Question

What is the expectation value of kinetic energy for a particle in a box of length ( L/2 ) in the ground eigenstate (n=1)? What about for the third excited eignestate (n=3). Explain the difference.

Answer 1

Wave function of a particle of mass m in a box of length a is given by:

where, n is the quantum state of the particle.

Now, for our system, a = L/2

In case one, n = 1

Now, as kinetic energy is given by:

in Quantum mechanics,

or

and expectation value of p²/2m will be given by:

which is, in case 1 , n= 1 and putting value of p^2/2m , we get:

or

or

which is required value of expectation value of kinetic energy when n= 1. case.

Now, when n = 3, we have

and expectation value of kinetic energy is:

or

or

which is required value of expectation value at n = 3.

Hence,