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]\displaystyle{ V(x) = \infty \;\; x<0 }[/math]
- [math]\displaystyle{ V(x) =\left \{ {-V_0 \;\;\;\; 0<x <a \atop 0 \;\;\;\; x>a} \right . }[/math]
V_0 > 0 and E>0. Assume particles are incident from [math]\displaystyle{ 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]\displaystyle{ V(x) = \infty \;\; x<0 }[/math]
- [math]\displaystyle{ V(x) =\left \{ {V_o \;\;\;\; 0<x <a \atop 0 \;\;\;\; x>a} \right . }[/math]
show that the transmission coefficient is
- [math]\displaystyle{ T = \frac{1}{1+ \frac{V_o^2 \sinh^2(k_2a)}{4E(V_o-E)}} }[/math]
when [math]\displaystyle{ E < V_o }[/math]
Assume particles are incident from [math]\displaystyle{ x = -\infty }[/math] and [math]\displaystyle{ k_2^2 = 2m(V_o-E)/\hbar^2 }[/math].
4.) 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]\displaystyle{ V_{bound} =\left \{ {-V_o \;\;\;\; r <r_o \atop \frac{A}{r} \;\;\;\; r>r_o} \right . }[/math]
Once outside the nucleus, the alpha particle is repelled via Coulombs law
- [math]\displaystyle{ V_{outside} \approx \frac{2(Z-2)e^2}{r} \equiv \frac{A}{r} }[/math]
The original nucleus had a charge [math]\displaystyle{ Ze }[/math] and the alpha particle has a charge [math]\displaystyle{ 2e }[/math].
Use the WKB approximation to show that the transmissivity (T : transmission coefficient) is:
- [math]\displaystyle{ 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]
where
- [math]\displaystyle{ (v=\sqrt{2E/m}) }[/math] and [math]\displaystyle{ \sqrt{r_o/r}\equiv cos W }[/math] and [math]\displaystyle{ r_1 = \frac{A}{E} }[/math].