2. (20 pts total) At timet= 0, the wavefunction of a particle of mass min one dimension isψ(x,0) =(A(1-x2)for|x|<1,0otherwise.whereAis a normalization constant which we can take to be real andpositive.(a) (3 pts) Sketch a graph ofψ(x,0), labeling the places where thefunction intercepts the axes.(b) (3 pts) Compute a value of the constantAthat normalizes thiswave function.(c) (3 pts) Compute the expectation value of positionhxiand theexpectation value of momentumhpi. (Note: All of the expectationvalues to be computed in this problem are att= 0.)(d) (3 pts) Compute the expectation value of position squared,hx2i.(e) (5 pts) Compute the expectation value of momentum squared,hp2i.(f) (3 pts) Compute the standard deviations of position and momen-tum,σxandσp.Then check that the uncertainty principle isobeyed.Isσxσpat the uncertainty-principle limit, or above it?Could you have predicted which would be true without doing anycalculations? (See Townsend Example 6.2.)3. (20 pts total) Using operator methods we showed (a) that a quantumstate|ψithat is normalized att= 0 remains normalized at all timesand (b) that the classical relationdx/dt=p/mholds in quantummechanics for expectation values. In equations,ddthψ|ψi=0,(1)ddthψ|ˆx|ψi=1mhψ|ˆp|ψi.(2)In this problem you will prove these two identities for a particle in onedimension “the old fashioned way”, by direct calculations using thewave function. 2
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