# First order perturbation theory

• Joe_UK
In summary, the perturbation on the normal hydrogenic Hamiltonian is represented by the difference between the perturbed Hamiltonian and the original Hamiltonian. This can be calculated by subtracting the two Hamiltonians in the two different regions, r > r_0 and r \le r_0.
Joe_UK
The potential of an electron in the field of a nucleus is:

-Ze^2/(4 Pi Epsilon0 r) r > r0
-Ze^2/(4 Pi Epsilon0 r0) r <= r0

where r0 is the fixed radius of the nucleus.

What is the pertubation on the normal hydrogenic Hamiltonian?

Calculate the change in energy of the 1s state to the first order.

Hint: If r0 is small e^(-r0/a0) ~ 1-r0/a0

-----

So I know the normal hydrogen Hamiltonian potential part is -Ze^2/(4 Pi Epsilon0 r) and I have been fiddling around with trying different terms added onto this such as:

-Ze^2/(4 Pi Epsilon0 (r0-r)) or -Ze^2/(4 Pi Epsilon0 r0/r)

However I can't find a pertubation that really seems to make sense or fit the original conditions. I think that the pertubation should also possibly be a constant not dependant on r, but I can't seem to find any at all that fit. I tried

-Ze^2/(4 Pi Epsilon0 r0)

but this just gave a potential 2x too big at r0. I know how to find the change in energy by <u*|H'|u> once I have the pertubation, but its just eluding me.

Cheers,
Joe

Well, the perturbation is what you add on to the original Hamiltonian to give the perturbed Hamiltonian:
$$H = H_0 + H'$$
You can easily rearrange that to
$$H' = H - H_0$$
But you know $H$ and $H_0$, so it should be a simple matter to find $H'$ by subtracting one from the other. It's slightly "unsimple" (I hesitate to say complicated) because $H$ is a piecewise function, with different definitions in two regions. All that really means, though, is that you need to calculate $H'$ separately in each of the two regions. Do one subtraction for $r > r_0$ and one for $r \le r_0$.

## 1. What is first order perturbation theory?

First order perturbation theory is a method used in quantum mechanics to approximate the energy levels and wavefunctions of a system when a small perturbation is applied. It is based on the assumption that the system can be separated into an unperturbed part, whose energy levels and wavefunctions are known, and a perturbation that slightly modifies the system's Hamiltonian.

## 2. When is first order perturbation theory applicable?

First order perturbation theory is applicable when the perturbation is small compared to the unperturbed part of the system and when the energy levels of the unperturbed system are known. It is also applicable when the perturbation is time-independent, meaning that it does not change with time.

## 3. How is first order perturbation theory used in practice?

To use first order perturbation theory, one must first determine the unperturbed energy levels and wavefunctions of the system. Then, the perturbation is added to the Hamiltonian, and the new energy levels and wavefunctions are calculated using the first order perturbation formula. This allows for an approximation of the system's behavior under the perturbation.

## 4. What are the limitations of first order perturbation theory?

First order perturbation theory is limited by the assumption that the perturbation is small compared to the unperturbed part of the system. If this assumption is not valid, higher order perturbation theory methods may be necessary. Additionally, first order perturbation theory is only applicable to time-independent perturbations.

## 5. What are some real-world applications of first order perturbation theory?

First order perturbation theory is used in many areas of physics, including atomic and molecular physics, solid state physics, and quantum chemistry. It is also used in engineering applications, such as in the study of electric circuits and mechanical systems. Additionally, first order perturbation theory has been used in the analysis of financial markets and economic systems.

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