Time independent perturbation theory

[cks]

H=H0 + lambda * W

lambda << 1 must hold and the matrix elements of W are comparable in magnitude to those of H0.

More precisely, the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0.

I don't understand what is the meaning of " the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0".

(the above explanation are obtained from the SChaum's Outlines of Quantum Mechanics)

 

[cks]

the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0".

Let's say the matrix W=[2.2 3.1 4.1; 4.1 5.3 6.0; 7.3 8.2 9.3] (matlab code)

let's say the eigenvalues of H0 are 1 2 3 4 5 6 7 8 9

the matrix element 2.2 is roughly the same as the difference of the eigenvalues of 3-1. Am I understanding this correctly?

the matrix elements of W are of the "same magnitude"(don't understand what same magnitude means?) as the difference(difference? difference between which eigenvalues, in my example, there are 9 eigenvalues, which minus which is the difference the author is talking?) between the eigenvalues of H0".

 

[denian]

Time independent perturbation theory is a powerful tool used in quantum mechanics to approximate the behavior of a system when a small perturbation is introduced. In this case, the perturbation is represented by the term lambda * W, where lambda is a small parameter and W is a perturbation operator.

The condition lambda << 1 ensures that the perturbation is small enough to be treated as a perturbation rather than a significant change to the system. This allows us to use perturbation theory to calculate the corrections to the energy levels and wavefunctions of the system.

The statement that "the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0" means that the perturbation operator W is comparable in size to the difference between the energy levels of the unperturbed system, H0. This condition is important because it ensures that the perturbation is strong enough to cause significant changes in the system, but not too strong that it completely dominates the behavior of the system.

In summary, time independent perturbation theory is a useful tool in quantum mechanics when dealing with small perturbations to a system. The conditions of lambda << 1 and comparable matrix elements of W to the energy differences of H0 are crucial in ensuring the accuracy and applicability of this theory.