Question

The table gives relative values for three situations for the barrier tunneling experiment of Figs. 38-16 and 38-17. Rank the situations according to the probability of the electron tunneling through the barrier, greatest first.

	Electron Energy	Barrier Height	Barrier Thickness
(a)	E	5E	L
(b)	E	17E	L/2
(c)	E	2E	2L

Short answer

The rank is $T_{\left(a\right)}=T_{\left(b\right)}=T_{\left(c\right)}$

Step-by-step solution

Describe the expression for probability of electron

The probability that is given by the transmission coefficient T is as follows.

$\mathit{T}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{b}\mathbf{L}}$

where, $\mathit{b}\mathbf{=}\sqrt{\frac{\mathbf{8}{\mathbf{\pi }}^{\mathbf{2}}\mathbf{m}\left(U-E\right)}{{\mathbf{h}}^{\mathbf{2}}}}$, U is barrier of height, and L is thickness.

According to these equations, the transmission coefficient decreases as the potential height increase, and also decreases as the energy of the incident beam decreases, it can be observed that, it depends on the value of $\sqrt{\mathbf{U}\mathbf{-}\mathbf{E}}$, as the value of $\sqrt{\mathbf{U}\mathbf{-}\mathbf{E}}$increases the transmission coefficient decrease. Also the transmission coefficient depends on width of the barrier, so,$\mathit{T}\mathbf{\propto }{\mathit{e}}^{\mathbf{-}\mathbf{L}\sqrt{\mathbf{U}\mathbf{-}\mathbf{E}}}$

Rank the the probability of the electron tunneling through the barrier:

a) The transmission coefficient for the case (a) is given by,

$$\begin{gathered} T_{\left(a\right)}\propto e^{-L\sqrt{4E}} \\ \propto e^{-2L\sqrt{E}} \end{gathered}$$

b) The transmission coefficient for the case (b) is given by,

$$\begin{gathered} T_{\left(b\right)}\propto e^{-\frac{L}{2}\sqrt{16E}} \\ \propto e^{-2L\sqrt{E}} \end{gathered}$$

c) The transmission coefficient for the case (c) is given by,

$$\begin{gathered} T_{\left(c\right)}\propto e^{-2L\sqrt{2E-E}} \\ \propto e^{-2L\sqrt{E}} \end{gathered}$$

Therefore, all the transmission coefficients are equal.

Hence, the rank is $T_{\left(a\right)}=T_{\left(b\right)}=T_{\left(c\right)}$.