Calculating energy separation between two states

[IHateMayonnaise]

Homework Statement

If the energy separation of two states is known for some atom, how can the same energy separation be calculated for the same two states for an ionized atom? i.e. if we know that the difference between the $1s^2$ and the $1s2s$ states in Helium is something like 25eV, how could I use this information to approximate the energy difference between the same two levels of an ionized atom with the same number of electrons, like $Li^+$, or $Be^{2+}$?

Homework Equations

None

The Attempt at a Solution

Other than doing extensive calculations using perturbation theory and the variational principle, I have no idea how to do this problem. The prompt asks to estimate the energy separation..so even if we know the $\ell$ and $n$ of the two states in question, without plugging it all into the wave function for hydrogen (assuming that this is even valid) and finding the normalization constant and looking up the spherical harmonics how is this possible?

Clearly, the energy separation between the $1s^2$ and $1s2s$ state of an ionized atom are going to be less that of helium...but as far as a quantitative estimate?? Does anyone have any hints or ideas?

IHateMayonnaise

 

[lanedance]

what do you think will have the largest effect on the energy of the state?

consider hydrogen, then He⁺, and Li²⁺... and the effect of Z the energy separation for 2 given states... not a silver bullet but hopefully it helps...

 

[lanedance]

Clearly, the energy separation between the $1s^2$ and $1s2s$ state of an ionized atom are going to be less that of helium...but as far as a quantitative estimate?? Does anyone have any hints or ideas?

also why do you think it is less? I'd probably think its more at first glance

 

[IHateMayonnaise]

what do you think will have the largest effect on the energy of the state?

consider hydrogen, then He⁺, and Li²⁺... and the effect of Z the energy separation for 2 given states... not a silver bullet but hopefully it helps...

Well, the largest impact would be what state the atom is in. I.e. how hard is it to ionize one (or both) of the electrons in this case.

The energy that seprates Hydrogen from singly ionized helium (for ex.) is that obviously there is a difference of one electron. That electron, although it is screened by the inner one, adds to the total binding energy.

also why do you think it is less? I'd probably think its more at first glance

I would think that the energy separation between the levels (orbitals) would be less since the extremely charged nucleus would pull them closer together..?

Thanks for your reply, btw

 

[lanedance]

average separation distance & energy level differnece are 2 very differnt things... and energy and "distance" can often have an inverse relationship, think of a praticle de broglie wavelength.

so in the hydrogen & Li+ case:
write out the simple energy for energy level as function primary quantum number n, how does E scale as Zis increased?

I thiink of it as extra proton greatly increase steepness and depth of the potential well the electron must reside in leading to higher energies

now you must think of your 2 electrons states, with different Zs. if we ignore nuclear spins, then you are considering exactly the same state transition only with a different Z

 

[IHateMayonnaise]

average separation distance & energy level differnece are 2 very differnt things... and energy and "distance" can often have an inverse relationship, think of a praticle de broglie wavelength.

so in the hydrogen & Li+ case:
write out the simple energy for energy level as function primary quantum number n, how does E scale as Zis increased?

E increases as Z increases. For ex,

$$E_n=E_R\left(\frac{Z^2}{n^2}\right)$$

In the case of hydrogenic atoms, which is not the case here.

I thiink of it as extra proton greatly increase steepness and depth of the potential well the electron must reside in leading to higher energies

now you must think of your 2 electrons states, with different Zs. if we ignore nuclear spins, then you are considering exactly the same state transition only with a different Z

So..then the energy separation will be the same, just deeper?