Perturbation Theory: Calculating 2nd Approx of E in Hydrogen 2s State

[Gavroy]

hi

i want to calculate the second approximation of the energy by a potential V between two hydrogen atoms in a 2s state, but I do not know how to apply pertubation theory correctly?

Landau Lifgarbagez says:

$$E_n^2= \sum_m ' \frac{|V_m_n|^2}{E_n^0-E_m^0}$$

(where the prime means that the term with m=n is omitted from the sum)

my problem is, that if i take n=2, as it is a 2s-state, shall I start with the term m=1, leave out m=2 and go on with m=3, m=4, m=5... in the sum

or does this in a 2s-state mean, that i leave out the m=1 and m=2 state and start with m=3, m=4, m=5...

 

[A. Neumaier]

hi

i want to calculate the second approximation of the energy by a potential V between two hydrogen atoms in a 2s state, but I do not know how to apply pertubation theory correctly?

Landau Lifgarbagez says:

$$E_n^2= \sum_m ' \frac{|V_m_n|^2}{E_n^0-E_m^0}$$

(where the prime means that the term with m=n is omitted from the sum)

my problem is, that if i take n=2, as it is a 2s-state, shall I start with the term m=1, leave out m=2 and go on with m=3, m=4, m=5... in the sum

or does this in a 2s-state mean, that i leave out the m=1 and m=2 state and start with m=3, m=4, m=5...

The sum is over all m distinct from n. The labeling of the states is the same as how you label them in the unperturbed case (where you have a free choice).

 

[Gavroy]

okay thank you

do you know whether it should only be summed over all principal quantum numbers or do i need to sum over all principal, azimuthal and magnetic quantum numbers?

i am not quite sure about this, cause this kind of pertubation theory deals with the undegenerated case, doesn't it?

 

[A. Neumaier]

okay thank you

do you know whether it should only be summed over all principal quantum numbers or do i need to sum over all principal, azimuthal and magnetic quantum numbers?

i am not quite sure about this, cause this kind of pertubation theory deals with the undegenerated case, doesn't it?

If your system has degeneracies in the unperturbed system but fewer in the perturbed system then this simple formula simply doesn't apply anymore,a nd you need to use the formulas for degenerate perturbation theory.

If your perturbed system has the same symmetries as the unperturbed one, you can restrict to a subspace where all quantum numbers corresponding to the symmetry group are fixed, and then solve the eigenvalue problem on this subspace. In this case, you only sum over the eigenvalues distinct from n with these quantum numbers fixed.

 

[Gavroy]

thank you for your explanation, but actually, i should have referred to my real problem, sorry

http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/vanderWaals.pdf"

it is on page 3

in this example, they calculate the van der waals forces between two hydrogen atoms in the 1s-state.

so my question is: why are they allowed to sum over all quantum numbers n,l,m (and for the other hydrogen atom n',l',m') ?

however, in the case of van der waals interaction between the 1s and 2p-state they solve the secular equation(page 5)?

sorry, that i behave that stupidly, but could you explain to me the reason why they :
in the first case, are able to sum over all quantum numbers
and in the second case have to use a different kind of pertubation theory?
actually, i assume, that i just do not know, what is necessary that the amount of degeneracies is reduced.
(and by the way: sorry for my english, but i am from germany and I still go to school)

 

[A. Neumaier]

thank you for your explanation, but actually, i should have referred to my real problem, sorry

http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/vanderWaals.pdf"

it is on page 3

in this example, they calculate the van der waals forces between two hydrogen atoms in the 1s-state.

so my question is: why are they allowed to sum over all quantum numbers n,l,m (and for the other hydrogen atom n',l',m') ?

They are not anly allowed, they have to!

however, in the case of van der waals interaction between the 1s and 2p-state they solve the secular equation(page 5)?

This is because they only treat here the degenerate part..