Chapter 3:Particle in an infinite potential well

Particle in an infinite potential well

Particle of mass m and fixed total energy E confined to a relatively small segment of one dimensional space between x = 0 and x = a. In terms of the potential energy we can view that the particle is trapped in an infinitely deep one dimensional potential well with constant for 0 < x < a

Fig.2.2

Now time independent Schrödinger equation is

Boundary conditions

Solving    which can also be represented as

A + exp (+jkx) + A - exp (-jkx)

Since , as vanishes for all x, so

discrete values.

Therefore, modes or eigen functions

for , Energy

So particle can have only a few discrete energy states/levels

Fig.2.3

Wave function

standing wave

particle is bouncing back and forth between the walls of the potential well.

average value of momentum at any x is 0. So to calculate momentum we have to isolate forward wave going in +x or backward wave going in -x.

For +x wave,   and -x going wave Discrete points lie on the continuous parabola of a free particle.

Fig.2.4

The integer n is called the quantum number

Value of from the equation that