Physics 375 - Homework 5
1. The Fourier transform and the inverse Fourier transform are given in equations 2.103.
Calculate the Fourier transform of
x < 0 = x0
a) You have already used it.
Now consider
L - Derive this integral by completing the square and using the result for the Gaussian integral. Hint: If you have not done this before, change variables to y = a[x + b/2a] and substitute b. For a free particle, assume that it is initially a localized lump described by the wavefunction x.0 = e^where e and a are constants. Normalize the wavefunction and calculate the Fourier transform of Px.0 (Equation 2.103). The integral from part a will be useful.
Now calculate x, the wavefunction for any time. This is the inverse Fourier transform of k) Equation 2.100. Again, you will find your result of part a) useful. The answer is given in part b) of Problem 222 of the textbook.
d) Calculate the probability density Px and express it in terms of a/1+r where -2ah/m. Sketch again as a cSketch or plot the function of x. Discuss wh
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