Difference between revisions of "NucPhys I HomeworkProblems"
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=Chapter 2= | =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. | 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. | ||
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:<math>V(x) =\left \{ {-V_0 \;\;\;\; 0<x <a \atop 0 \;\;\;\; x>a} \right .</math> | :<math>V(x) =\left \{ {-V_0 \;\;\;\; 0<x <a \atop 0 \;\;\;\; x>a} \right .</math> | ||
− | V_0 > 0 | + | <math>V_0 > 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<0</math> | |
− | :<math>V(x) =\left \{ { | + | :<math>V(x) =\left \{ {V_o \;\;\;\; 0<x <a \atop 0 \;\;\;\; x>a} \right .</math> |
show that the transmission coefficient is | show that the transmission coefficient is | ||
− | : T = \frac{1}{1+ \frac{ | + | : <math>T = \frac{1}{1+ \frac{V_o^2 \sinh^2(k_2a)}{4E(V_o-E)}}</math> |
when <math>E < V_o</math> | when <math>E < 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><r^2> = 3R^2/5</math> | ||
+ | |||
+ | 2.) Using the definition of the form factor <math>F(q)</math> and probably an integral table, calculate <math>F(q)</math> when | ||
+ | |||
+ | a.):<math>\rho(r) =\left \{ {\rho_0 \;\;\;\; r<R \atop 0 \;\;\;\; r>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 == | ||
+ | 1.) | ||
+ | a.) find the binding energy difference between O-15 and N-15 | ||
+ | |||
+ | b.) compute the nuclear radius of O-15 and N-15 assuming the above binding energy is due to the coulomb energy. | ||
+ | |||
+ | |||
+ | 2.) Muonic X-rays | ||
+ | |||
+ | a.) Calculate the energies of muonic K<math>_{\alpha}</math>-line X-rays from Fe assuming a point nucleus and using a one-electron model.. | ||
+ | |||
+ | b.) Calculate the energy correction <math>(\Delta E)</math> due to the finite nuclear size. | ||
+ | |||
+ | 3.) Find the binding energy using the fit equation B(Z,A) from the semi-empirical mass formula for | ||
+ | |||
+ | a.) Ne-21 | ||
+ | |||
+ | b.) Fe-57 | ||
+ | |||
+ | c.) Bi-209 | ||
+ | |||
+ | d.) Fm-256 | ||
+ | |||
+ | 4.) Find the neutron separation energies for | ||
+ | |||
+ | a.) Li-7 | ||
+ | |||
+ | b.) Zr-91 | ||
+ | |||
+ | c.) U-236 | ||
+ | |||
+ | 5.) Find the proton separation energies for | ||
+ | |||
+ | a.) Ne-20 | ||
+ | |||
+ | b.) Mn-55 | ||
+ | |||
+ | c.) Au-197 | ||
+ | |||
+ | == Set 6 == | ||
+ | |||
+ | 1.) Assume a neutron may be described as a proton with a negative pion <math>(\pi^-)</math> in an <math>\ell =1</math> orbital state. | ||
+ | |||
+ | What would be the orbital magnetic dipole moment of this system <math>(s_{\pi} = 0)</math>? | ||
+ | |||
+ | 2.) Assume that the proton magnetic moment is due to the rotational motion of a positive spherical uniform charge distribution of radius<math> R</math> spinning about its axis with angular speed <math>\omega</math>. | ||
+ | |||
+ | a.) Integrate the charge distribution to show that : | ||
+ | |||
+ | : <math>\mu = \frac{1}{5} e \omega R^2</math> | ||
+ | |||
+ | (hint: <math>\mu = i A</math>) | ||
+ | |||
+ | b.) show that | ||
+ | |||
+ | :<math>\mu = \frac{e s}{2 m}</math> | ||
+ | |||
+ | using the classical relationship between angular momentum and rotational speed for the spin. | ||
+ | |||
+ | == Set 7 == | ||
+ | |||
+ | |||
+ | 1.) Solving the transcendental equation for the deuteron | ||
+ | |||
+ | a.) Assume the 3-D square well approximates the deuteron system such that the well width is 2.2 fm. Using boundary conditions show that | ||
+ | |||
+ | :<math>\frac{k_1}{k_2} = -\tan(k_1R)</math> | ||
+ | |||
+ | where | ||
+ | :<math>k_1^2 = \frac{2 m}{\hbar^2} (V+E)</math> | ||
+ | and | ||
+ | :<math>k_2^2 = -\frac{2 m}{\hbar^2}E</math> | ||
+ | :<math>E = -2.224 MeV < 0</math> : bound state | ||
+ | |||
+ | b.) Rewrite the transcendental equation for the deuteron in the form | ||
+ | |||
+ | :<math>x=-tan(bx)</math> | ||
+ | |||
+ | and show that | ||
+ | :<math>b \approx 0.46</math> | ||
+ | |||
+ | when R = 2 fm. | ||
+ | |||
+ | |||
+ | ;Use the reduced mass for the deuteron system. | ||
+ | |||
+ | c.) Solve the transcendental equation for <math>x</math> using an iterative technique. | ||
+ | |||
+ | ; I got x = 3.93xxxxxx | ||
+ | |||
+ | == Set 500 == | ||
+ | |||
+ | |||
+ | 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 <r_o \atop \frac{A}{r} \;\;\;\; r>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>. | ||
− | [http:// | + | [http://wiki.iac.isu.edu/index.php/Forest_NucPhys_I Go Back][[ Forest_NucPhys_I]] |
Latest revision as of 17:48, 18 March 2009
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:
. Assume particles are incident from , don't try to normalize but do express the wave function in terms of one coefficient.
Set 2
1.) Given the following barrier potential
show that the transmission coefficient is
when
Assume particles are incident from
and .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
2.) Using the definition of the form factor
and probably an integral table, calculate whena.):
b.)
c.)
Set 5
1.) a.) find the binding energy difference between O-15 and N-15
b.) compute the nuclear radius of O-15 and N-15 assuming the above binding energy is due to the coulomb energy.
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
due to the finite nuclear size.3.) Find the binding energy using the fit equation B(Z,A) from the semi-empirical mass formula for
a.) Ne-21
b.) Fe-57
c.) Bi-209
d.) Fm-256
4.) Find the neutron separation energies for
a.) Li-7
b.) Zr-91
c.) U-236
5.) Find the proton separation energies for
a.) Ne-20
b.) Mn-55
c.) Au-197
Set 6
1.) Assume a neutron may be described as a proton with a negative pion
in an orbital state.What would be the orbital magnetic dipole moment of this system
?2.) Assume that the proton magnetic moment is due to the rotational motion of a positive spherical uniform charge distribution of radius
spinning about its axis with angular speed .a.) Integrate the charge distribution to show that :
(hint:
)b.) show that
using the classical relationship between angular momentum and rotational speed for the spin.
Set 7
1.) Solving the transcendental equation for the deuteron
a.) Assume the 3-D square well approximates the deuteron system such that the well width is 2.2 fm. Using boundary conditions show that
where
and
- : bound state
b.) Rewrite the transcendental equation for the deuteron in the form
and show that
when R = 2 fm.
- Use the reduced mass for the deuteron system.
c.) Solve the transcendental equation for
using an iterative technique.- I got x = 3.93xxxxxx
Set 500
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.
Once outside the nucleus, the alpha particle is repelled via Coulombs law
The original nucleus had a charge
and the alpha particle has a charge .Use the WKB approximation to show that the transmissivity (T : transmission coefficient) is:
- Media:GamowFormula.pdf Gamow's formula
where
- and and .