Solve Ground State Energy Level of Proton in Al Nucleus - 100MeV, 5fm

[Slayer537]

I've been working at this problem for about an hour and can't seem to make any progress. Any help would greatly be appreciated.

Homework Statement

Estimate the ground state energy level of a proton in the Al nucleus which has a potential energy of 100 MeV. Compare your answer to that calculated from the infinite square model. The radius of the Al nucleus is 5 fm.

2. The attempt at a solution

I thought that for the first part of the question this equation should be used

E_n = n²*h²/(8*m*L²)

However, I was getting nowhere close to the answer of 1.72 MeV. For the second part I figure that it would involve Schrodinger's equation and and this equation:

$$\psi$$ = (2/L)^1/2*sin(n*pi*x/L)

Oddly enough using the first equation and using the diameter instead of the radius I got the right answer for the second part of the question of 2.05 Mev; however, I don't think that I solved it correctly.

 

[nickjer]

The first part isn't an infinite well. The 2nd part is an infinite well, the first equation you listed is the energy levels for an infinite well, that is why it worked. Also, "L" is the width of the well which is the diameter and not the radius (that is why you got the right answer using diameter).

The first part sounds like you will be using a finite potential well. Unless you are learning some other method like the shell model.

 

[Slayer537]

The first part isn't an infinite well. The 2nd part is an infinite well, the first equation you listed is the energy levels for an infinite well, that is why it worked. Also, "L" is the width of the well which is the diameter and not the radius (that is why you got the right answer using diameter).

The first part sounds like you will be using a finite potential well. Unless you are learning some other method like the shell model.

Thanks, that explains the second part of the question. Still can't figure out how to do the first part. We have done finite potential well, but not shell. I looked up the equations in my book and think that I should use Schrodinger's time independent equation:

-(ħ/2m)(d²/dx²)Ψ(x)+U(x)Ψ(x)=EΨ(x)

Where I would solve for E. Could I then use this for Ψ(x) :

Ψ(x)=(2/L)^1/2sin(pi*x/L)

? If so what value would I use for x, or am I still missing something?

 

[Slayer537]

Never mind. I just figured it out.

First solve for δ:

δ=ħ/(2*m*U)^1/2

Then use δ to solve for Energy, making sure to use diameter, not radius:

E=pi²*ħ²/(2*m*L²)

---> E = 1.72 MeV

 

[paranormal]

Thank you for reaching out for help with this problem. It seems like you are on the right track in using the equations for energy levels in a potential well and the wavefunction for a particle in a box. However, there are a few things to keep in mind when solving this problem.

First, make sure you are using the correct units for all the variables in the equations. In this case, the potential energy is given in MeV, so you will need to convert it to joules before using it in the equation.

Second, the equation you used for the energy levels in a potential well is for a one-dimensional system, but the problem is asking for the energy level of a proton in a three-dimensional nucleus. This means you will need to use the three-dimensional version of the equation, which takes into account the volume of the nucleus.

Lastly, for the second part of the problem, you are correct in using Schrodinger's equation to calculate the energy level of a particle in a box. However, make sure you are using the correct boundary conditions for a particle in a three-dimensional box, which will involve the volume of the nucleus as well.

I hope this helps you make progress on the problem. If you are still having trouble, I suggest consulting with your instructor or a classmate for further assistance. Good luck!