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###### 3. Consider the wave function (x, t) = Ae-2 -ut Where A, 2, and are positive...

3. Consider the wave function (x, t) = Ae-2 -ut Where A, 2, and are positive real constants. (a) Normalize Y. (b) Determine the expectation values of x and x?. (c) Find the standard deviation of x. Sketch the graph of V', as a function of x, and mark the points (x) + a) and (x) -o to illustrate the sense in which represents the spread" in x. What is the probability that the particle would be found outside this range? 4. Let Pen(t) be the probability of finding the particle in the range (a<x<b), at time t. (a) Show that de = 1(a,t) – J(6,1) Where |(2,4) ( - ) What are the units of ](x, t)?[is called the probability current, because it tells you the rate at which probability is "flowing past the point x. f Pen(t) is increasing, then more probability is flowing into the region at one end than flows out at the other.) (b) Find the probability current for the wave function in the previous problem (No. 3).

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