Question

A particle is constrained to move in a one dimensional space between two infinitely high barriers located a distance a apart.

a.) Using the uncertainty principle find an expression for the zero-point (minimum) energy that the particle.

b.) Calculate the minimum kinetic energy, in eV, that an electron trapped in this well can have, if a = 10^(-10)m.

( 3 marks)

c) Calculate the minimum kinetic energy, in eV, that a 100 mg bead trapped in this well can have, if a = 2.0 cm.

Answer 1

a)
If the particle is trapped between two walls separated by a distance a, then the uncertainty in the position is
  
So, the uncertainty in the momentum is
  
So, if for the minimum energy, we consider the magnitude of momentum to be of the order of magnitude of the uncertainty in the momentum. So, we get,

b)
For an electron, m = 9.11 * 10^{-31} kg, a = 10^{-10} m, so we get the minimum kinetic energy is
  
  
c)
For a bead of mass m = 100 mg, and a = 2 cm, we get