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

In summary: The first case is a special case where there are no degeneracies. In the second case, there are degeneracies and therefore, a different type of perturbation theory must be used to account for them. In summary, the conversation discusses the application of perturbation theory to calculate the second approximation of energy by a potential V between two hydrogen atoms in different states. The formula used for this is E_n^2= \sum_m ' \frac{|V_m_n|^2}{
  • #1
Gavroy
235
0
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:

[tex] E_n^2= \sum_m ' \frac{|V_m_n|^2}{E_n^0-E_m^0} [/tex]

(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...
 
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  • #2
Gavroy said:
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:

[tex] E_n^2= \sum_m ' \frac{|V_m_n|^2}{E_n^0-E_m^0} [/tex]

(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).
 
  • #3
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?
 
  • #4
Gavroy said:
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.
 
  • #5
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" [Broken]

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)
 
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  • #6
Gavroy said:
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" [Broken]

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!
Gavroy said:
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..
 
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What is perturbation theory?

Perturbation theory is a mathematical method used to approximate the solutions of a problem that cannot be solved exactly. It involves adding a small perturbation to a known system and studying the effects of this perturbation on the overall system.

How is perturbation theory used in calculating the 2nd approximation of energy in the hydrogen 2s state?

In order to calculate the 2nd approximation of energy in the hydrogen 2s state, perturbation theory is used to consider the effects of the electron-electron interaction on the energy levels of the system. This interaction is treated as a small perturbation to the known energy levels of the hydrogen atom, allowing for a more accurate calculation of the 2nd approximation of energy in the 2s state.

What are the main assumptions made in perturbation theory?

The main assumptions made in perturbation theory are that the perturbation is small, and that the perturbed system can be described by the same equations as the unperturbed system. Additionally, it is assumed that the perturbation does not significantly affect the overall behavior of the system.

What is the significance of calculating the 2nd approximation of energy in the hydrogen 2s state?

Calculating the 2nd approximation of energy in the hydrogen 2s state is important in understanding the behavior of the hydrogen atom and its energy levels. This information can also be applied to other systems, such as multi-electron atoms, by providing a better understanding of the effects of electron-electron interactions on the overall energy levels.

Is perturbation theory limited to calculating the 2nd approximation of energy in the hydrogen 2s state?

No, perturbation theory can be applied to a wide range of systems and problems, and is not limited to calculating the 2nd approximation of energy in the hydrogen 2s state. It can be used to study the effects of small perturbations on any known system and can be applied to other physical phenomena, such as quantum field theory and statistical mechanics.

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