# Question on Time-independent perturbation theory: I am confused

• ani4physics
In summary, the conversation discusses time-independent perturbation theory and how it affects the energy and wave function of an atom in ground state. The question is whether the average energy is calculated using E0 = E0(0) + E0(1) + E0(2) + ... or [<0](0) + <0](1) + <0](2) + ...] [ H0 + lambda V] [[0>(0) + [0>(1) + [0>(2) + ...]. The speaker believes they should be equal, but in the second formula, terms such as <0(1)] H0 [0(1)> appear which do not appear in the first equation. The speaker also
ani4physics
We all know from time-independent perturbation theory that if we have an atom in ground state [0>, and when a time-independent perturbation acts on it, the energy of the ground state gets shifted and the ground state wave function also gets modified. Using Time-independent Schroedinger eq.,

[H0 + lambda . V] [0> = [E0] [0>, where V is the perturbation hamiltonian.

Now we expand E0 as E0 = E0(0) + lambda . E0(1) + lambda^2. E0(2) +...
and [0> as [0> = [0>(0) + lambda. [0>(1) + lambda^2. [0>(2) + ...

Then we compare powers of lambda, left multiply with <0] to get the energy corrections and so on.

My question is, is the average energy of the atom E0 = E0(0) + E0(1) + E0(2) + ...

or is it [<0](0) + <0](1) + <0](2) + ...] [ H0 + lambda V] [[0>(0) + [0>(1) + [0>(2) + ...]

Which one is it? They give different result.

They shouldn't be different to the order you calculated the wavefunction. However, it is a general theorem that the wavefunction of order n is sufficient to calculate the energy to order 2n+1 with the second formula you gave.

DrDu said:
They shouldn't be different to the order you calculated the wavefunction. However, it is a general theorem that the wavefunction of order n is sufficient to calculate the energy to order 2n+1 with the second formula you gave.

Yeah I was hoping they should be equal. But in the second formula we get terms like <0(1)] H0 [0(1)>, which do not appear in the equation E = E0(0) + E0(1) + E0(2) +..., because

E0(2) = <0] V [0(1)>

Be carefull, because your original wavefunction and the perturbed wavefunction do not have the same norm, i.e., although you may chose <0][sum_n O(n)>=1, <sum_n O(n)][sum_n O(n)> >1. That is you have to divide your last formula in #1 by <sum_n O(n)][sum_n O(n)>.

## 1. What is time-independent perturbation theory?

Time-independent perturbation theory is a mathematical method used in quantum mechanics to solve for the energy levels and wavefunctions of a perturbed system. It is based on the assumption that the perturbation, or disturbance, acting on the system does not depend on time.

## 2. How does time-independent perturbation theory differ from time-dependent perturbation theory?

Time-independent perturbation theory is used for systems in which the perturbation does not change over time, while time-dependent perturbation theory is used for systems in which the perturbation varies with time. Time-independent perturbation theory is also simpler and easier to solve compared to time-dependent perturbation theory.

## 3. What are the assumptions made in time-independent perturbation theory?

The main assumptions made in time-independent perturbation theory are that the perturbation is small and that the perturbed system can be approximated as a sum of the unperturbed system and a small perturbation term. Additionally, it assumes that the perturbation is time-independent and that the perturbation does not cause any degeneracy in the energy levels of the system.

## 4. How is time-independent perturbation theory applied in practice?

To apply time-independent perturbation theory, the Hamiltonian of the perturbed system is written as a sum of the Hamiltonian of the unperturbed system and a small perturbation term. Then, the unperturbed energy levels and wavefunctions are used to calculate the corrections to the energy levels and wavefunctions caused by the perturbation. These corrections are then added to the unperturbed energy levels to obtain the perturbed energy levels.

## 5. What are some examples of systems where time-independent perturbation theory is used?

Time-independent perturbation theory is commonly used in quantum mechanics to study systems such as atoms, molecules, and solids. It is also used in other fields, such as nuclear physics, to study the effects of perturbations on systems such as nuclei. Additionally, it is used in theoretical chemistry to calculate the effects of electron correlation on molecular systems.

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