Question

Verify that the reflection and transmission probabilities given in equation (6-7) add to 1.

Short answer

The values of T and R in equations (6-7) add up to 1

Step-by-step solution

Concept involved

The ratio of amplitudes of transmitted and incident waves is called theTransmission coefficient.

Similarly, the ratio of amplitudes of reflected and incident waves is called theReflection coefficient

Formula used

The Transmission coefficient (T) and the Reflection coefficient (R) are given by the formulas:

$$\begin{gathered} T=4\frac{\sqrt{E\left(E-U_0\right)}}{{\left(\sqrt{E+\sqrt{E-U_0}}\right)}^2} \\ R=\frac{{\left(\sqrt{E-\sqrt{E-U_0}}\right)}^2}{{\left(\sqrt{E+\sqrt{E-U_0}}\right)}^2} \end{gathered}$$

Calculation

$$\begin{gathered} T+R=4\frac{\sqrt{E\left(E-U_0\right)}}{{\left(\sqrt{E}+\sqrt{E-U_0}\right)}^2}+\frac{\sqrt{E\left(E-U_0\right)}}{{\left(\sqrt{E}+\sqrt{E-U_0}\right)}^2} \\ T+R=\frac{E+\left(E-U_0\right)-2\sqrt{E\left(E-U_0\right)}+4\sqrt{E\left(E-U_0\right)}}{{\left(\sqrt{E}+\sqrt{E-U_0}\right)}^2} \\ T=R\frac{{\left(\sqrt{E}\right)}^2+{\left(\sqrt{E-U_0}\right)}^2+2\sqrt{E\left(E-U_0\right)}}{{\left(\sqrt{E}+\sqrt{E-U_0}\right)}^2} \\ T+R=\frac{{\left(\sqrt{E}+\sqrt{E-U_0}\right)}^2}{{\left(\sqrt{E}+\sqrt{E-U_0}\right)}^2} \\ \Rightarrow T+R=1 \end{gathered}$$

So, it can be clearly seen that the values of T and R in equations (6-7) add up to 1

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