Quantum Mechanics/Perturbation Theory

Introduction
Perturbation Theory is an extremely important method of seeing how a Quantum System will be affected by a small change in the potential. And as such the Hamiltonian.

Perturbation Theory revolves around expressing the Potential as multiple (generally two) separate Potentials, then seeing how the second affects the system.

It allows us to get good approximations for systems where the Eigenstates are not all easily findable.

In the real life not many hamiltonians are exactly solvable. Most of the real life situations requires some approximation methods to solve their hamiltonians. Perturbation theory is one among them. Perturbation means small disturbance. Remember that the hamiltonian of a system is nothing but the total energy of that system. Some external factors can always affect the energy of the system and its behaviour. To analyse a system's energy, if we dont know the exact way of solution, then we can study the effects of external factors(perturbation) on the hamiltonian. For an example, we can study the elastic properties of a spring by the application of load on that. By this way, we can approximately predict the properties of the concerned systems. Perturbation applied to a system is of two types: time dependent and time independent and hence the theory. We have to split the hamiltonian into two parts. One part is a hamiltonian whose solution we know exactly and the other part is the perturbation term. By this way we can solve the problems with a very good approximation. For an example of this method in quantum mechanics, we can use the hamiltonian of the hydrogen atom to solve the problem of helium ion.

Energy
The first order correction to energy is given by $$ \epsilon_n ^{(1)} = \langle n|V|n\rangle = \int \Psi_n ^{(0)^\star}(x) \hat V(x) \Psi_n ^{(0)}(x) dx$$

The first order correction to the wave function is $$ C_k ^{(1)}= \frac{\langle k|V|n \rangle}{\epsilon_n ^{(0)}-\epsilon_k ^{(0)} } $$

Proof
Let $$ \hat H_0 $$ be a Hamiltonian where all eigenstates are known. $$ \hat H_0 \Psi_n ^{(0)} = \epsilon _n ^{(0)} \Psi_n ^{(0)} \qquad n \in\mathbb N \qquad (1)$$

$$ (\hat H_0 + \hat V) \Psi = \epsilon \Psi \qquad (2)$$

Now represent $$ \Psi $$ as a linear combination of all its parts, $$ \Psi = \sum c_m \Psi_m^{(0)}$$ and put this into (2).