m)}\psi _\lambda ^{\mu _1}(\beta ) \prod _{s=2}^m\left[ \psi ^{(s-1)}(\beta )\right] ^{\mu _s}, \end{aligned}$$

(B13)

where the coefficients ${\mathcal {C}}_{\mu _1,\mu _2,\ldots ,\mu _m}^{(m)}$ are identified, from Eqs. (B12) and (B13), as follows:

$$\begin{aligned} {\mathcal {C}}_{\mu _1,\mu _2,\ldots ,\mu _m}^{(m)} = m!\prod _{s=1}^m \frac{1}{\mu _s!}\left( \frac{1}{s!}\right) ^{\mu _s}. \end{aligned}$$

(B14)

This completes the determination of Eqs. (11) and (12).