三角势垒下的隧穿时间The Tunnel Time in a Triangular Potential

DOI: 10.12677/AAM.2021.104130, PDF, HTML, XML, 下载: 13  浏览: 39

Abstract: Quantum tunneling is a pure quantum phenomenon without any classical analogy. Whether it is instantaneous process or a process with finite time is an intensively debated issue since the early days of quantum mechanics. In this paper, we calculate the Wigner phase time and dwell time for a particle tunneling through a triangle barrier. We find that with the change of wave number k, the transmission phase time and the reflection phase show two opposite trends, while the dwell time changes the same as the transmission phase time. However, the phase time and dwell time both appear peak and oscillate.

1. 引言

2. 三角势垒下薛定谔方程的解

${V}_{tri}=\left\{\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z<0或z\ge l\\ \alpha z\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le z (1)

Figure 1. Particles pass through a triangular barrier

$i\hslash \frac{\partial }{\partial t}\Psi \left(z,t\right)=\left[-\frac{{\hslash }^{2}}{2m}+V\left(z\right)\right]\Psi \left(z,t\right)$ (2)

${{\psi }^{″}}_{E}\left(z\right)+\frac{2mE}{{\hslash }^{2}}\left(1-\frac{V\left(z\right)}{E}\right){\psi }_{E}\left(z\right)=0$ (3)

${\psi }_{I}={\text{e}}^{ikz}+R{\text{e}}^{-ikz},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z<0$ (4)

${\psi }_{II}={c}_{1}\text{Ai}\left[-{E}_{D}+{z}_{D}\right]+{c}_{2}\text{Bi}\left[-{E}_{D}+{z}_{D}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le z (5)

${\psi }_{III}=T{\text{e}}^{-ikz},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}l\le z$ (6)

$1+R={c}_{1}\text{Ai}\left[-{E}_{D}\right]+{c}_{2}\text{Bi}\left[-{E}_{D}\right]$ (7)

$ik\left(1-R\right)=\frac{1}{{L}_{C}}\left({c}_{1}Ai\prime \left[-{E}_{D}\right]+{c}_{2}B{i}^{\prime }\left[-{E}_{D}\right]\right)$ (8)

$T{\text{e}}^{ikl}={c}_{1}\text{Ai}\left[{l}_{d}-{E}_{D}\right]+{c}_{2}\text{Bi}\left[{l}_{d}-{E}_{D}\right]$ (9)

$ikT{\text{e}}^{ikl}=\frac{1}{{L}_{C}}\left({c}_{1}Ai\prime \left[{l}_{d}-{E}_{D}\right]+{c}_{2}B{i}^{\prime }\left[{l}_{d}-{E}_{D}\right]\right)$ (10)

$R=1+\frac{-2+2ik{L}_{C}P}{M+k{L}_{C}\left(iF+k{L}_{C}Q\right)};T=\frac{2i{\text{e}}^{-ikl}Jk{L}_{C}}{M+k{L}_{C}\left(iF+k{L}_{C}Q\right)}$

${c}_{1}=-\frac{2k{L}_{C}\left(iB{i}^{\prime }\left[{l}_{d}-{E}_{D}\right]+\text{Bi}\left[{l}_{d}-{E}_{D}\right]k{L}_{C}\right)}{M+k{L}_{C}\left(iF+k{L}_{C}Q\right)}$ (11)

${c}_{2}=\frac{2k{L}_{C}\left(iAi\prime \left[{l}_{d}-{E}_{D}\right]+\text{Ai}\left[{l}_{d}-{E}_{D}\right]k{L}_{C}\right)}{M+k{L}_{C}\left(iF+k{L}_{C}Q\right)}$

(7)~(10)式中的M、P、Q、J、F分别为：

$M=Ai\prime \left[-{E}_{D}+{l}_{d}\right]B{i}^{\prime }\left[-{E}_{D}\right]-Ai\prime \left[-{E}_{D}\right]B{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]$

$P=-\text{Bi}\left[-{E}_{D}+{l}_{d}\right]A{i}^{\prime }\left[-{E}_{D}\right]+\text{Ai}\left[-{E}_{D}+{l}_{d}\right]B{i}^{\prime }\left[-{E}_{D}\right]$

$Q=\text{Ai}\left[-{E}_{D}+{l}_{d}\right]\text{Bi}\left[-{E}_{D}\right]-\text{Ai}\left[-{E}_{D}\right]\text{Bi}\left[-{E}_{D}+{l}_{d}\right]$ (12)

$J=\text{Bi}\left[-{E}_{D}+{l}_{d}\right]A{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]-\text{Ai}\left[-{E}_{D}+{l}_{d}\right]B{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]$

$\begin{array}{c}F=\text{Bi}\left[-{E}_{D}+{l}_{d}\right]A{i}^{\prime }\left[-{E}_{D}\right]+\text{Bi}\left[-{E}_{D}\right]A{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]\\ \text{\hspace{0.17em}}-\text{Ai}\left[-{E}_{D}+{l}_{d}\right]B{i}^{\prime }\left[-{E}_{D}\right]-\text{Ai}\left[-{E}_{D}\right]B{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]\end{array}$

Figure 2. Wave function modulo as a function of z

3. 三角势垒的相位时间

$T\left(k\right){\text{e}}^{i\left[\alpha \left(k\right)+kx-E\left(k\right)t/\hslash \right]}$ (13)

$R\left(k\right){\text{e}}^{i\left[\beta \left(k\right)+kx-E\left(k\right)t/\hslash \right]}$ (14)

$\frac{\text{d}\alpha }{\text{d}k}+{x}_{p}\left(t\right)-\frac{1}{\hslash }\frac{\text{d}E}{\text{d}k}t=0$ (15)

$\delta {\tau }_{T}^{P}=\hslash \frac{\text{d}\alpha }{\text{d}E}=\frac{1}{v\left(k\right)}\frac{\text{d}\alpha }{\text{d}k}$ (16)

$\delta {\tau }_{R}^{P}=\hslash \frac{\text{d}\beta }{\text{d}E}=\frac{1}{v\left(k\right)}\frac{\text{d}\beta }{\text{d}k}$ (17)

${\tau }_{T}^{P}=\frac{-{g}^{2}\left({F}^{2}{k}^{2}l+FgM+{g}^{2}l{M}^{2}+2F{k}^{2}lP\right)+Fg{k}^{2}Q+{k}^{4}l{Q}^{2}}{{F}^{2}{g}^{2}{k}^{3}+k{\left({g}^{2}M+{k}^{2}Q\right)}^{2}}$ (18)

$\begin{array}{c}H=\text{Bi}\left[-{E}_{D}+{l}_{d}\right]A{i}^{\prime }\left[-{E}_{D}\right]-\text{Bi}\left[-{E}_{D}\right]A{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]\\ \text{\hspace{0.17em}}-\text{Ai}\left[-{E}_{D}+{l}_{d}\right]B{i}^{\prime }\left[-{E}_{D}\right]+\text{Ai}\left[-{E}_{D}\right]B{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]\end{array}$

$G=\text{Bi}\left[-{E}_{D}\right]A{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]-\text{Ai}\left[-{E}_{D}\right]B{i}^{\prime }\left[-{E}_{D}+{l}_{d}\right]$

${q}_{1}=iFgk-{g}^{2}M-{k}^{2}Q$ , ${q}_{2}=-iFgk-{g}^{2}M-{k}^{2}Q$ (19)

${q}_{3}=-gM+ikP$ , ${q}_{4}=-gM-ikP$

${q}_{5}=-\frac{{k}^{2}}{{g}^{3}}+l$ , ${q}_{6}=-\frac{2{k}^{2}}{{g}^{4}}+\frac{2l}{g}$

$\begin{array}{c}{\tau }_{R}^{p}=\left(i\left(1-\frac{2g{q}_{4}}{{q}_{2}}\right)\left(-{q}_{2}\left({q}_{2}-2g{q}_{4}\right)\left(-2i{k}^{4}Q{q}_{1}-2{g}^{2}{k}^{2}\left(-iM{q}_{1}+\left(G+H\right)k{q}_{3}\right)\\ \text{\hspace{0.17em}}\text{ }\text{ }+2g{k}^{3}\left(G{q}_{1}+ikQ{q}_{3}\right)-{g}^{4}\left(2kQ{q}_{3}+P{q}_{1}\left(i+2k{q}_{5}\right)\right)+{g}^{6}kP{q}_{3}{q}_{6}\\ \text{ }\text{ }\text{ }\text{\hspace{0.17em}}-i{g}^{5}{q}_{3}\left(F-{k}^{2}Q{q}_{6}\right)\right)+{q}_{1}\left({q}_{1}-2g{q}_{3}\right)\left(2i{k}^{4}Q{q}_{2}-2{g}^{2}{k}^{2}\left(iM{q}_{2}+\left(G+H\right)k{q}_{4}\right)\\ \text{\hspace{0.17em}}\text{ }\text{ }\text{ }+2g{k}^{3}\left(G{q}_{2}-ikQ{q}_{4}\right)+{g}^{4}\left(-2kQ{q}_{4}+P{q}_{2}\left(i-2k{q}_{5}\right)\right)\\ \begin{array}{c}{}_{}{}^{}\\ \end{array}+{g}^{6}kP{q}_{4}{q}_{7}+i{g}^{5}{q}_{4}\left(F-{k}^{2}Q{q}_{6}\right)\right)\right)\right)/\left({g}^{3}{q}_{1}\left(k{q}_{1}-2gk{q}_{3}\right){\left({q}_{2}-2g{q}_{4}\right)}^{2}\right)\end{array}$ (20)

Figure 3. Phase time as a function of wave number k

Table 1. Transmission time peaks and corresponding k values at different heights with the same barrier widths

Table 2. Transmission time peaks and corresponding k values at different width with the same barrier height

4. 三角势垒的驻留时间

${\tau }_{D}\left({x}_{1},{x}_{2},k\right)=\frac{1}{j\left(k\right)}{\int }_{{x}_{1}}^{{x}_{2}}\text{d}x{|\Psi \left(x,k\right)|}^{2}$ (21)

$\begin{array}{c}{\tau }_{D}=\frac{1}{k}\left({l}_{1}+{l}_{1}\left(1-\frac{2g{q}_{3}}{{q}_{1}}\right)\left(1-\frac{2g{q}_{4}}{{q}_{2}}\right)+\frac{{\text{e}}^{-ik{l}_{1}}\left(1+\frac{{\text{e}}^{2ik{l}_{1}}\left({q}_{1}-2g{q}_{3}\right)}{{q}_{1}}-\frac{2g{q}_{4}}{{q}_{2}}\right)\mathrm{sin}\left[k{l}_{1}\right]}{k}\\ \text{\hspace{0.17em}}-\frac{4{g}^{2}{k}^{2}\left(l-{l}_{2}\right){\left(\text{Bi}\left[{l}_{d}-{E}_{D}\right]A{i}^{\prime }\left[{l}_{d}-{E}_{D}\right]-\text{Ai}\left[{l}_{d}-{E}_{D}\right]B{i}^{\prime }\left[{l}_{d}-{E}_{D}\right]\right)2}^{}}{{F}^{2}{g}^{2}{k}^{2}+{\left({g}^{2}M+{k}^{2}Q\right)}^{2}}\\ \text{\hspace{0.17em}}-\frac{8}{Ecg{q}_{1}{q}_{2}}\left(E\text{Ai}\left[-{E}_{D}\right]\text{Bi}\left[-{E}_{D}\right]-E\text{Ai}\left[{l}_{d}-{E}_{D}\right]\text{Bi}\left[{l}_{d}-{E}_{D}\right]+Ecgl\text{Ai}\left[{l}_{d}-{E}_{D}\right]\text{Bi}\left[{l}_{d}-{E}_{D}\right]\\ \text{\hspace{0.17em}}+EcAi\prime \left[-{E}_{D}\right]B{i}^{\prime }\left[-{E}_{D}\right]-EcAi\prime \left[{l}_{d}-{E}_{D}\right]B{i}^{\prime }\left[{l}_{d}-{E}_{D}\right]\right)\left({k}^{4}\text{Ai}\left[{l}_{d}-{E}_{D}\right]\text{Bi}\left[{l}_{d}-{E}_{D}\right]\end{array}$

$\begin{array}{c}\text{\hspace{0.17em}}\text{ }+{g}^{2}{k}^{2}Ai\prime \left[{l}_{d}-{E}_{D}\right]B{i}^{\prime }\left[{l}_{d}-{E}_{D}\right]\right)+\frac{1}{{g}^{3}{q}_{1}{q}_{2}}4{k}^{2}\left(\left({k}^{2}\text{Ai}{\left[-{E}_{D}\right]}^{2}+{g}^{2}\left(\left(-{k}^{2}+gl\right)\text{Ai}{\left[{l}_{d}-{E}_{D}\right]}^{2}\\ \text{\hspace{0.17em}}\text{ }+Ai\prime {\left[-{E}_{D}\right]}^{2}-Ai\prime {\left[{l}_{d}-{E}_{D}\right]}^{2}\right)\right)\left({k}^{2}\text{Bi}{\left[{l}_{d}-{E}_{D}\right]}^{2}+{g}^{2}B{i}^{\prime }{\left[{l}_{d}-{E}_{D}\right]}^{2}\right)\\ \text{\hspace{0.17em}}\text{ }+\left({k}^{2}\text{Ai}{\left[{l}_{d}-{E}_{D}\right]}^{2}+{g}^{2}Ai\prime {\left[{l}_{d}-{E}_{D}\right]}^{2}\right)\left({k}^{2}\text{Bi}{\left[-{E}_{D}\right]}^{2}+{g}^{2}\left(\left(-{k}^{2}+gl\right)\text{Bi}{\left[{l}_{d}-{E}_{D}\right]}^{2}\\ \begin{array}{c}\text{ }\\ \text{ }\\ \text{ }\\ \text{ }\end{array}+B{i}^{\prime }{\left[-{E}_{D}\right]}^{2}-B{i}^{\prime }{\left[{l}_{d}-{E}_{D}\right]}^{2}\right)\right)\right)\right)\end{array}$ (22)

Figure 4. The dwell time as a function of wave number k

Table 3. Dwell time peaks and corresponding k values at different width with the same barrier height

Table 4. Dwell time peaks and corresponding k values at different width with the same barrier height

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