For a general wave pulse neither E nor p (i.eneither $\mathit{\omega }$nor k) are weIldefined. But they have approximate values ${\mathbf{E}}_{\mathbf{0}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{p}}_{\mathbf{0}}$. Although it comprisemany plane waves, the general pulse has an overall phase velocity corresponding to these values.

${\mathit{v}}_{\mathbf{p}\mathbf{h}\mathbf{a}\mathbf{s}\mathbf{e}}\mathbf{=}\frac{{\mathbf{\omega }}_{\mathbf{0}}}{{\mathbf{k}}_{\mathbf{0}}}\mathbf{=}\frac{{\mathbf{E}}_{\mathbf{0}}\mathbf{/}\mathbf{h}}{{\mathbf{p}}_{\mathbf{0}}\mathbf{/}\mathbf{h}}\mathbf{=}\frac{{\mathbf{E}}_{\mathbf{0}}}{{\mathbf{p}}_{\mathbf{0}}}$

If the pulse describes a large massive particle. The uncertainties are reasonably small, and the particle may be said to have energy ${\mathit{E}}_{\mathbf{0}}$and momentum ${\mathit{p}}_{\mathbf{0}}$.Using the relativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater than c and given by

${\mathit{v}}_{\mathbf{p}\mathbf{h}\mathbf{a}\mathbf{s}\mathbf{e}}\mathbf{=}\frac{{\mathbf{c}}^{\mathbf{2}}}{{\mathbf{u}}_{\mathbf{p}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{i}\mathbf{c}\mathbf{l}\mathbf{e}}}$

Note that the phase velocity is greatest for particles whose speed is least.

Your friend has just finished classical physics and can’t wait to know what lies ahead. Keeping extraneous ideas and postulates to a minimum, Explain the process of Quantum-mechanical tunneling.

Exercise 54 gives a rough lifetime for a particle trapped particle to escape an enclosure by tunneling.

(a) Consider an electron. Given that $\mathbf{W}\mathbf{=}\mathbf{100}\mathbf{\text{\hspace{0.17em}nm}}\mathbf{,}\mathbf{L}\mathbf{=}\mathbf{1}\mathbf{\text{\hspace{0.17em}nm\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}}{\mathbf{U}}_{\mathbf{0}}\mathbf{=}\mathbf{5}\mathbf{\text{\hspace{0.17em}eV}}$, first verify that the ${\mathbf{E}}_{\mathbf{GS}}\mathbf{<}\mathbf{<}{\mathbf{U}}_{\mathbf{0}}$ assumption holds, then evaluate the lifetime.

(b) Repeat part (a), but for a $\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{\mu }\mathbf{g}$particle, with $\mathbf{W}\mathbf{=}\mathbf{1}\mathbf{nm}\mathbf{,}\mathbf{L}\mathbf{=}\mathbf{1}\mathbf{\mu }\mathbf{m}$ , and a barrier height ${\mathbf{U}}_{\mathbf{0}}$ that equals the energy the particle would have if its speed were just $\mathbf{1}\mathbf{\text{\hspace{0.17em}mm per year}}$.

Show that $\mathit{\psi }\left(x\right)\mathbf{=}\mathit{A}\mathbf{\text{'}}{\mathit{e}}^{\mathbf{i}\mathbf{k}\mathbf{x}}\mathbf{+}\mathit{B}\mathbf{\text{'}}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{k}\mathbf{x}}$ is equivalent to $\mathit{\psi }\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{A}\mathit{s}\mathit{i}\mathit{n}\mathit{k}\mathit{x}\mathbf{+}\mathit{B}\mathit{c}\mathit{o}\mathit{s}\mathit{k}\mathit{x}$, provided that $\mathit{A}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left(B-iA\right)$ $\mathit{B}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left(B+iA\right)$.

Suppose the tunneling probability is ${\mathbf{10}}^{\mathbf{-}\mathbf{12}}$for a wide barrier when E is $\frac{\mathbf{1}}{\mathbf{100}}{\mathit{U}}_{\mathbf{0}}$

(a) About how much smaller would it be if ’E’ were instead $\frac{\mathbf{1}}{\mathbf{1000}}{\mathit{U}}_{\mathbf{0}}$ ?

(b) If this case does not support the general rule that transmission probability is a sensitive function of E, what makes it exceptional?

Exercise 39 gives a condition for resonant tunneling through two barriers separated by a space width of $\mathbf{2}\mathbf{s}$, expressed I terms of factor $\mathbf{\beta }$ given in exercise 30. Show that in the limit in which barrier width $\mathbf{L}\mathbf{\to }\mathbf{\infty }$, this condition becomes exactly energy quantization condition (5.22) for finite well. Thus, resonant tunneling occurs at the quantized energies of intervening well.