What is the Probability for Deuteron Hi in a Square-Well Potential?

[komigen]

hi,
Need help to calculate the probability for the proton and neutron in the deuteron to be found beyond the range of the nucleon-nucleon force, assuming a square-well potential.

I used schrödinger equation and solved for the r<R, and r>R. then took into acount the continuity condition at r=R. Then couldn't know what to do.

Would be great if someone could help me

 

[AndyUrquijo]

What is the problem? You don't have enough equations?

Remember the function is continuous up to the first derivative (if the well has a finite depth). You can also discard the "non-physical" terms.

 

[komigen]

What is the problem? You don't have enough equations?

Remember the function is continuous up to the first derivative (if the well has a finite depth). You can also discard the "non-physical" terms.

The problem is that i don't know how to calculate the probability usind the continuity condition at r=R.
The equation for r<R: u₁(r)=Asink₁r
And for r>R we have : u₂(r)=Ce^-k₂r

k₁=squere root of (2m(E+V₀)/h²)
k₂=squere root of (-2mE/h²).

I then put ,
u₁(R)=u₂(R), and,
u₁'(R)=u₂'(R).
These to conditions should in some way help me to calculate the probability for the proton and neutron in the deuteron to be found beyond the range of the nucleon-nucleon force.

I think like this, because it's the probability beyoned the range , i.e r>R, then it's maybe u₂(r) that I should took the squere of the absolutbelopp of the calculate the probability, but then i should know the constant C.
It means that I need to calculate lu₂(r)l². is that right? then how can I calculate this probability?

 

[AndyUrquijo]

Now that I see it I noticed, that the continuity in the derivative doesn't help you here. You only need to use the continuity on the function. The probability of finding the particle anywhere should be one, right? Use this normalization property to get the last variable. Now you have well-defined functions and you can proceed to find the probability of finding the nucleons out of the strong force range. To do this, evaluate the integral of the probability density over the according region.
$$\int_R^\infty \left|\Psi \left( r \right) \right|^2dr$$

I asume R is the nuclear force range? If this is true, then the range of the integral is domain of your $$u_2$$

Edit: Oh and you are using the coupled coordinate for the separation between the nucleons. So you need to use the reduced mass $$\mu$$

 

[komigen]

Now that I see it I noticed, that the continuity in the derivative doesn't help you here. You only need to use the continuity on the function. The probability of finding the particle anywhere should be one, right? Use this normalization property to get the last variable. Now you have well-defined functions and you can proceed to find the probability of finding the nucleons out of the strong force range. To do this, evaluate the integral of the probability density over the according region.
$$\int_R^\infty \left|\Psi \left( r \right) \right|^2dr$$

I asume R is the nuclear force range? If this is true, then the range of the integral is domain of your $$u_2$$

Edit: Oh and you are using the coupled coordinate for the separation between the nucleons. So you need to use the reduced mass $$\mu$$

oh, thanks a lot, yeah R is the nuclear force range. If then understood right then i use the normalzation for u2(r) to get the variable C. Then calculate the probability using this equation 2 right? then it becomes hard to find an primitive function for this equation in this intervall, well?
I have also another question, feels like I mix probability with the average value of position, i.e. <x>, what does the average value give for information? and way canot I use it ?

 

[AndyUrquijo]

The normalization must be done for the two functions. Or rather, for u₁ in r < R and u₂ for r > R. The sum of this two integrals is 1. With this, you should have all your coefficients defined. Then with this you can calculate the second term numerically.

The value <a> gives you an averaged value of a variable. This is basic probability theory, just put average value on wikipedia. In some cases it tells you significant information, but sometimes it doesn't. For example, in a simple hydrogen atom the value <r> of the electron is the bohr radius. But the average value of some axial coordinate <z> would give you a 0 (even with x and y set to 0). Which is not that much helpful, because the electron never reaches this point inside the proton.

 

[komigen]

thanx a lot, understand now :)