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Fractional Double δ-Potential in Fractional Dimensional Space
DOI: 10.12677/MP.2021.114011, PDF, HTML, XML, 下载: 220  浏览: 343  国家自然科学基金支持

Abstract: In this paper, we study the fractional-order quantum mechanics problems in the fractional dimen-sional space. The fractional Schrödinger equation with Riesz fractional derivative in dimensional space is considered. By using the Fourier transform in the fractional dimensional space, the fractional Schrödinger equation with fractional double δ-potential well in fractional dimensional space is solved and obtained the wave function with the form of Fox’s H functions and the energy eigenvalue. In addition, by using the properties of Fox’s H functions, we study the asymptotic properties of the wave function when the independent variable and the double delta potential interval a tending to zero and infinity, and give the specific asymptotic expressions. It is found that the behavior of wave function is a negative power law function that contains space di-mension under two kinds of infinite trends, the close relationship between space dimension and wave function, fractional calculus and negative power law is revealed.

1. 引言

1918年，Hausdorff引入了分数维的概念，在Mandelbrot发现了分形几何之后，这个概念变得更加的重要，他利用分数的概念用标度法计算出了分数维和整数维之间的关系 [1]。与分形几何和分数维空间相关的是分数导数和分数积分领域，这些研究最近已经在许多领域中得到应用，包括分数哈密顿系统、混沌动力学、分形和复杂介质物理学以及标度现象等的研究 [2] [3] [4] [5] [6]。在量子物理学中，Feynman首次将分形概念成功用于量子力学，Feynman和Hibbs重新将非相对论性量子力学表述为布朗路径上的路径积分 [7]。之后，Laskin在路径积分中用Lévy路径代替布朗路径，将Feynman路径积分推广到Lévy路径积分，到得到了含有Riesz分数导数的空间分数阶薛定谔方程，建立了分数阶量子力学 [8] [9] [10]。

2. 傅里叶变换与薛定谔方程

${\text{d}}^{\lambda }x=\frac{{\pi }^{\lambda /\text{2}}|x|}{\Gamma \left(\lambda /\text{2}\right)}\text{d}x$ (1)

$g\left(k\right)=F\left(f\left(x\right)\right)=\int f\left(x\right){\text{e}}^{ikx}{\text{d}}^{\lambda }x$ (2)

$f\left(x\right)={F}^{-1}\left(g\left(k\right)\right)={\left(\frac{1}{2\pi }\right)}^{\lambda }\int g\left(k\right){\text{e}}^{-ikx}{\text{d}}^{\lambda }k$ (3)

$\lambda$ 维分数空间中的广义狄拉克δ函数定义为

${\delta }^{\lambda }\left(x-{x}^{\prime }\right)={\left(\frac{1}{2\pi }\right)}^{\lambda }\int {\text{e}}^{ik\left(x-{x}^{\prime }\right)}{\text{d}}^{\lambda }k$ (4)

${\int }_{-\infty }^{+\infty }f\left(x\right){\delta }^{\lambda }\left(x-{x}_{0}\right){\text{d}}^{\lambda }x=f\left({x}_{0}\right)$ (5)

$0<\lambda <1$，维分数空间中的广义狄拉克δ函数满足以下恒等式 [20]

${\delta }^{\lambda }\left(x-{x}_{0}\right)=\underset{\epsilon \to \infty }{\mathrm{lim}}{\epsilon }^{\lambda }{\text{e}}^{-\pi {\epsilon }^{2}{\left(x-{x}_{0}\right)}^{2}}$ (6)

${\int }_{-\infty }^{+\infty }f\left(x\right){\delta }^{\lambda }\left(x-{x}_{0}\right){\text{d}}^{\lambda }x={\int }_{-\infty }^{+\infty }\underset{\epsilon \to \infty }{\mathrm{lim}}{\epsilon }^{\lambda }{\text{e}}^{-\pi {\epsilon }^{2}{\left(x-{x}_{0}\right)}^{2}}f\left(x\right){\text{d}}^{\lambda }x$ (7)

${\int }_{-\infty }^{+\infty }h\left(x+{x}_{0}\right){\text{d}}^{\lambda }x={\int }_{-\infty }^{+\infty }h\left(x\right){\text{d}}^{\lambda }x$ (8)

${\int }_{-\infty }^{+\infty }f\left(x\right){\delta }^{\lambda }\left(x-{x}_{0}\right){\text{d}}^{\lambda }x=\underset{\epsilon \to \infty }{\mathrm{lim}}{\int }_{-\infty }^{+\infty }{\epsilon }^{\lambda }{\text{e}}^{-\pi {\epsilon }^{2}{x}^{2}}f\left(x+{x}_{0}\right){\text{d}}^{\lambda }x$ (9)

${\int }_{0}^{\infty }{\text{e}}^{-k{x}^{m}}{x}^{n}\text{d}x=\frac{1}{m}{k}^{-\left(n+1\right)/m}\Gamma \left(\frac{n+1}{m}\right)$ (10)

${\int }_{-\infty }^{+\infty }f\left(x\right){\delta }^{\lambda }\left(x-{x}_{0}\right){\text{d}}^{\lambda }x=\frac{1}{\Gamma \left(\lambda /2\right)}\underset{\epsilon \to \infty }{\mathrm{lim}}f\left({x}_{0}\right){\int }_{0}^{+\infty }{y}^{\lambda /2-1}{\text{e}}^{-y}\text{d}y=f\left({x}_{0}\right)$ (11)

${\int }_{-\infty }^{\infty }f\left(x\right){\delta }^{\lambda }\left(x\right){\text{d}}^{\lambda }x=f\left(0\right)$ (12)

$f\left(x\right)=1$ 时，

${\int }_{-\infty }^{\infty }{\delta }^{\lambda }\left(x\right){\text{d}}^{\lambda }x=1$ (13)

$\varphi \left(p\right)={\int }_{-\infty }^{\infty }\phi \left(x\right){\text{e}}^{ipx/\hslash }{\text{d}}^{\lambda }x$ (14)

$\phi \left(x\right)={\left(\frac{1}{2\pi \hslash }\right)}^{\lambda }{\int }_{-\infty }^{\infty }\varphi \left(p\right){\text{e}}^{-ipx/\hslash }{\text{d}}^{\lambda }p$ (15)

${\left(2\pi \hslash \right)}^{\lambda }E\varphi \left(p\right)={\left(2\pi \hslash \right)}^{\lambda }{D}_{\alpha }{|p|}^{\alpha }\varphi \left(p\right)+{\int }_{-\infty }^{+\infty }{\text{e}}^{-ix\left(p-{p}^{\prime }\right)/\hslash }V\left(x\right){\text{d}}^{\lambda }x\varphi \left({p}^{\prime }\right){\text{d}}^{\lambda }{p}^{\prime }$ (16)

3. 分数阶双δ-势

$V\left(x\right)=-\gamma \left[{\delta }^{\lambda }\left(x\right)+{\delta }^{\lambda }\left(x-a\right)\right]$ (17)

$\begin{array}{l}{\left(2\pi \hslash \right)}^{\lambda }E\varphi \left(p\right)={\left(2\pi \hslash \right)}^{\lambda }{D}_{\alpha }{|p|}^{\alpha }\varphi \left(p\right)-\gamma {\int }_{-\infty }^{+\infty }{\text{e}}^{-ix\left(p-{p}^{\prime }\right)/\hslash }{\delta }^{\lambda }\left(x\right){\text{d}}^{\lambda }x\varphi \left({p}^{\prime }\right){\text{d}}^{\lambda }{p}^{\prime }\\ \text{}-\gamma {\int }_{-\infty }^{+\infty }{\text{e}}^{-ix\left(p-{p}^{\prime }\right)/\hslash }{\delta }^{\lambda }\left(x-a\right){\text{d}}^{\lambda }x\varphi \left({p}^{\prime }\right){\text{d}}^{\lambda }{p}^{\prime }\end{array}$ (18)

${\left(2\pi \hslash \right)}^{\lambda }\left({D}_{\alpha }{|p|}^{\alpha }-E\right)\varphi \left(p\right)=\gamma \left[{\int }_{-\infty }^{+\infty }\varphi \left({p}^{\prime }\right){\text{d}}^{\lambda }{p}^{\prime }+{\int }_{-\infty }^{+\infty }{\text{e}}^{-ia\left(p-{p}^{\prime }\right)/\hslash }\varphi \left({p}^{\prime }\right){\text{d}}^{\lambda }{p}^{\prime }\right]$ (19)

$\varphi \left(p\right)=\frac{\gamma \left(K\left(0\right)+{\text{e}}^{-iap/\hslash }K\left(a\right)\right)}{{D}_{\alpha }{|p|}^{\alpha }-E}$ (20)

$K\left(a\right)=\frac{1}{{\left(2\pi \hslash \right)}^{2}}{\int }_{-\infty }^{+\infty }{\text{e}}^{iaq/\hslash }\varphi \left(q\right){\text{d}}^{\lambda }q$ (21)

$K\left(0\right)=\frac{1}{{\left(2\pi \hslash \right)}^{2}}{\int }_{-\infty }^{+\infty }\varphi \left(q\right){\text{d}}^{\lambda }q$ (22)

$\varphi \left(p\right)$ 代入方程(18)联立 $K\left(0\right)$$K\left(a\right)$

${N}^{2}{\int }_{-\infty }^{\infty }\frac{{\text{e}}^{-iap/\hslash }}{{D}_{\alpha }{|p|}^{\alpha }-E}{\text{d}}^{\lambda }p={\int }_{-\infty }^{\infty }\frac{{\text{e}}^{iap/\hslash }}{{D}_{\alpha }{|p|}^{\alpha }-E}{\text{d}}^{\lambda }p$ (23)

$\frac{\pi \hslash }{aE\alpha }{\left(\frac{{D}_{\alpha }}{-E}\right)}^{-\left(\frac{\lambda -1}{\alpha }\right)}{H}_{2,3}^{2,1}\left[a{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(1-\left(\lambda -1\right)/\alpha ,1/\alpha \right),\left(1,1/2\right)\\ \left(1,1\right),\left(1-\left(\lambda -1\right)/\alpha ,1/\alpha \right),\left(1,1/2\right)\end{array}\right]=0$ (24)

$\varphi \left(p\right)$ 表达式(19)代入到 $K\left(0\right)$$K\left(a\right)$ 中有

$K\left(0\right)=\frac{\gamma }{{\left(2\pi \hslash \right)}^{2}}\left[I\left(0\right)K\left(0\right)+I\left(a/\hslash \right)K\left(a\right)\right]$ (25)

$K\left(a\right)=\frac{\gamma }{{\left(2\pi \hslash \right)}^{2}}\left[I\left(0\right)K\left(a\right)+I\left(a/\hslash \right)K\left(0\right)\right]$ (26)

$I\left(\omega \right)={\int }_{\text{0}}^{\infty }\frac{\mathrm{cos}p\omega /\hslash }{{D}_{\alpha }{|p|}^{\alpha }-E}{\text{d}}^{\lambda }p$ (27)

$\varphi \left(p\right)=\frac{\gamma K\left(0\right)\left(1+{\text{e}}^{-iap/t}\right)}{{D}_{\alpha }{|p|}^{\alpha }-E}$ (28)

$\phi \left(x\right)=\frac{\gamma K\left(0\right)}{{\left(2\pi \hslash \right)}^{2}}\left[{\int }_{-\infty }^{\infty }\frac{{\text{e}}^{-ipx/\hslash }}{{D}_{\alpha }{|p|}^{\alpha }-E}{\text{d}}^{\lambda }p+{\int }_{-\infty }^{\infty }\frac{{\text{e}}^{-ip\left(x+a\right)/\hslash }}{{D}_{\alpha }{|p|}^{\alpha }-E}{\text{d}}^{\lambda }p\right]$ (29)

$\phi \left(x\right)={\phi }_{1}\left(x\right)+{\phi }_{1}\left(x+a\right)$ (30)

${\phi }_{1}\left(x\right)={C}_{\alpha }^{\lambda }\frac{\pi \hslash }{\alpha }{H}_{2,3}^{2,1}\left[|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]$ (31)

${C}_{\alpha }^{\lambda }=\frac{\gamma K\left(0\right)}{{\left(2\pi \hslash \right)}^{2\lambda }}\frac{2{\pi }^{\lambda }/2}{\Gamma \left(\lambda /2\right)}\frac{1}{{\left({D}_{\alpha }\right)}^{\lambda -1/\alpha }{\left(-E\right)}^{\alpha +1-\lambda /\alpha }}$

$\begin{array}{l}\phi \left(x\right)={C}_{\alpha }^{\lambda }\frac{\pi \hslash }{\alpha }{H}_{2,3}^{2,1}\left[|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]\\ \text{}+{C}_{\alpha }^{\lambda }\frac{\pi \hslash }{\alpha }{H}_{2,3}^{2,1}\left[|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]\end{array}$ (32)

4. 波函数的渐进性质

$|x|\to \infty$ 时，利用Fox’s H函数的渐进性质 [22]

${H}_{2,3}^{2,1}\left[|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]\approx {\left(|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{-\lambda }$ (33)

${H}_{2,3}^{2,1}\left[|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]\approx {\left(|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{-\lambda }$ (34)

$\phi \left(x\right)\approx {C}_{\alpha }^{\lambda }\frac{\pi \hslash }{\alpha }\left[{\left(|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{-\lambda }+{\left(|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{-\lambda }\right]$ (35)

${H}_{2,3}^{2,1}\left[|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]\approx {\left(|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{0}$ (36)

${H}_{2,3}^{2,1}\left[|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]\approx {\left(|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{0}$ (37)

$\begin{array}{c}\phi \left(x\right)\approx {C}_{\alpha }^{\lambda }\frac{\pi \hslash }{\alpha }\left[{\left(|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{0}+{\left(|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{0}\right]\\ \approx \frac{\gamma \pi \hslash K\left(0\right)}{\alpha {\left(2\pi \hslash \right)}^{2\lambda }}\frac{2{\pi }^{\lambda }/2}{\Gamma \left(\lambda /2\right)}\frac{1}{{\left({D}_{\alpha }\right)}^{\lambda -1/\alpha }{\left(-E\right)}^{\alpha +1-\lambda /\alpha }}\end{array}$ (38)

$|x|\to 0$ 时， $a\to \infty$ 时，由Fox’s H函数的渐进性质我们得到

${H}_{2,3}^{2,1}\left[|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]\approx {\left(|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{0}$ (39)

${H}_{2,3}^{2,1}\left[|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}|\begin{array}{c}\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\\ \left(0,1\right),\left(\left(\alpha -\lambda \right)/\alpha ,1/\alpha \right),\left(1/2,1/2\right)\end{array}\right]\approx {\left(|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{-\lambda }$ (40)

$\begin{array}{c}\phi \left(x\right)\approx {C}_{\alpha }^{\lambda }\frac{\pi \hslash }{\alpha }\left[{\left(|x|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{0}+{\left(|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{-\lambda }\right]\\ \approx {C}_{\alpha }^{\lambda }\frac{\pi \hslash }{\alpha }\left[1+{\left(|x+a|{\left(\frac{{D}_{\alpha }{\hslash }^{\alpha }}{-E}\right)}^{\frac{-1}{\alpha }}\right)}^{-\lambda }\right]\end{array}$ (41)

5. 结论

NOTES

*通讯作者。

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