Experiement: Potential Energy Diagrams

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.

1. What will be the range of motion of the particle when subject to this potential energy function?
- The range of motion will be between -5 and 5 cm.

2. Clearly state why the particle can not travel more than 5 cm from the origin.
- The particle can not travel more than 5cm from the origin because the particle does not have enough energy to go over the potential energy 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?
- There is a higher probability of detecting the particle between -5 and 0 cm because there it will ahve less kinetic energy, making the particle slower.

4. What will happen to the range of motion of the particle if its energy is doubled?
-The range of motion will increase if the energy is doubled.

5. Clearly describe the shape of the graph of the particle's kinetic energy vs. position
- The shape will be a concave down parabola.

6. Assume we measure the position of the particle at several random times. Where will the particle most likely be detected?
-The particle will most likely be found at the edges.


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.

1. If the potential well was infinitely deep, determine the ground state energy. Is this also the ground state energy in the finite well?
- E1(infinite) = (n^2*h^2)/(8*m*L^2)
        = 1*(6.63*10^-34)^2 / (8* 1.67*10^-27 *(10*10^-15)^2)1
        = 3.29*10^-32 J = 2.05 MeV
 E1 (finite) = 1.80 MeV

2. 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?
-E4(infinite) = E1 * 2^2= 8.20 MeV
 E4 (finite) = 6.80 MeV

3. Since the wave function 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 finite well will be less than that of the infinite well because there is a greater probability of tunneling.

4. 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 second exited state will decrease if the depth is decreased because there will be less tunneling.

5. What will happen to the penetration depth as the mass of the particle is increased?
- If the mass of the particle is increased, the depth of the penetration will decrease.