Experiment 14: Potential Wells & Potential Energy Diagrams

Part I Potential Wells:

A particle is trapped in a one-dimensional region of space by a potential energy function which is zero between positions zero and L, and equal to U0 at all other positions. This is referred to as a potential well of depth U0. 

Examine a proton in a potential well of depth 50 MeV and width 10 x 10-15 m.

Question 1: Infinite Well

If the potential well was infinitely deep, determine the ground state energy. Is this also the ground state energy in the finite well?

E_1(infinite well) = (1)^2(h)^2/(8*(mass of proton)*(10*10^-15) = 2.05MeV

E_1(finite well) = 1.8MeV 

The ground state energy of an infinite well is more than the ground state energy of a finite well.

Question 2: First Excited State

If the potential well was infinitely deep, determine the energy of the first excited state (n = 2). Is this also the energy of the first excited state in the finite well?

E_2(infinite) = 4E_1 = 8.20MeV.

E_2(finite) = 6.8 MeV

The energy of the first excited state in the finite well is not the same as the one in the infinite well.

Question 3: "Forbidden" Regions

Since the wavefunction can penetrate into the "forbidden" regions, will the energy of the first excited state in the finite well to be greater than or less than the energy of the first excited state in the infinite well? Why? 

The energy in the first excited state in the finite well is less than the first excited state in the infinite well due to the greater probability of tunneling.

Question 4: More Shallow Well

Will the energy of the n = 3 state increase or decrease if the depth of the potential well is decreased from 50 MeV to 25 MeV? Why? 

The energy of the n=3 state decreases if the potential well is decreased from 50MeV to 25MeV due to less tunneling.

Question 5: Penetration Depth

What will happen to the penetration depth as the mass of the particle is increased?

As the mass of the particle increases the penetration depth decreases 

Part II Potential Energy Diagrams:

A particle of energy 12 x 10-7 J moves in a region of space in which the potential energy is 10 x 10-7 J between the points -5 cm and 0 cm, zero between the points 0 cm and +5 cm, and 20 x 10-7 J everywhere else.

Question 1: Range of Motion
What will be the range of motion of the particle when subject to this potential energy function?
The particle is between +-5 cm.

Question 2: Turning Points
Clearly state why the particle can not travel more than 5 cm from the origin.

The energy that the particle has is less than the energy at the top of the well.

Question 3: Probability of Detection
Assume we measure the position of the particle at several random times. Is there a higher probability of detecting the particle between -5 cm and 0 cm or between 0 cm and +5 cm?

The particle is most likely to be found between -5cm and 0cm because the particle has less kinetic energy at U_1 it moves slower, thus spending more time there.

Question 4: Range of Motion
What will happen to the range of motion of the particle if its energy is doubled?


The range of motion increases

Question 5: Kinetic Energy
Clearly describe the shape of the graph of the particle's kinetic energy vs. position.

The shape is an concave down parabola.

Question 6: Most Likely Location(s)
Assume we measure the position of the particle at several random times. Where will the particle most likely be detected?

The edges.