- #1

James_1978

- 37

- 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} J-s##

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} J-s##

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.