Experiment 14: Potential Energy Diagrams & Potential Wells

Purpose: to use a computational model to analyze a potential energy diagram and well.

Part A) 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?

·         Range of motion is between -5 and 5 cm 

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

• the particle doesn’t have enough energy to surpass the potential barrier.

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?

·         Probability of detection is higher between -5 and 0 because the particle spends more time there (less kinetic energy).

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

·         E=1/2kx^x, 2E>sqrt2x

Question 5: Kinetic Energy
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

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?

• Most detected at extrema since kinetic energy is minimum

Part B) 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?

·         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