Quantum Tunneling Time: Relativistic Extensions

Abstract

Several years ago, in quantum mechanics, Davies proposed a method to calculate particle’s traveling time with the phase difference of wave function. The method is convenient for calculating the sojourn time inside a potential step and the tunneling time through a potential hill. We extend Davies’ non-relativistic calculation to relativistic quantum mechanics, with and without particle-antiparticle creation, using Klein–Gordon equation and Dirac Equation, for different forms of energy-momentum relation. The extension is successful only when the particle and antiparticle creation/annihilation effect is negligible.

Appendix: Derivation of Expression (39)

Using the continuity conditions of the wave function (37) at x=0 and x=a, we can derive

$$\begin{aligned} D &= \frac{-4iMNe^{-ika}}{(N-iM)^2e^{pa}-(N+iM)^2e^{-pa}} \\ &= \frac{-4MN[-2MN(e^{pa}+e^{-pa})+i(N^2-M^2)(e^{pa}-e^{-pa})]e^{-ika}}{|(N^2-M^2)(e^{pa}-e^{-pa})-2iMN(e{pa}+e^{-pa})|^{2}}. \end{aligned}$$

(A.1)

For notational simplicity we have defined M=k/p, N=(E−m)/(E−V−m). Making use of this result and following the same logic in Sect. 3.2, one can prove that the phase change is

$$ \delta(E)=\arctan \biggl[\frac{M^2-N^2}{2MN}\tanh(pa) \biggr]. $$

(A.2)

Consequently, the tunneling time is found to be

$$ T=\frac{ (\frac{M^2-N^2}{2MN} )'\tanh(pa)+\frac{M^2-N^2}{2MN}[\tanh(pa)]'}{1+ (\frac{M^2-N^2}{2MN} )^2\tanh^2(pa)}. $$

(A.3)

Note that

$$\begin{aligned} \begin{aligned} \frac{1}{1+ (\frac{M^2-N^2}{2MN} )^2\tanh^2(pa)}&=\frac{4M^2N^2\cosh^2(pa)}{\cosh^2(pa)(M^2+N^2)^2-(M^2-N^2)^2}, \\ \biggl(\frac{M^2-N^2}{2MN} \biggr)'&=\frac{M^2+N^2}{2M^2N^2}\times \frac{m(E-m)(p^2+k^2)}{p^3k(m+V-E)}, \\ \bigl[\tanh(pa) \bigr]' &= \frac{a(V-E)}{p\cosh^2(pa)}, \end{aligned} \end{aligned}$$

(A.4)

we can rewrite (A.3) as

$$\begin{aligned} T =& \frac{2}{\cosh^2(pa)(M^2+N^2)^2-(M^2-N^2)^2} \\ &{} \times \biggl[\frac{(M^2+N^2)(p^2+k^2)(E-m)m\sinh(2pa)}{2p^3k(m+V-E)}+MN\bigl(M^2-N^2 \bigr)\frac{a(V-E)}{p} \biggr] \\ =&\bigl\{2(E-m) (m+V-E)\bigr\} \bigl\{\cosh^2(pa) \bigl[k^2(E-V-m)^2+p^2(E-m)^2 \bigr]^2 \\ &{}-\bigl[k^2(E-V-m)^2-p^2(E-m)^2 \bigr]^2\bigr\}^{-1} \\ &{} \times \biggl\{\frac{\sinh(2pa)}{2pk}\bigl[k^2(E-V-m)^2+p^2(E-m)^2 \bigr]\bigl(p^2+k^2\bigr) \\ &{}+ka(E-V)\bigl[k^2(E-V-m)^2-p^2(E-m)^2 \bigr] \biggr\}. \end{aligned}$$

(A.5)