NucPhys I HomeworkProblems

Chapter 2

1.) In your own words, describe what the "Standard Model" of physics is using 2 to 5 paragraphs. Your paragraphs should include the concepts of the 4 fundamental forces of physics, QED, and QCD at a minimum. Any sentences in which a string of 5 or more words match a sentence that is found on the internet will be evidence of cheating.

2.) Solve the Schrodinger equation for the following potential:

[math]V(x) = \infty \;\; x\lt 0[/math]

[math]V(x) =\left \{ {-V_0 \;\;\;\; 0\lt x \lt a \atop 0 \;\;\;\; x\gt a} \right .[/math]

V_0 > 0 and E>0. Assume particles are incident from [math]x = -\infty[/math], don't try to normalize but do express the wave function in terms of one coefficient.

3.) Given the following barrier potential

[math]V(x) = 0 \;\; x\lt 0[/math]

[math]V(x) =\left \{ {V_o \;\;\;\; 0\lt x \lt a \atop 0 \;\;\;\; x\gt a} \right .[/math]

show that the transmission coefficient is

[math]T = \frac{1}{1+ \frac{V_o^2 \sinh^2(k_2a)}{4E(V_o-E)}}[/math]

when [math]E \lt V_o[/math]

Assume particles are incident from [math]x = -\infty[/math] and [math]k_2^2 = 2m(V_o-E)/\hbar^2[/math].

4.) Starting with Shrodinger's time-independent equation, derive the wave functions for a 2-D simple harmonic oscillator. Your derivation should include a description of the boundary conditions that are applied but you do not need to normalize the wave function.

5.) Several nuclei decay by the emmission of an alpha particle. An alpha particle (He-4) is a tighlty bound nuclear containing 2 protons and 2 neutrons in which the energy needed to remove one neutron is 20.5 MeV. One model for this decay process views the alpha particle as being bound to the nucleus via a spherical potential well.

[math]V_{bound} =\left \{ {-V_o \;\;\;\; r \lt r_o \atop \frac{A}{r} \;\;\;\; r\gt r_o} \right .[/math]

Once outside the nucleus, the alpha particle is repelled via Coulombs law

[math]V_{outside} \approx \frac{2(Z-2)e^2}{r} \equiv \frac{A}{r}[/math]

The original nucleus had a charge [math]Ze[/math] and the alpha particle has a charge [math]2e[/math].

Use the WKB approximation to show that the transmissivity (T : transmission coefficient) is:

[math]T = e^{-\frac{2}{\hbar} \int_{r_o}^{r_1} \sqrt{2m \left ( \frac{A}{r} - E\right )}dr} = e^{-\frac{2A}{\hbar v} (2W-\sin(2W) )}[/math] Gamow's formula Media:GamowFormula.pdf

where

[math](v=\sqrt{2E/m})[/math] and [math]\sqrt{r_o/r}\equiv cos W[/math] and [math]r_1 = \frac{A}{E}[/math].

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