# 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

 [1] Trixler, F. (2013) Quantum Tunnelling to the Origin and Evolution of Life. Current Organic Chemistry, 17, 1758-1770. https://doi.org/10.2174/13852728113179990083 [2] Griffiths, D.J. (2016) Introduction to Quantum Mechanics. Cambridge University Press, Cambridge. [3] Condon, M. (1931) Quantum Mechanics of Collision Processes I. Scattering of Particles in a Definite Force Field. Review of Modern Physics, 3, 43-88. https://doi.org/10.1103/RevModPhys.3.43 [4] MacColl, L.A. (1932) Note on the Transmission and Reflection of Wave Packets by Potential Barriers. Physical Review, 40, 621-626. https://doi.org/10.1103/PhysRev.40.621 [5] Donoso, A. and Martens, C.C. (2001) Quantum Tunneling Using Entangled Classical Trajectories. Physical Review Letters, 87, Article ID: 223202. https://doi.org/10.1103/PhysRevLett.87.223202 [6] Pauli, W. (1933) Die allgemeinen Prinzipien der Wellenmechanik. In: Quantentheorie, Springer, Berlin, Handbuch der Physik, Vol. 24, 83-272. https://doi.org/10.1007/978-3-642-52619-0_2 [7] Hartman, T.E. (1962) Tunneling of a Wave Packet. Journal of Applied Physics, 33, 3427-3433. https://doi.org/10.1063/1.1702424 [8] Wigner, E.P. (1955) Lower Limit for the Energy Derivative of the Scattering Phase Shift. Physical Review, 98, 145-147. https://doi.org/10.1103/PhysRev.98.145 [9] Smith, F.T. (1960) Lifetime Matrix in Collision Theory. Physical Review, 119, 2098-2098. https://doi.org/10.1103/PhysRev.119.2098.4 [10] Buttiker, M. (1983) Larmor Precession and the Traversal Time for Tunneling. Physical Review B, 27, 6178-6188. https://doi.org/10.1103/PhysRevB.27.6178 [11] Keller, U., Gallmann, L., et al. (2014) Ultrafast Resolution of Tunneling Delay Time. Optica, 1, 343. https://doi.org/10.1364/OPTICA.1.000343 [12] Litvinyuk, I.V., Sang, R.T., et al. (2019) Attosecond Angular Streaking and Tunnelling Time in Atomic Hydrogen. Nature, 568, 75-77. https://doi.org/10.1038/s41586-019-1028-3 [13] Ramos, R., Spierings, D., Racicot, L. and Steinberg, A.M. (2020) Measurement of the Time Spent by a Tunneling Atom within the Barrier Region. Nature, 583, 529-532. https://doi.org/10.1038/s41586-020-2490-7 [14] Landauer, R. and Martin, Th. (1994) Barrier Interaction Time in Tunneling. Reviews of Modern Physics, 66, 217. https://doi.org/10.1103/RevModPhys.66.217 [15] Peres, A. (1980) Measurement of Time by Quantum Clocks. American Journal of Physics, 48, 552. https://doi.org/10.1119/1.12061 [16] Büttikerm, M. and Landauer, R. (1982) Traversal Time for Tunneling. Physical Review Letters, 49, 1739-1742. https://doi.org/10.1103/PhysRevLett.49.1739