Stationary Perturbation Theory

[tommy01]

Hi together...

When reading Sakurai's Modern Quantum Mechanics i found two problems in the chapter "Approximation Methods" in section "Time-Independent Perturbation Theory: Nondegenerate Case"

First:
The unperturbed Schrödinger equation reads
$$H_0 | n^{(0)}\rangle=E_n^{(0)} |n^{(0)}\rangle ~~~~~~ (1)$$
whereas the perturbed looks like
$$(E_n^{(0)} - H_0)|n \rangle = ( \lambda V - \Delta_n)|n\rangle ~~~~~~ (2)$$
with $$\Delta_n = E_n - E_n^{(0)}$$

The Equation is inverted and we arrive at
$$|n\rangle = \frac{1}{E_n^{(0)} - H_0} \Phi_n ( \lambda V - \Delta_n)|n\rangle$$ with the projection operator $$\Phi_n$$ to ensure that the inverse is well defined.

Then Sakurai says this can't be the correct form because as lambda approaches zero the perturbed state ket has to approach the unperturbed ket. Then he says even for lambda not equal to zero we can add a solution to the homogeneous equation (1) which is $$|n^{(0)}\rangle$$ and so the result is

$$|n\rangle = |n^{(0)}\rangle + \frac{1}{E_n^{(0)} - H_0} \Phi_n ( \lambda V - \Delta_n)|n\rangle$$

And there lies the problem. In my opinion (2) is also a homogeneous equation (but not the same as (1)) and you can't add a solution to a homogeneous equation to a different homogeneous equation. Or am I wrong?

Second:
The formalism then leads to an expresion for the energy shift
$$\Delta_n=\lambda V_{nn} + \lambda^2 \sum_{k\neq n} \frac{|V_{nk}|^2}{E_n^{(0)}-E_k^{(0)}} + ...$$

Then a special case of the no-level crossing theorem is stated.
Say we have 4 energy levels i, j, k and l in increasing order of magnitude.
Then Sakurai states that two levels connected by perturbation tend to repel each other. Inserting i and j in the equation above leads to a negative energy shift of i and a positive shift of j analogous for k and l.

But when we compare j and k they also repel each other which means j gets lower and k gets higher. In contradiction to the observations above.
Now i think there isn't really a contradiction because Sakurai only considered terms in second order of lambda and neglected other parts of the sum.

But then how he can deduce the statement that no two levels cross when connected by perturbation form the argumentation above?

I hope i could explain my two problems satisfying.

thanks in advance.
greetings.

 

[azntoon]

In regards to the first problem, you are correct in that equation (2) is a homogeneous equation. However, what Sakurai is trying to show is that the perturbed state ket should approach the unperturbed ket when lambda is equal to zero. To do this, it is necessary to add the solution to the homogeneous equation (1), |n^(0)> to the RHS of equation (2). This is why he adds the term, |n^(0)>, to the equation.As for your second problem, the no-level crossing theorem states that two levels which are connected by perturbation will not cross each other. In other words, one level will always be higher than the other. This is because the energy shift due to the perturbation is always negative for one level and positive for the other. Thus, when you consider levels i and j, the energy shift for i will be negative while the energy shift for j will be positive. Similarly, the energy shift for k will be positive and the energy shift for l will be negative. Therefore, j and k will both be pushed away from each other, with j getting lower and k getting higher. This is why the no-level crossing theorem holds.

 

[Entropia]

Hello,

Thank you for sharing your thoughts and questions about stationary perturbation theory. You have raised some valid points and I would like to address them in the following response.

Firstly, you are correct in noting that (2) is also a homogeneous equation, but it is not the same as (1). This is because (1) is the unperturbed Schrödinger equation, while (2) is the perturbed version. Therefore, adding a solution to (1) to (2) is not allowed, as they are different equations. However, what Sakurai is suggesting is to add a solution to the homogeneous version of (2), which would be of the form (E_n^(0) - H_0)|n'\rangle = 0, where |n'\rangle is a different state. This will not change the solution to (2), but it will ensure that the inverse is well-defined.

Moving on to your second point, you are correct in noting that the expression for the energy shift is only valid up to second order in λ. Therefore, it is possible that the levels j and k may cross if higher order terms are taken into account. However, the no-level crossing theorem states that in the absence of degeneracy, no two levels will cross each other. This means that even if higher order terms are taken into account, the levels will not cross. This is because the energy shift is always positive, leading to a repulsion between levels.

I hope this addresses your concerns and helps clarify any confusion you may have had. Stationary perturbation theory is a powerful tool for approximating solutions to the Schrödinger equation, but it is important to understand its limitations and assumptions. Keep exploring and questioning to deepen your understanding of this fundamental concept in quantum mechanics. Best of luck in your studies.