Time Development of the Probability Density

1. Introduction

Consider a proton that is in a force-free region. At time t=0 it has an initial wave packet that has a Gaussian shape and is described by the expression

                                         |Ψ(x,0)⟩ = (2πσ2)-1/4 Exp(-x2/4σ2)            

where σ is the width of the distribution. We want to investigate the behavior of this proton as a function of time. More specifically what happens to the probability density as time marches forward?

2. Theoretical Background

In this section we develop the ideas needed to expand the initial wave packet in terms of the eigenfunctions. Answer the following series of questions.

(a) What are the eigenfunctions and eigenvalues of the proton in the force-free region? What are the eigenvalues in terms of the wave number k?

(b) What is the general expression for the expansion of the initial wave packet in terms of the eigenfunctions?

(c) Recall that we have already confronted the question in part 2.b. What was the result we found for the expansion 'coefficients', b(k)?

(d) What is the initial wave packet in terms of the specific form of b(k) you recorded for part 2.c? Express your answer in terms of an integral.

3. Expression for the Time Development

Now we generate the explicit mathematical forms of the wave packet for all times.

(a) How is your result for part 2.d modified to include times t>0?

(b) In part 2.a you wrote down the eigenvalues for the free particle in terms of k, the wave number. Make sure you incorporate that result into your expression for part 3.a.  What is the final expression for wave function for all times? Your answer should be in the form of an integral. If it is not consult your instructor.

(c) Before we go on and investigate the time evolution of the wave packet, you should predict what you think will happen to the initial, Gaussian, wave packet for t>0?

4. Observing the Wave Function

Now we investigate the evolution of the wave packet in time by plotting it for different values of the time.

(a) Your answer to part 3.b should be in the form of an integral where the integrand is a function of the wave number k and the integral is over k from negative infinity to positive infinity. To perform this integral use the syntax for the Integrate command that is demonstrated below or use the palettes.

Integrate[Exp[-k],{k,0,Infinity}]

(b) You must also make some choices of the constants and units in the problem, namely the width of the initial Gaussian and factor multiplying the time t. Set the width equal to unity and use the constants list below.

Some constants:
       
                hbarc = 197 MeV-fm
                σ = 1 fm
                c = 3 x 108 m/s
                1 fm = 10-15 m
                mproton = 938 MeV/c2
                
It might be useful to use units of 10-22 s for time.

(c) Use the result of the previous integration in part 4.a to define a Mathematica function for the wave function for all times. A useful feature of Mathematica can be used here so you don't have to enter the function 'by hand'. You can highlight the result of the integration in part 4.a and then copy and paste it into a new cell. You have to add the appropriate syntax at the beginning of the cell to define the function. 

Notice the wave function depends on x and t. The syntax for functions that depend of multiple variables and how to plot them for a fixed value of one of the variables is shown below.

f[x_,t_] := (x^2)*Exp[-x*x]*Exp[-t]
Plot[f[x,1], {x,-5,5},
              PlotRange->{{-5,5},{0,1}},
              Frame->True,
              FrameLabel->{"x","f(x)",
                  "A title",""}]

(d) To demonstrate the time evolution, plot the probability density for several values of the time. The probability density is the product of the wave function and its complex conjugate. To perform that calculation use the Mathematica syntax shown below.

f[x]*Conjugate[f[x]]

(e) How does the nature of the probability density change with time? What is the average momentum of the wave packet and how does it change with time?

(f) Include a purpose, answers to questions,  figures, and a discussion in your report.