Fall 2016 Homework Assignments
Normalize the following wavefunctions (i.e. find the normalization constants A) and determine the expectation values for <x> and <p> corresponding to them. ψ(x) = A exp[-(x-xo)2/a2] ψ(x) = A exp[-(x-xo)2/a2] exp(i x po/ℏ) The variables a, xo, and po are real constants. What physical quantities do they correspond to?
A particle is confined to a one dimensional box with zero potential in the region -L/2 < x < L/2 and infinite potential elsewhere. 1. Find the normalization constants and the expansion coefficients cn for the series Ψ(x,0) = Σn cn un(x) corresponding to the wavefunctions Ψ(x,0) below: (All finite only in the interval -L/2 < x < L/2 .) (a) Ψ(x,0) = A [ exp(-α |x| ) - exp(-α L/2 ) ] (b) Ψ(x,0) = A x (|x| -L/2 ) 2. If Ψ(x,0) = A [ u1(x) + 2 u2(x) ] find <p>t, the expectation value of the momentum as a function of time.
Consider the Harmonic oscillator raising and lowering operators a+ and a-. 1. Note that the position and momentum operators are proportional to (a+ + a-) and (a+ - a-). Express x and P in terms of a+ and a- 2. Expand the quantities x4 and P4 in terms of a+ and a-. (Be careful: a+ and a- do not commute!) 3. Using the commutation relation for a+ and a-, rearrange the above quantities so that in each term all the a+ are on the left and all the a- are on the right. For example, a-a-a+a+ = a- ( 1 + a+a- ) a+ = a-a+ + a-a+a-a+ = ( 1 + a+a-) + ( 1 + a+a-) ( 1 + a+a-) = 2 + 3a+a- + a+a-a+a- = 2 + 3a+a- + a+ ( 1 + a+a- ) a- = 2 + 4a+a- + a+a+a-a-
Using the radial Hydrogenic wavefunctions in Table 4.7 of your textbook, show the orhonormality of the states |n,l,m> below: |1 0 0> , |2 0 0> and |3 0 0> . Note that all three states have l=0 and m=0 so that we know that the integral of the angular part ∫ Y00*(θ,φ) Y00(θ,φ) dΩ will be equal to 1 in all cases.
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.