 #1
James_1978
 36
 3
 Homework Statement:
 I believe I solved the transcendental equation but the plot does not make sense.
 Relevant Equations:

##k_{1} \cot{k_{1}R} = k_{2}##
##k_{1} = \frac{\sqrt{2m(E+V_{o})}}{\hbar}##
##k_{2} = \frac{\sqrt{2mE}}{\hbar}##
##x = \tan{bx}##
##x = \sqrt{\frac{(V_{o} + E)}{E}}##
Dear Forum,
I am trying to solve a problem (4.6) from the introductory nuclear physics textbook by Krane. The problem is as follows:
Solving the deuteron using the radial equations gives the transcendental function,
##k_{1} \cot{k_{1}R} = k_{2}##
Were
##k_{1} = \frac{\sqrt{2m(E+V_{o})}}{\hbar}##
And
##k_{2} = \frac{\sqrt{2mE}}{\hbar}##
That gives the relations between and R. Show that this equation can be written in the form,
##x = \tan{bx}##
Where
##x = \sqrt{\frac{(V_{o} + E)}{E}}##
Evaluate the parameter b for R = 2fm. Note that is the reduced mass. Solve the transcendental equation.
When rearranging we get ##b## as.
##b = \frac{\sqrt{2mE}}{\hbar}*R##
For the reduced mass ##m = \frac{1.67x10^{27}}{2} kg##
For ##\hbar = 1.054x10^{34} Js##
For ##E = 2.22 MeV##
We are suppose to see that when solving the transcendental equation we get ##V_{o} = 36 MeV##. However we must have something wrong because the function does not clearly show how you infer the ##V_{o} = 36 MeV##. Any help is appreciated.
I am trying to solve a problem (4.6) from the introductory nuclear physics textbook by Krane. The problem is as follows:
Solving the deuteron using the radial equations gives the transcendental function,
##k_{1} \cot{k_{1}R} = k_{2}##
Were
##k_{1} = \frac{\sqrt{2m(E+V_{o})}}{\hbar}##
And
##k_{2} = \frac{\sqrt{2mE}}{\hbar}##
That gives the relations between and R. Show that this equation can be written in the form,
##x = \tan{bx}##
Where
##x = \sqrt{\frac{(V_{o} + E)}{E}}##
Evaluate the parameter b for R = 2fm. Note that is the reduced mass. Solve the transcendental equation.
When rearranging we get ##b## as.
##b = \frac{\sqrt{2mE}}{\hbar}*R##
For the reduced mass ##m = \frac{1.67x10^{27}}{2} kg##
For ##\hbar = 1.054x10^{34} Js##
For ##E = 2.22 MeV##
We are suppose to see that when solving the transcendental equation we get ##V_{o} = 36 MeV##. However we must have something wrong because the function does not clearly show how you infer the ##V_{o} = 36 MeV##. Any help is appreciated.