Fall 2016 Homework Assignments

**Homework 1: (Due Tuesday, October 3)**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-x

_{o})^{2}/a^{2}] ψ(x) = A exp[-(x-x_{o})^{2}/a^{2}] exp(i x p_{o}/ℏ) The variables a, x_{o}, and p_{o}are real constants. What physical quantities do they correspond to?**Homework 2: (Due Thursday, October 20)**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 c

_{n}for the series Ψ(x,0) = Σ_{n}c_{n}u_{n}(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 [ u_{1}(x) + 2 u_{2}(x) ] find <p>_{t}, the expectation value of the momentum as a function of time.**Homework 3: (Due Tuesday, October 25)**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 x^{4}and P^{4}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_{-}**Homework 4: (Due Thursday, December 29)**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 ∫ Y

_{0}^{0*}(θ,φ) Y_{0}^{0}(θ,φ) dΩ will be equal to 1 in all cases.

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