Chapter 4: Finite Potential well

Finite Potential well

Fig.2.5

U(x) = 0 for 0 < x < a

U(x) = U 0 otherwise

We assume a particle with energy confined to the potential well

For 0 < x < a,

For x <0,

For x > 0,

The three sets of solution are

.....(1)

 

Now we set up the boundary conditions.

Far from the potential well, wave functions must go to zero as the probability of finding the particle away from potential region = 0

Therefore,

must be continuous at well boundary

  must be continuous at well boundaries x = 0 & x = a

From boundary conditions 

And we get four simultaneous equations

elimination and we obtain

if identically.

Therefore,

or

This is a transcendental equation

We define a dimensionless quantity

is constant of the system.

and

from definition of k & .

Eigen value equation becomes

                                              Fig. 2.6

Let then

if intersect at one points

intersect at 2 points

intersect at 3 points

So for,

there are m points.

In the limit (E finite). .

Finite potential

In finite potential wells, we talked about Eigen value equation

            (1)

for a given system constant

We have a discrete no. of solutions where LHS or RHS of (1) intersect.

                                           Fig.2.7

Wave Function of Finite Potential Wells

For each intersection a value of particular and

outside the well of potential

One can clearly see from this that there is a finite probability of existence outside in the classically forbidden region (classically a particle with an energy     cannot exist outside the well). If the potential well is slightly modified as in the figure below.

                                             Fig.2.8

The wave function will be nonzero in region C also.

Thus the particle will have a finite probability of existing or coming in region C past the potential barrier B

In C the particle can appear as a free particle.

This quantum mechanical phenomenon of passing through a barrier is known as tunneling. The wave function is different inside and outside.

Therefore there exists a finite probability of reflection at the well walls, called quantum mechanical reflection at the well walls. In analogy to optics it may be looked at as two partially reflecting mirrors, where an infinite potential well could be visualized as two 100% reflecting mirrors.

It is used in Tunnel diodes and operation of many other solid-state devices. One of the usage of this phenomenon for high frequency decive is called Resonant Tunnelling Diode (RTD), which would be used as an oscillator and even as an amplifier.