Chapter 7, Problem 7D.12P

Interpretation Introduction

Interpretation:

The expression for the transmission probability has to be derived.  The expression for the condition when transmission probability reduces to $T\approx 16\epsilon \left(1-\epsilon \right){\text{e}}^{-\kappa W}$ when $\kappa W$ is much higher than $1$ has to be shown.

Concept introduction:

In quantum mechanics, the wavefunction is given by $\psi$.  The wavefunction contains all the information about the state of the system.  The wavefunction is the function of the coordinates of particles and time.

Answer to Problem 7D.12P

The expression for the transmission probability is shown below.

    $T={\left(\frac{{\left({\text{e}}^{\kappa W}-{\text{e}}^{-\kappa W}\right)}^2}{16\epsilon \left(1-\epsilon \right)}+1\right)}^{-1}$

When $\kappa W\gg 1$, then the expression for the transmission probability is shown below.

    $T=16\epsilon \left(1-\epsilon \right){\text{e}}^{2\kappa W}$

Explanation of Solution

The transmission probability is given by the expression shown below.

    $T=\frac{{\left|A\text{'}\right|}^2}{{\left|A\right|}^2}$                                                                                                        (1)

Where,

• $A$ and $A\text{'}$ are the coefficients.

At the edge of barrier $\left(x=0\right)$, the relationship between the coefficients $A$, $B$, $C$, and $D$ is given by the expression as shown below.

    $A+B=C+D$                                                                                               (2)

At the edge of barrier $\left(x=W\right)$, the relationship between the coefficients $A\text{'}$, $C$, and $D$ is given by the expression as shown below.

    $C{\text{e}}^{\kappa W}+D{\text{e}}^{-\kappa W}=A\text{'}{\text{e}}^{ikW}$                                                                                 (3)

Where,

• $W$ is the potential energy width.
• $\kappa$ is a constant.

At the edge of barrier $\left(x=0\right)$, the slope is given by the expression as shown below.

    $ikA-ikB=\kappa C-\kappa D$                                                                                      (4)

Where,

• $k$ is a constant.

At the edge of barrier $\left(x=0\right)$, the slope is given by the expression as shown below.    $\kappa C{\text{e}}^{\kappa W}-\kappa D{\text{e}}^{-\kappa W}=ikA\text{'}{\text{e}}^{ikW}$                                                                          (5)

Rearrange the equation (2) for the value of $B$.

    $B=C+D-A$                                                                                               (6)

Rearrange the equation (4) for the value of $B$.

    $B=\frac{ikA-\kappa C+\kappa D}{ik}$                                                                                        (7)

Substitute the value of equation (6) in equation (7).

    $\begin{array}{c}C+D-A=\frac{ikA-\kappa C+\kappa D}{ik}\\ C+D-A=A-\frac{\kappa C}{ik}+\frac{\kappa D}{ik}\end{array}$                                                                           (8)

Rearrange the equation (8) for the value of $C$.

    $C=\frac{2Aik+D\left(\kappa -ik\right)}{\kappa +ik}$                                                                                   (9)

Rearrange the equation (3) for the value of $A\text{'}$.

    $\begin{array}{c}A\text{'}=\frac{C{\text{e}}^{\kappa W}+D{\text{e}}^{-\kappa W}}{{\text{e}}^{ikW}}\\ A\text{'}=\left(C{\text{e}}^{\kappa W}+D{\text{e}}^{-\kappa W}\right){\text{e}}^{-ikW}\end{array}$                                                                           (10)

Rearrange the equation (5) for the value of $A\text{'}$.

    $A\text{'}=\frac{\kappa {\text{e}}^{-ikW}\left(C{\text{e}}^{\kappa W}-D{\text{e}}^{-\kappa W}\right)}{ik}$                                                                        (11)

Substitute the value of equation (10) in equation (11).

    $\left(C{\text{e}}^{\kappa W}+D{\text{e}}^{-\kappa W}\right){\text{e}}^{-ikW}=\frac{\kappa {\text{e}}^{-ikW}\left(C{\text{e}}^{\kappa W}-D{\text{e}}^{-\kappa W}\right)}{ik}$                                          (12)

Rearrange the equation (12) for the value of $C$.

  $\begin{array}{c}C=\frac{\left(\frac{\kappa }{ik}+1\right)D{\text{e}}^{-2\kappa W}}{\frac{\kappa }{ik}-1}\\ C=\frac{\left(\kappa +ik\right)D{\text{e}}^{-2\kappa W}}{\kappa -ik}\end{array}$                                                                                    (13)

Substitute the value of equation (9) in the equation (13).

    $\frac{\left(\kappa +ik\right)D{\text{e}}^{-2\kappa W}}{\kappa -ik}=\frac{2Aik+D\left(\kappa -ik\right)}{\kappa +ik}$                                                           (14)

Rearrange the equation (14) for the value of $D$.

    $D=\frac{2Aik\left(\kappa -ik\right)}{{\left(\kappa +ik\right)}^2{\text{e}}^{-2\kappa W}-{\left(\kappa -ik\right)}^2}$                                                                    (15)

Substitute the value of $D$ from equation (15) to the equation (13).

    $\begin{array}{c}C=\frac{\left(\kappa +ik\right)\left(\frac{2Aik\left(\kappa -ik\right)}{{\left(\kappa +ik\right)}^2{\text{e}}^{-2\kappa W}-{\left(\kappa -ik\right)}^2}\right){\text{e}}^{-2\kappa W}}{\kappa -ik}\\ C=\frac{2Aik\left(\kappa +ik\right){\text{e}}^{-2\kappa W}}{{\left(\kappa +ik\right)}^2{\text{e}}^{-2\kappa W}-{\left(\kappa -ik\right)}^2}\end{array}$                                          (16)

Substitute the values of $D$ from equation (15) and $C$ from equation (16) to the equation (10).

    $\begin{array}{c}A\text{'}=\left(\left(\frac{2Aik\left(\kappa +ik\right){\text{e}}^{-2\kappa W}}{{\left(\kappa +ik\right)}^2{\text{e}}^{-2\kappa W}-{\left(\kappa -ik\right)}^2}\right){\text{e}}^{\kappa W}+\left(\frac{2Aik\left(\kappa -ik\right)}{{\left(\kappa +ik\right)}^2{\text{e}}^{-2\kappa W}-{\left(\kappa -ik\right)}^2}\right){\text{e}}^{-\kappa W}\right){\text{e}}^{-ikW}\\ A\text{'}=\frac{2Aik}{{\left(\kappa +ik\right)}^2{\text{e}}^{-2\kappa W}-{\left(\kappa -ik\right)}^2}\left(\left(\kappa +ik\right){\text{e}}^{-\kappa W}+\left(\kappa -ik\right){\text{e}}^{-\kappa W}\right){\text{e}}^{-ikW}\\ A\text{'}=\frac{4Ai\kappa k{\text{e}}^{-\kappa W}{\text{e}}^{-ikW}}{{\left(\kappa +ik\right)}^2{\text{e}}^{-2\kappa W}-{\left(\kappa -ik\right)}^2}\\ A\text{'}=\frac{4Ai\kappa k{\text{e}}^{-ikW}}{{\left(\kappa +ik\right)}^2{\text{e}}^{-\kappa W}-{\left(\kappa -ik\right)}^2{\text{e}}^{\kappa W}}\end{array}$

Substitute the value of $A\text{'}$ from above equation in the equation (1).

    $\begin{array}{c}T=\frac{{\left|\frac{4Ai\kappa k{\text{e}}^{-ikW}}{{\left(\kappa +ik\right)}^2{\text{e}}^{-\kappa W}-{\left(\kappa -ik\right)}^2{\text{e}}^{\kappa W}}\right|}^2}{{\left|A\right|}^2}\\ =\left(\frac{4i\kappa k{\text{e}}^{-ikW}}{{\left(\kappa +ik\right)}^2{\text{e}}^{-\kappa W}-{\left(\kappa -ik\right)}^2{\text{e}}^{\kappa W}}\right)\left(\frac{-4i\kappa k{\text{e}}^{ikW}}{{\left(\kappa +ik\right)}^2{\text{e}}^{-\kappa W}-{\left(\kappa -ik\right)}^2{\text{e}}^{\kappa W}}\right)\\ =\frac{-16i^2{\kappa }^2{k}^2}{{\left({\left(\kappa +ik\right)}^2{\text{e}}^{-\kappa W}-{\left(\kappa -ik\right)}^2{\text{e}}^{\kappa W}\right)}^2}\end{array}$

Substitute the value $i^2=-1$ on the above equation.

    $\begin{array}{c}T=\frac{-16\left(-1\right){\kappa }^2{k}^2}{{\left({\left(\kappa +ik\right)}^2{\text{e}}^{-\kappa W}-{\left(\kappa -ik\right)}^2{\text{e}}^{\kappa W}\right)}^2}\\ =\frac{16{\kappa }^2{k}^2}{{\left({\left(\kappa +ik\right)}^2{\text{e}}^{-\kappa W}-{\left(\kappa -ik\right)}^2{\text{e}}^{\kappa W}\right)}^2}\end{array}$

The above equation is further solved to get the expression shown below.

    $\begin{array}{c}T=\frac{16{\kappa }^2{k}^2}{{\left({\kappa }^2+k^2\right)}^2{\left({\text{e}}^{\kappa W}-{\text{e}}^{-\kappa W}\right)}^2+16{\kappa }^2{k}^2}\\ ={\left(\frac{{\left({\kappa }^2+k^2\right)}^2{\left({\text{e}}^{\kappa W}-{\text{e}}^{-\kappa W}\right)}^2+16{\kappa }^2{k}^2}{16{\kappa }^2{k}^2}\right)}^{-1}\\ ={\left(\frac{{\left({\kappa }^2+k^2\right)}^2{\left({\text{e}}^{\kappa W}-{\text{e}}^{-\kappa W}\right)}^2}{16{\kappa }^2{k}^2}+1\right)}^{-1}\end{array}$                                                 (17)

The constants $\kappa$ and $k$ can be written in the terms of energy as shown below.

    $\begin{array}{c}\kappa =\frac{{\left[2m\left(V_0-E\right)\right]}^{1/2}}{\hslash }\\ k=\frac{{\left[2mE\right]}^{1/2}}{\hslash }\end{array}$

Where,

• $E$ is the total energy.
• $V$ is the potential energy.
• $m$ is the mass.
• $\hslash$ is a constant.

Substitute the value of $\kappa$ and $k$ in the equation (17).

    $T={\left(\frac{{\left({\left(\frac{{\left[2m\left(V_0-E\right)\right]}^{1/2}}{\hslash }\right)}^2+{\left(\frac{{\left[2mE\right]}^{1/2}}{\hslash }\right)}^2\right)}^2{\left({\text{e}}^{\kappa W}-{\text{e}}^{-\kappa W}\right)}^2}{16{\left(\frac{{\left[2m\left(V_0-E\right)\right]}^{1/2}}{\hslash }\right)}^2{\left(\frac{{\left[2mE\right]}^{1/2}}{\hslash }\right)}^2}+1\right)}^{-1}$

The above equation is further simplified to get an expression shown below.

    $\begin{array}{c}T={\left(\frac{\left(V_0{}^2\right){\left({\text{e}}^{\kappa W}-{\text{e}}^{-\kappa W}\right)}^2}{16E\left(V_0-E\right)}+1\right)}^{-1}\\ T={\left(\frac{{\left({\text{e}}^{\kappa W}-{\text{e}}^{-\kappa W}\right)}^2}{16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)}+1\right)}^{-1}\end{array}$                                                                    (20)

The ratio of total energy and potential energy is given by the expression shown below.

    $\epsilon =\frac{E}{V_0}$

Substitute the value of $\frac{E}{V_0}$ in the equation (20).

  $T={\left(\frac{{\left({\text{e}}^{\kappa W}-{\text{e}}^{-\kappa W}\right)}^2}{16\epsilon \left(1-\epsilon \right)}+1\right)}^{-1}$

When $\kappa W\gg 1$, then the ${\text{e}}^{\kappa W}$ term is very negligible and the ${\text{e}}^{-\kappa W}$ term is very large in comparison to $1$.  Therefore, the above expression can be simplified as shown below.

    $\begin{array}{c}T={\left(\frac{{\left(-{\text{e}}^{-\kappa W}\right)}^2}{16\epsilon \left(1-\epsilon \right)}\right)}^{-1}\\ ={\left(\frac{{\text{e}}^{-2\kappa W}}{16\epsilon \left(1-\epsilon \right)}\right)}^{-1}\\ =\frac{16\epsilon \left(1-\epsilon \right)}{{\text{e}}^{-2\kappa W}}\\ =16\epsilon \left(1-\epsilon \right){\text{e}}^{2\kappa W}\end{array}$