Interaction between two excited hydrogen atoms

[luxiaolei]

hi all.

i am trying to calculate the interaction between two excited hydrogen atoms, using degenerate perturbation theory. but comes up a few problems, any helps will be greatly appreciated.

the perturbation has the form:

H`=e³/R³(X_aX_b+Y_aY_b-2Z_aZ_b)
where the R is the distancee between two atoms and a , b subscripts donates atom a and b respectively.

i need to calculate the matrix elements have the form:
(<ψ_aψ_b H` ψ_aψ_b>

first thing is how to construct the total wave functions of the two atoms system? they add together or times together? and why? what is the physics meaning of:
ψ_total=ψ_aψ_b
ψ_total=ψ_a+ψ_b

the second is, if using the exzact form, and ignor the radial funtion, is it possible to evaluate the matrix element? i.e
for one atom in its 2 0 0 state and the other in its 2 1 0 state. then wavefunctions equal to theirs spherical harrmonics which are:

2 0 0 state = 1/(2* square root pi )
2 1 0 state = square root 3 /(2* square root pi) * cosθ

couples to one atom in 2 1 0 state and the other in 2 1 1 state

2 1 1 state = squre root (3)/2 * 1/(squre root (2* pi)) * sinθ*e^i∅

the first term yields:

∫∫∫2 0 0 state * X_a * 2 1 0 state dr sinθ dθ d∅ *∫∫∫2 1 0 state * X_b * 2 1 1 state dr sinθ dθ d∅

since x= rsinθcos∅ and we have ignored the radial part, how can I evaluate dr ?
above all, am I right for constructing the matrix element like the above way?

Apologise for such a messy question, it is my first time to put equations up here, and I haven't figured out how to construct a complex equation.

Thank you in advance.

 

[eljose79]

Hello,

Thank you for your question. Constructing the total wave function for a system of two atoms can be done in two ways: by adding the individual wave functions or by multiplying them. The choice depends on the physical situation and the type of interaction between the atoms.

If the atoms are far apart and there is no interaction between them, then the total wave function would be the product of the individual wave functions. This is because the atoms are considered to be in a superposition state where their individual states are independent of each other. In this case, the physics meaning of ψtotal = ψaψb is that the wave function describes the probability of finding atom a in state ψa and atom b in state ψb.

However, if the atoms are close enough that there is interaction between them, then the total wave function would be the sum of the individual wave functions. This is because the atoms are now considered to be entangled and their states are not independent of each other. In this case, the physics meaning of ψtotal = ψa+ψb is that the wave function describes the probability of finding atom a in state ψa or atom b in state ψb.

To evaluate the matrix elements, it is not possible to ignore the radial function as it is an integral part of the wave function. The wave function includes both the radial and angular parts, and they cannot be separated. The radial part represents the probability of finding the electrons at a certain distance from the nucleus, while the angular part represents the probability of finding the electrons in a certain direction. Therefore, you cannot ignore the radial part when calculating the matrix elements.

I hope this helps. Good luck with your calculations!