Potential step

Potential step \(( E< V_0)\) :

consider a flux of non-interacting particle having same mass 'm' and energy 'E' is incident on the step of the form,

$$V(x)=0\; for \; x\leq 0$$

$$= V_0\; for \; x>0\dotsm(1)$$

as shown in figure,

According to classical mechanics if the energy of incident particle E is less than the chance of finding particle inside step (i.e. x>0) is zero. The Quantum mechanical behaviour of particle is obtained from the solution of one dimensional time independent Schrodinger equation.

$$\frac{-\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)= E\psi(x)\dotsm(2)$$

From (2):

$$\frac{d^2\psi(x)}{dx^2}+\frac{2m}{\hbar}[E-V(x)]\psi(x)=0\dotsm(3)$$

In region (1) \( x<0, v(x)=0, \psi(x)= \psi_I(x)\)

$$\frac{d^2\psi_I(x)}{dx^2}+\frac{2m}{\hbar^2}E\psi_1(x)=0$$

$$or,\; \frac{d^2\psi_I(x)}{dx^2}+K_1^2 \psi_I(x)=0\dotsm(4)$$

Where,

$$K_1=\sqrt{\frac{2mE}{\hbar^2}}\dotsm(5)$$

solution of equation (4) is, oscillating,

$$\psi_1(x)= Ae^{ik_1 x}+ Be^{-ik_1x}\dotsm(6)$$

In region (II) \(V(x)= V_0>E\; \psi(x)=\psi_{II}(x)\)

Equation (3) become,

$$\frac{d^2\psi_{II}(x)}{dx^2}+\frac{2m}{\hbar^2}(E-V_0)\psi_{II}(x)=0$$

$$or,\; \frac{d^2\psi_{II}(x)}{dx^2}-\frac{2m}{\hbar^2}(V_0-E)\psi_{II}(x)=0$$

$$or,\; \frac{d^2\psi_{II}(x)}{dx^2}-K_2^2 \psi_{II}(x)=0\dotsm(7)$$

Where,

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

The solution of equation (7) is non-oscillatory and given by,

$$\psi_{II}(x)= ce^{-k_2 x}+ De^{k_2 x}\dotsm(9)$$

For, x>0, since \(e^{k_2 x}\) gives divergent result for probability density so D must be zero. [ D=0]

$$\therefore\; \psi_{II}(x)= Ce^{-k_2 x}\dotsm(10)$$

Using boundary condition at x=0

$$i.e,\;\; \psi_I(x)|_{x=0}= \psi_{II}(x)|_{x=0} \Rightarrow A+B=C\dotsm(11)$$

And

$$\frac{d\psi_I(x)}{dx}\biggl|_{x=0}= \frac{d}{dx}\psi_{II}(x)\biggr|_{x=0}$$

$$\Rightarrow ik_1(A-B)= -k_ 2 C$$

$$\Rightarrow (A-B)= \frac{-k_2}{ik_1}c\dotsm(12)$$

Adding (11) and (12)

$$A=\biggl[1-\frac{k_2}{k_1}\biggr]\frac{c}{s}$$

$$\therefore\; \frac{c}{A}= \frac{2}{\biggl(1-\frac{k_2}{ik_1}\biggr)}\dotsm(13)$$

Dividing (11) by A we get,

$$\frac AA + \frac BA=\frac CA$$

$$or,\; \frac BA= \frac CA- 1 $$

$$=\frac{2}{1-\frac{k_2}{ik_1}}-1$$

$$=\frac{2-1+\frac{k_2}{ik_1}}{1-\frac{k_2}{ik_1}}$$

$$=\frac{1+\frac{k_2}{ik_1}}{1-\frac{k_2}{ik_1}}$$

$$=\frac{ik_1+ k_2}{ik_1-k_2}$$

$$\therefore\; \frac{B}{A}= \frac{ik_1+K_2}{ik_1-k_2}\dotsm(14)$$

The probability current density in region (I)

$$\vec J_1(x)=\frac{\hbar}{m}K_1|A|^2 - \frac{\hbar K_1}{m}|B|^2$$

$$\therefore\; J_{inc}(x)= \frac{\hbar k_1}{m}|A|^2$$

$$J_{ref}(x)=\frac{\hbar k_1}{m}|B|^2\dotsm(15)$$

Similarly, probability current density in region (II)

$$J_{II}(x)=0\;\; \; \; [ \psi_{II}(x) is real ]$$

Probability of reflection or reflection coefficient (R):

$$R= \frac{J_{ref}(x)}{J_{inc}(x)}=\biggl|\frac BA\biggr|^2$$

$$=\biggl(\frac BA\biggr)^* \biggl(\frac BA\biggr)$$

$$=\biggl(\frac{ik_1+k_2}{ik_1-k_2}\biggr)^*\biggl(\frac{ik_1+k_2}{ik_1-k_2}\biggr)$$

$$=\biggl(\frac{k_2-ik_1}{-k_2-ik_1}\biggr)\frac{(ik_1+k_2)}{(ik_1-k_2)}$$

$$=\frac{(k_2-ik_1)(ik_1+k_2}{(-)(ik_1+k_2)(-)(k_2-ik_1)}$$

$$=1$$

$$\therefore\; R=1\dotsm(16)$$

There 100% chance of probability of reflection in region (I) i.e x<0.

Translation Coefficient (T):-

$$T=\frac{J_{II}(x)}{J_{inc}(x)}$$

$$=\frac{0}{J_{in}(x)}$$

$$=0$$

$$\therefore\; T=0\dotsm(17)$$

The wave function in region (II)

$$\rho_{II}(x)= \psi_{II}^*(x)\psi_{II}(x)$$

$$\rho_{II}(x)= |c|^2 e^{-2k_2 x}\dotsm(8)$$

From (18) we can say that the probability density in region (II) (x>0) for finite value of x is non-zero.

According to classical mechanics region (II) (x>0) is forbidden region. Because momentum of particle is imaginary in that region.

$$i.e.\;\; P= \sqrt{2m(E-V)} ( Imaginary)$$

Classically, the probability of finding particle in region (II) is zero.

But, Quantum mechanically existance of wave function indicates that there is finite chance of finding particle in region (II). This value of probability is called leakage probability.

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.