Potential Energy Diagrams and Potential Wells

The objective of the activity is to simulate the quantum effects of a particle trapped in a high-energy potential. 

Procedure:
ActivPhysics: Part VI Modern Physics, Activities 20.1 and 20.3
http://wps.aw.com/aw_young_physics_11/13/3510/898597.cw/index.html

Analysis

Potential Energy Diagrams:

1. What will be the range of motion of the particle when subject to this potential energy function?   
Range of motion is between -5 and 5 cm.   
2. Clearly state why the particle can not travel more than 5 cm from the origin.  
The particle doesn’t have enough energy to surpass the potential barrier.
3. 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?  
Probability of detection is higher between -5 and 0 because the particle spends more time there (less kinetic energy).
4. What will happen to the range of motion of the particle if its energy is doubled?
E=1/2kx^x, 2E>sqrt2x 
5. Clearly describe the shape of the graph of the particle's kinetic energy vs. position.

Kinetic energy vs potential is an upside down parabola with vertex at x=0 
6. Assume we measure the position of the particle at several random times. Where will the particle most likely be detected?

Most detected at extrema since kinetic energy is minimum 

Potential Wells: 

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?

1   The ground state energy in an infinite well in greater than 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?

     Energy of infinite=8.4 MeV. Not allowed in finite 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

     Energy infinite>energy finite; wavelength infinite<energy finite

 

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?

     If U decreases, then wavelength decreases

 

Question 5: Penetration Depth
What will happen to the penetration depth as the mass of the particle is increased?

     If m increases, then penetration length decreases