Experiment 14: Classical Harmonic Oscillator

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: What will be the range of motion of the particle when subject to this potential energy function?

The range of motion is between +-5 cm.

Question 2: 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: 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 finding the particle from -5 to 0 cm because it has a higher potential energy on that side.

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

The range of motion of the particle will increase.

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

The graph is an inverted parabola 

Question 6: Assume we measure the position of the particle at several random times. Where will the particle most likely be detected?

The particles will most likely be found at the ends where the boundaries are located.

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

Energy in infinite well: 2.05 MeV

Energy in finite well: 1.8MeV

The energy in the infinite well is greater than the energy in the finite well. 

Question 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?

Energy in first excited state of infinite well = 8.4 MeV

Energy in first excited state of finite well = 6.8 MeV

The energy of the first excited state in the infinite well is greater than the energy of the first excited state in the finite well.

Question 3: 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 of the finite well is less than the energy of the first excited state in the infinite well. The is because in the finite well, there is a greater probability of tunneling than there is in the infinite well.

Question 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 n=3 state decreases if the potential well is decreased from 50MeV to 25MeV. This is because there is less of a probability tunneling.

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

As the mass of the particle is increased, the penetration depth decreases.