# What is Degenerate perturbation theory: Definition and 34 Discussions

In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.
Degeneracy plays a fundamental role in quantum statistical mechanics. For an N-particle system in three dimensions, a single energy level may correspond to several different wave functions or energy states. These degenerate states at the same level are all equally probable of being filled. The number of such states gives the degeneracy of a particular energy level.

View More On Wikipedia.org
1. ### I Doubt in understanding degenerate perturbation theory

McIntyre, quantum mechanics,pg360 Suppose states ##\left|2^{(0)}\right\rangle## and ##\left|3^{(0)}\right\rangle## are degenerate eigenstates of unperturbed Hamiltonian ##H## Author writes: "The first-order perturbation equation we want to solve is ##...
2. ### Diagonalizing q1ˆ3q2ˆ3 with Degenerate Perturbation Theory

I tried to use the degenerated perturbation theory but I'm having problems when it comes to diagonalizing the perturbation q1ˆ3q2ˆ3 which I think I need to find the first order correction.
3. ### A Degenerate Perturbation Theory: Correction to the eigenstates

Given the unperturbed Hamiltonian ##H^0## and a small perturbating potential V. We have solved the original problem and have gotten a set of eigenvectors and eigenvalues of ##H^0##, and, say, two are degenerate: $$H^0 \ket a = E^0 \ket a$$ $$H^0 \ket b = E^0 \ket b$$ Let's make them...

12. ### Wigner-Eckart theorem in Stark effect

Hi. I'm reviewing some past qualifying exams and stumbled on something i can't figure out, probably because I'm still confused about the Wigner-Eckart theorem.. So, the set-up is just degenerate perturbation theory for constant electric field along z on the n = 2 hydrogen states. That's a...
13. ### Spin-orbit Interaction & Degenerate Perturbation Theory

Hello! This is my first time posting, so please correct me if I have done anything incorrectly. There's something that I don't understand about the spin-orbit interaction. First of all I know that [\hat{S} \cdot \hat{L}, \hat{L_z}] \ne 0 [\hat{S} \cdot \hat{L}, \hat{S_z}] \ne 0 so this means...
14. ### Degenerate Perturbation Theory

Homework Statement We have spin-1 particle in zero magnetic field. Eigenstates and eigenvalue of operator \hat S_z is - \hbar |-1> , 0 |0> and \hbar |+1> . Calculate the first order of splitting which results from the application of a weak magnetic field in the x direction. Homework...
15. ### Conmutative Hermitian operator in degenerate perturbation theory

Hi. In 2-fold degenerate perturbation theory we can find appropiate "unperturbate" wavefunctions by looking for simultaneous eigenvectors (with different eigenvalues) of and H° and another Hermitian operator A that conmutes with H° and H'. Suppose we have the eingenvalues of H° are ##E_n =...
16. ### Sakurai Degenerate Perturbation Theory: projection operators

Hi, So, I am working through section 5.2 of Sakurai's book which is "Time Independent Perturbation Theory: The Degenerate Case", and I see a few equations I'm having some trouble reconciling with probably because of notation. These are equations 5.2.3, 5.2.4, 5.2.5 and 5.2.7. First, we...
17. ### Degenerate Perturbation Theory: Two Spin 1/2 Particles

So I know this might be a lot to read but I am having a very hard time understanding how to use the formulas in degenerate perturbation theory. Here is the problem I am on. Homework Statement A system of two spin-1/2 particles is described by the following Hamiltonian...
18. ### Degenerate Perturbation Theory

http://farside.ph.utexas.edu/teaching/qm/lectures/node53.html So I was reading this and I don't understand how he goes from 658 to 661 using the completeness relation. In 661 if you use the completeness relaton can you get rid of the I n,l''>s by doing the outer product and ignoring the...
19. ### Degenerate perturbation theory (Sakurai's textbook)

In the theory of degenerate perturbation in Sakurai’s textbook, Modern Quantum Mechanics Chapter 5, the perturbed Hamiltonian is H|l\rangle=(H_0 +\lambda V) |l\rangle =E|l\rangle which is written as 0=(E-H_0-\lambda V) |l\rangle (the formula (5.2.2)). By projecting P_1 from the left (P_1=1-P_0...
20. ### Degenerate Perturbation Theory (Particle in 3D box)

Homework Statement Consider a particle confined in a cubical box with the sides of length L each. Obtain the general solution to the eigenvalues and the corresponding eigenfunctions. Compute the degeneracy of the first excited state. A perturbation is applied having the form H' = V from 0...
21. ### Degenerate perturbation theory degeneracy not lifted by perturbation

Hi, I have an equation of the form (-i \lambda \frac{d}{dr}\sigma_z+\Delta(r)\sigma_x) g =(\epsilon + \frac{\mu \hbar^2}{2mr^2}) g where \sigma refers to the Pauli matrices, g is a two component complex vector and the term on the right hand side of the equation is small compared to the other...
22. ### Weak field zeeman effect, why don't we still use degenerate perturbation theory?

the spin orbit coupling removes the degeneracy but not completely, should we still use the degenerate perturbation theory. is it because of relativistic corrections? Thanks!
23. ### Degenerate Perturbation Theory

Hi I am reading about Degenerate Perburbation Theory, and I have come across a question. We all know that the good quantum numbers in DPT are basically the eigenstates of the conserved quantity under the perburbation. As Griffiths he says in his book: "... look around for some hermitian...
24. ### Degenerate Perturbation Theory Wavefunction Correction

Hi, If we have a non degenerate solution to a Hamiltonian and we perturb it with a perturbation V, we get the new solution by |\psi_{n}^{(1)}> = \sum \frac{<\psi_{m}^{(0)}|V|\psi_{n}^{(0)}>}{E_n^{(0)} - E_m^{(0)}}\psi_m^{(0)} where we sum over all m such that m\neq n. When we do the same...
25. ### Degenerate perturbation theory help

So in time-independent degenerate perturbation theory we say that we can construct a set of wavefunctions that diagonalize the perturbation Hamiltonian (H') from the degenerate subspaces of the unperturbed Hamiltonian (Ho). Since the original eigenstates are degenerate, combinations of them are...
26. ### Degenerate Perturbation Theory: Operator commuting with Ho and H'

Homework Statement Question is: Prove the following: Let A be a Hermitian operator that commutes with H0 and perturbation H'. If two degenerate states have distinct eigenvalues for A, then the matrix element of perturbation between them is zero! The real problem is I don't understand...
27. ### Degenerate perturbation theory

Homework Statement Hi, i have put the question, my attempt and actual answer in the attached picture. My answer is not quite right; firstly why is the second term a minus lambda, and where does the O(lamdba^2) come from? Homework Equations The Attempt at a Solution
28. ### How Does Degenerate Perturbation Theory Apply to Matrix Elements?

Homework Statement Ok, so i have this online test to be completed by tomorrow and i have NO IDEA how to go about it, my notes are useless, they don't explain anything. On the up side all the questions seem to be on a very similar topic so if i could understand some key ideas then i should be...
29. ### Non-degenerate and degenerate perturbation theory

Consider a system of a rigid rotator together with a uniform E-field directing along z-axis. So to calculate the perturbed energy and wavefunction we have to use perturbation theory. But the book said we can use non-degenerate one to calculate the result. I wonder why. It is because the original...
30. ### Quantum Mechanics: Degenerate Perturbation Theory on square well

Homework Statement Hi I am trying to apply degenerate perturbation theory to a three dimensional square well v= 0 for x, y,z interval 0 to a, perturbed by H' = xyz (product) from 0 to a, otherwise infinite. I need to find the correction to energy of the first excited state which I know is...
31. ### Griffiths' explanation of degenerate perturbation theory

Hi all. I'm reading about time-independent perturbation theory for degenerate states in Griffiths' Introduction to QM. I have a question on the things he writes in chapter 6.2, page 269. What does he mean by the so-called "good" linear combinations? I hope you can shed some light on...
32. ### Degenerate Perturbation Theory Question

Hello, This is a question on perturbation theory - which I am trying to apply to the following example. Homework Statement The two-dimensional infinitely deep square well (with sides at x=0,a; y=0,a) is perturbed by the potential V(x)=\alpha(x^{2}+y^{2}). What is the first-order correction...
33. ### Degenerate perturbation theory question

This's a question from Griffiths, about degenerate pertrubation theory: For \alpha=0, \beta=1 for instance, eq. 6.23 doesn't tell anything at all! What does it mean "determined up to normalization"?. Equations 6.21 and 6.23 involve 3 unknowns (\alpha, \beta, E^1), and Griffiths solved them...
34. ### Degenerate perturbation theory question

Can anybody explain what Griffiths means when he talks about "good eigenstates" in degenerate time-independent purturbation theory? Mathematically, I know he is just talking about the eigen-vectors of the W matrix (where Wij = <pis_i|H'|psi_j>). But what do the eigen-vectors physically...