NucPhys I HomeworkProblems

Chapter 2

Set 1

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]

[math]V_0 \gt 0[/math]. 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.

Set 2

1.) 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].

Set 3

1.) Starting with Shrodinger's time-independent equation, derive the wave functions for a 2-D simple harmonic oscillator. Your derivation should take advantage of separation of variables and you are not required to normalize the wave function.

Set 4

1.) Show that the mean-square charge radius of a uniformly charged sphere is [math]\lt r^2\gt = 3R^2/5[/math]

2.) Using the definition of the form factor [math]F(q^2)[/math] and probably an integral table, caluclate [math]F(q^2)[/math] when

a.):[math]\rho(r) =\left \{ {\rho_0 \;\;\;\; r\lt R \atop 0 \;\;\;\; r\gt R} \right .[/math]

b.) [math]\rho(r) = \rho_0 e^{- \alpha r}[/math]

c.) [math]\rho(r) = \rho_0 e^{- \alpha^2 r^2}[/math]

Set 5

2.) Muonic X-rays a.) Calculate the energies of muonic K-line X-rays from Fe assuming a point nucleus and using a one-electron model.. b.) Calculate the energy correction (\Delta E)

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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