Stationary Perturbation theory
Stationary perturbation theory is a mathematical method used in quantum mechanics to approximate the behaviour of a quantum system when subjected to a small external perturbation. The term "stationary" refers to the fact that the system is assumed to be in a stationary state, meaning its state does not change with time. This theory is particularly useful in situations where the perturbation is relatively small compared to the unperturbed system.
The basic idea is to start with a known solution to the unperturbed Hamiltonian (the operator representing the total energy of the system) and then use perturbation theory to find corrections to the energy levels and wavefunctions caused by the perturbation. The perturbation parameter is typically denoted by a symbol like λ and is assumed to be small.
Here's a brief outline of the method:
Unperturbed System (H₀): Begin with a quantum system described by an unperturbed Hamiltonian operator 0H0 for which the solutions are known. These solutions often correspond to eigenstates and eigenvalues.
Perturbation (H' or λV): Introduce a perturbation term ′H′ or λV, where λ is the perturbation parameter and V is the perturbing potential.
Perturbed Hamiltonian (H = H₀ + H'): Form the total Hamiltonian H by adding the unperturbed Hamiltonian to the perturbation term.
Perturbation Series: Express the wavefunction and energy of the perturbed system as a series expansion in powers of λ. The series typically takes the form: E=E(0)+λE(1)+λ2E(2)+…
Solution of Each Order: Solve the equations order by order, obtaining corrections to the wavefunction and energy at each order of λ.
Physical Interpretation: Analyze the physical significance of the corrections and determine whether the perturbation theory is convergent.
We consider a system whose Hamiltonian operator H can be split into two parts as
where Ho is the Hamiltonian operator whose eigen function and the eigen values are exactly known.
and eigen values
are known. So we are going to find eigen functions and eigen values of total Hamiltonian operator H as
The set of eigen kets of Ho constitutes othogonal basis. In perturbation theory we used the approximation
Writing eigen ket and eigen value in form of series as
Comparing the coefficient of like power of 𝜆 on both sides
Multiplying both side by <𝜓n(0)l
Since H' is very very small in comparison to Ho. So we can write
Multiplying both side by <𝜓n(0)l
First order correction to energy
Total energy with first order correction
Multiplying both sides by <𝜓m(0)l
Second order correction to energy
Second order correction to wavefunction
Multiplying by < 𝜓k(0) l
Stationary perturbation theory is a powerful tool when dealing with systems where the perturbation is small, allowing for a systematic and organized approach to finding corrections to the energy levels and wavefunctions. It's important to note that the method may not always converge for all systems, and its applicability depends on the specific characteristics of the problem at hand.
This note is a part of the Physics Repository.