HW06

ASSIGNED:     FRI 01 MAR
DUE:                  FRI 15 MAR

In all problems the Mathematica output is part of your grade and must be turned in with your paper-and-pencil calculations.

For the Mathematica calculations, such as in the Integrate, Simplify, FullSimplify commands, you need to specify that the constants such as a, σ, λ, etc., are real numbers greater than zero or else it will assume they are complex numbers and produce crazy looking answers. To do this read the "tutorial/UsingAssumptions" which you will find in the Mathematica Help / Documentation Center.

1. GRIFFITHS PROBLEM 3.4, PAGE 98. 

2. GRIFFITHS PROBLEM 3.5, PAGE 98.

3. GRIFFITHS PROBLEM 3.6, PAGE 100. Solve the second order differential equation by hand and apply periodic boundary conditions to get the quantization condition. Then do it again using the DSolve command in Mathematica. Make sure the results agree. Normalize the eigenfunctions over the range 0-->2Pi.

4. GRIFFITHS PROBLEM 3.8, 103. The range of integration is 0-->2Pi. For both parts you should be do the integrals by hand and then using the Integrate command in Mathematica and check the results.

5. GRIFFITHS PROBLEM 3.11, PAGE 109. For the Fourier transform integral use EQ18.75 in the Schaum's Outline (the password is "math"). The do the integrals again in Mathematica. If the Integrate command barfs try the FourierTransform command instead. You may find it convienent to replace the unit of position uncertianty  Δx2=ℏ/(2•m•ω) at the beginning of the calculation and the uncertianty Δp2=ℏ•m•ω/2 at the end of the calculation. This is a Minumum Uncertianty State (MUS)  so Δx•Δp=ℏ/2. Notice the Fourier Transform of a Gaussian is a Gaussian. Take units so that ℏ=m=1 so that Δx2=2•ω and then plot |Ψo(x,t)|2 as a function of x (the t drops out) for ω=0.1, 1.0, 10,  and then plot  |Φ(p,t)|2 as a function of p (the t drops out) for ω=0.1, 1.0, 10, to show that the Fourier Transform of a narrow tall Gaussian (|Ψo(x,t)|2 with ω=0.1) becomes a flat wide Gaussian (|Φ(p,t)|2  with ω=0.1) and vice versa. This is the Heisenberg Uncertainty Principle at work -- narrow spread in x gives large spread in p and vice versa. For ω=0.1 we have an x-squeezed MUS, for ω=1.0 we have a coherent MUS, and for ω=10 we have a p-squeezed MUS.
 
6. GRIFFITHS PROBLEM 3.12, PAGE 109.


Hilbert
 


"Physics is much too hard for physicists."

--- David Hilbert (1862–1943)



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