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.
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
##...
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.
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...
I was learning about Degenerate Perturbation Theory and I encountered the term 'Degenerate Subspace', I didn't really understand what it meant so I came here to ask - what does it mean? will it matter if i'll say 'Degenerate space' instead of 'Degenerate Subspace'? and subspace of what? (...
Hello! I am reading Griffiths and I reached the Degenerate Time Independent Perturbation Theory. When calculating the first correction to the energy, he talks about "good" states, which are the orthogonal degenerate states to which the system returns, once the perturbation is gone. I understand...
I'm struggling to understand degenerate perturbation theory. It's clear that in this case the 'normal' approximation method fails completely seeing as you get a divide by zero.
I follow the example for a two state system given in e.g D.J Griffiths "Introduction to Quantum Mechanics"
However...
I'm reading section 5.2 "Time-Independent Perturbation Theory: The Degenerate Case" of the book "Modern Quantum Mechanics" by Sakurai and Napolitano and I have trouble with some parts of the calculations.
At firsts he explains that there is a g-dimensional subspace(which he calls D) of...
Homework Statement
I did poorly on my exam, which I thought was very fair, and am now trying to understand certain aspects of perturbation theory. There are a total of three, semi related problems which i have questions about. They are mainly qualitative in nature and involve an intuitive...
Homework Statement
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The isotropic harmonic oscillator in 2 dimensions is described by the Hamiltonian $$\hat H_0 = \sum_i \left\{\frac{\hat{p_i}^2}{ 2m} + \frac{1}{2} m\omega^2 \hat{q_i}^2 \right\} ,$$ for ##i = 1, 2 ## and has energy eigenvalues ##E_n = (n + 1)\hbar \omega \equiv (n_1 +...
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...
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...
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...
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 =...
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...
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...
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...
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...
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...
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...
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!
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...
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...
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...
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...
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
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...
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...
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...
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...
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...
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...
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...