{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# ps06 - x that a classical particle can reach iF it has the...

This preview shows page 1. Sign up to view the full content.

Quantum Mechanics Problem Sheet 6 (Solutions are to be handed in for marking before the lecture on Monday, 28 Feb 2005) 1. Calculate the reflection and transmission probabilities for right-incident scattering from the potential V ( x ) = V 0 for x < 0 0 for x > 0 . at an energy E < V 0 . Find the probability density and the probability current density in the region x < 0. What can you say about where the reflection is taking place? (The strategy for tackling this problem is the same as in problem 2 of the previous problem sheet and in the examples of scattering from piecewise constant potentials that we looked at in the lecture.) 2. Consider a particle in the harmonic potential V ( x ) = 2 x 2 / 2. Its lowest energy eigenvalue is E 0 = ¯ hω/ 2 and the eigenfunction associated with this energy, ie the ground-state wave function, is φ 0 ( x ) = π ¯ h 1 / 4 exp - h x 2 . (a) Determine the limits of the classical motion in this potential (the “classical turning points”), ie the smallest and the largest values of
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x that a classical particle can reach iF it has the total energy E . (b) Assume that the wave Function oF the particle is the stationary state φ ( x ) exp(-i E t/ ¯ h ). Determine the probability (in the Form oF an integral) oF ²nding the particle outside the region where classical motion can occur. By making an appropriate change oF variable in the integral you obtain, show that the answer is independent oF m , ω , and ¯ h . (c) Calculate the variance Δˆ x in the ground-state oF the system and compare it to the limits oF the classical motion. Useful integral: Z ∞-∞ d z z 2 exp(-αz 2 ) = 1 2 r π α 3 . 3. Show that the three stationary wave Functions φ 1 ( x ) = C 1 exp(-x 2 ), φ 2 ( x ) = C 2 x exp(-x 2 ), and φ 3 ( x ) = C 3 (4 x 2-1) exp(-x 2 ) are all mutually orthogonal. Claudia Eberlein, 11 Jan 2005 1...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern