- With Clear Handwriting Please 1 (60 KiB) Viewed 27 times
with clear handwriting please
-
answerhappygod
- Site Admin
- Posts: 899604
- Joined: Mon Aug 02, 2021 8:13 am
with clear handwriting please
with clear handwriting please
The Harmonic Oscillator Consider a particle moving in the potential energy of the harmonic oscillator V = mw? x2 1. Given that y(x) = A xe-a x? is an eigenstate of the Hamiltonian of the system, find the constants a, A and the energy of this state. (Is this the ground state?). Do not copy results from notes. 2. For the state above y(x), calculate <x>,<x>,<p>,<p>. Make sure to write meaningful comments on your calculations and justify the answers. 3. For part 2 above, find the uncertainties Ax & Ap, then validate the Heisenberg uncertainty principle 4. Based on the energy you find in part 1, discuss the motion of the particle from classical point of view. In particular, find the turning points of the classical harmonic oscillator (XI & x2). 5. Draw the quantum mechanical probability density for the above y(x) then find the probability that the particle will be found between xi & x2 (from part 4). Write meaningful comments based on your calculation. 6. Using the raising and lowering operators a- & a. (as defined in class), write the Hamiltonian in terms of the raising and lowering operators. (This has been discussed in class.) 7. Find a++(x) and a y(x) for the state given in part 1. Write useful comments. 8. Calculate the commutation relations [a+, a.), [x, a.), [x?, a-), [p, a.), [p?, a+], [H, a.], [H, a-]. 9. Using the raising and lowering operators, repeat part 2 in calculating the expectation values <x>,<x< >,<p>,<p? >. Compare results. 10. Using the state (x) in part 1, find the next upper level state and the lower level state. (Use the raising and lowering operators.) Calculate their corresponding energies and check normalization constants. 11. Show that the states in part 1 and part 10 are orthogonal.
The Harmonic Oscillator Consider a particle moving in the potential energy of the harmonic oscillator V = mw? x2 1. Given that y(x) = A xe-a x? is an eigenstate of the Hamiltonian of the system, find the constants a, A and the energy of this state. (Is this the ground state?). Do not copy results from notes. 2. For the state above y(x), calculate <x>,<x>,<p>,<p>. Make sure to write meaningful comments on your calculations and justify the answers. 3. For part 2 above, find the uncertainties Ax & Ap, then validate the Heisenberg uncertainty principle 4. Based on the energy you find in part 1, discuss the motion of the particle from classical point of view. In particular, find the turning points of the classical harmonic oscillator (XI & x2). 5. Draw the quantum mechanical probability density for the above y(x) then find the probability that the particle will be found between xi & x2 (from part 4). Write meaningful comments based on your calculation. 6. Using the raising and lowering operators a- & a. (as defined in class), write the Hamiltonian in terms of the raising and lowering operators. (This has been discussed in class.) 7. Find a++(x) and a y(x) for the state given in part 1. Write useful comments. 8. Calculate the commutation relations [a+, a.), [x, a.), [x?, a-), [p, a.), [p?, a+], [H, a.], [H, a-]. 9. Using the raising and lowering operators, repeat part 2 in calculating the expectation values <x>,<x< >,<p>,<p? >. Compare results. 10. Using the state (x) in part 1, find the next upper level state and the lower level state. (Use the raising and lowering operators.) Calculate their corresponding energies and check normalization constants. 11. Show that the states in part 1 and part 10 are orthogonal.
Join a community of subject matter experts. Register for FREE to view solutions, replies, and use search function. Request answer by replying!