Reflection coefficient

Reflection coefficient (R) Or Probability of reflection:

It is defined as the ration of reflected probability current density to incident probability current density. It is denoted by R.

$$R=\biggl| \frac{B}{A}\biggr|^2= \frac{\vec J_{ref}}{\vec J_{inc}}$$

Using equation (12) , we get,

$$R=\biggl|\frac{K_1-k_2}{K_1+K_2}\biggr|^2= \frac{(K_1-K_2)^2}{(K_1+K_2)^2}\dotsm(15)$$

Similarly,

Transmission coefficient (T) or Probability of transmission:

It is defined as the ration of transmitted probability current density to incident probability current density. It is denoted by T.

$$T=\biggl|\frac{C}{A}\biggr|^2\frac{K_2}{K_1}=\frac{\vec J_{trans}}{\vec J_{inc}}$$

From equation (11), we get,

$$T=\biggl|\frac{2K_1}{K_1+K_2}\biggr|^2\frac{K_2}{K_1}$$

$$\therefore\; T=\frac{4K_1K_2}{(K_1+K_2})^2\dotsm(16)$$

Adding (15) and (16), we get,

$$R+T=\frac{(K_1-K_2)^2}{(K_1+K_2)^2}+\frac{4K_1K_2}{(K_1+K_2)^2}$$

$$=\frac{(K_1-K_2)^2+4K_1K_2}{(K_1+K_2)^2}$$

$$=\frac{(K_1+K_2)^2}{(K_1+K_2)^2}$$

$$\therefore\; R+T=1\dotsm(17)$$

Sum of reflected and transmitted coefficient is 1 ( i.e. 100%).

R and T in terms of energy (E) and probability \((V_0\)):-

We have,

$$K_1=\sqrt{\frac{2mE}{\hbar^2}}$$

And

$$K_2= \sqrt{\frac{2m(E-V_0)}{\hbar^2}}$$

$$\therefore\; T= \frac{4\sqrt{\frac{2mE}{\hbar^2}}\sqrt{\frac{2m(E-V_0)}{\hbar^2}}}{\biggl[\sqrt{\frac{2mE}{\hbar^2}}+\sqrt{\frac{2m(E-V_0)}{\hbar^2}}\biggr]^2}$$

$$=\frac{4\sqrt{E(E-V_0)}}{\biggl(\sqrt{E}+\sqrt{E-V_0}\biggr)^2}\dotsm(18)$$

Using value of \(K_1\) and \(K_2\) in (15) we get,

$$R=(\frac{\sqrt{E}-\sqrt{E-V_0})^2}{(\sqrt{E}-\sqrt{E-V_0})^2}\dotsm(19)$$

Classical Mechanics: \(E>V_0)\):-

According to classical mechanics if energy of incident particle is greater than the height of potential step. Particle is greater than the height of potential step, all particle should be transmitted in region II. The problem of reflection is zero ( T should be 1 and R should be zero).

Quantum Mechanics \(E>V_)\):-

For, \(V_0\ne 0\) and \(E>V_0\).

Values of reflection coefficient or probability of reflection (R) is less than 1. Bur \(R\ne 0\) and \(T_0=1\). In Quantum Mechanics there is a finite chance of reflection of high energy particle by energy potential step. This is due to wave nature associated with particle.

Graph of R and T \((E>V_0)\)

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Question:

Consider a flux of thousand electrons having same energy 50 ev incident on potential step of height 9ev. Calculate the value of transmission and reflection coefficient. And find the expected number of particle inside and outside potential state.

\(\Rightarrow\) Solution:

E= 10ev, \(V_0\)= 9ev

Case of \(E>V_0\)

$$R=\frac{(\sqrt{E}-\sqrt{E-V_0})^2}{(\sqrt{E}+\sqrt{E-V_0})^2}$$

$$=\frac{(\sqrt{10}-\sqrt{1})^2}{(\sqrt{10}+\sqrt{1})^2}= 0.26987$$

$$\therefore\; R\approx 0.2699\approx 26.99%$$

$$\therefore\; T= 1- R= 1-0.2699= 0.7301$$

$$\therefore T= 73.01%$$

Expected number of particle inside step ( transmitted)

$$N_1= N\times T= 1000\times 0.7301$$

$$\therefore\; N_1= 730$$

Expected number of particles reflected (outside step)

$$N_2= R\times 1000$$

$$or,\; N_2= 0.2699\times 1000/44$$

$$\therefore\; N_2\approx 270$$

Reference:

1. Mathews, P.M and K Venkatesan.A Text Book of Quantum Mechanics.New Delhi: Tata McGraw Hill Publishing Co. Ltd, 1997.
2. Merzbacher, E.Quantum Mechanics .New York: John Wiley, 1969.
3. Prakash, S and S Salauja.Quantum Mechanics.Kedar Nath Ram Nath Publishing Co, 2002.
4. Singh, S.P, M.K and K Singh.Quantum Mechanics.Chand & Company Ltd., 2002.