Energy level approximation

In summary, the problem requires estimating the lowest energy of an electron in a hydrogen atom using the equation for effective potential at its minimum. This is done by evaluating the equation for a specific value of orbital angular momentum, which gives a rough estimate. To improve the estimate, a second order Taylor expansion of the effective potential about a specific value of momentum is performed. However, it is unclear how this expansion will help with the estimation as the value at the specific momentum remains the same. Further manipulation may be needed to improve the estimate.
  • #1
Mjolnir
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Homework Statement


in the first part of the problem we were told to estimate the lowest energy of a electron in a hydrogen atom for a certain orbital angular momentum l by evaluating the equation for effective potential at it's minimum. That is to say that for any given l the minimum energy state has n = l + 1. If you then evaluate V-eff = l(l+1)/2p^2 -1/p for l = n - 1 and p=p0=l(l+1) this should be a rough estimate of the actual energy. Now this is easy enough to evaluate and gives not too large of an overestimate; the next part of the problem, however, has me a little confused. For the same case I'm supposed improve the estimate by making a second order taylor expansion of V-eff about p0. The thing is I really don't see how a taylor expansion is supposed to help me, especially considering that the value at p0 will be no different than that of the original function.


Homework Equations


V-eff = l(l+1)/2p^2 -1/p
exact energy eigenvalues = [tex]\lambda[/tex]n = 1/(2n^2), n = 1, 2, 3, ...



The Attempt at a Solution


Finding the taylor series is simple enough, giving you
(n^2-n-p)^2/(n*(n-1))^3 - 1 /(2n*(n-1))
The problem is that I don't know what to do with this now that I have it.
 
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  • #2
I feel like there should be some additional manipulation that I'm missing but I don't know what. The value of the function at p0 will still be the same, so it doesn't seem like it should help me improve the estimate.
 

1. What is energy level approximation?

Energy level approximation is a method used in quantum mechanics to simplify the calculation of the energy levels of a system. It involves approximating the potential energy of a system as a series of steps or "levels" instead of a continuous function.

2. Why is energy level approximation used?

Energy level approximation is used because it simplifies the calculations involved in determining the energy levels of a system, making it easier to analyze and understand complex physical systems.

3. How accurate is energy level approximation?

The accuracy of energy level approximation depends on the complexity of the system being studied and the level of approximation used. In general, it provides a good estimate of the energy levels, but may not be exact.

4. Can energy level approximation be used for all systems?

No, energy level approximation is typically used for systems with a discrete set of energy levels, such as atoms, molecules, and particles in a box. It is not suitable for systems with continuous energy spectra, such as free particles or solids.

5. Are there any limitations to energy level approximation?

Yes, energy level approximation has limitations and may not accurately describe the behavior of a system under certain conditions. It is important for scientists to carefully consider the assumptions and approximations made when using this method, and to use more advanced techniques when necessary.

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