Degeneracy Definition and 147 Threads

Electron degeneracy pressure is a particular manifestation of the more general phenomenon of quantum degeneracy pressure. The Pauli exclusion principle disallows two identical half-integer spin particles (electrons and all other fermions) from simultaneously occupying the same quantum state. The result is an emergent pressure against compression of matter into smaller volumes of space. Electron degeneracy pressure results from the same underlying mechanism that defines the electron orbital structure of elemental matter. For bulk matter with no net electric charge, the attraction between electrons and nuclei exceeds (at any scale) the mutual repulsion of electrons plus the mutual repulsion of nuclei; so in absence of electron degeneracy pressure, the matter would collapse into a single nucleus. In 1967, Freeman Dyson showed that solid matter is stabilized by quantum degeneracy pressure rather than electrostatic repulsion. Because of this, electron degeneracy creates a barrier to the gravitational collapse of dying stars and is responsible for the formation of white dwarfs.

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  1. blue_leaf77

    Does Commutativity Always Guarantee Shared Eigenkets?

    Let's denote ## \mathbf{p} ## and ## \Pi ## as the momentum and parity operators respectively. It's known that ## \mathbf{p} ## doesn't commute with ## \Pi ##, so they do not share the same set of eigenkets (plane wave doesn't have parity). But I just calculated that ##[\mathbf{p}^2,\Pi] = 0##...
  2. W

    How is Degeneracy Resolved by Building a CSCO?

    When finding a complete set of commuting observables, the goal, as I understand it, is to specify enough observables in the set such that, when given an eigenvalue of a state from each observable's matrix, these eigenvalues uniquely identify a corresponding eigenvector. It seems that every...
  3. ShayanJ

    Spin-3/2 particle and degeneracy in excited state

    Homework Statement Consider a particle with mass m in the following 1D potential: V(x)=\left\{ \begin{array}{lr} mgx \ \ \ x>0 \\ \infty \ \ \ \ \ \ \ x\leq 0 \end{array} \right. What is its minimum energy calculated using the uncertainty relation? Homework Equations \Delta x \Delta p...
  4. L

    A simple proof involving degeneracy and commutators

    Homework Statement In the absence of degeneracy, prove that a sufficient condition for the equation below (1), where \left|a'\right> is an eigenket of A, et al., is (2) or (3). Homework Equations \sum_{b'} \left<c'|b'\right>\left<b'|a'\right>\left<a'|b'\right>\left<b'|c'\right> = \sum_{b',b''}...
  5. K

    Density Matrix for Spin 1/2 particle in a magnetic field

    Hi everyone! I am trying to create the density matrix for a spin-1/2 particle that is in thermal equilibrium at temperature T, and in a constant magnetic field oriented in the x-direction. This is a fairly straightforward process, but I'm getting stuck on one little part. Before starting I...
  6. R

    Description of the Electron Degeneracy Principle

    I understand that the electron degeneracy principle states that no two electrons can occupy the same space at once. However, I do not think I clearly understand the physics behind that. I talked to my physics teacher about this and he said it had to do with the electron spin; I thought it was...
  7. T

    Statistics: Degeneracy and Multiplicity

    Homework Statement Consider drawing one card from a deck with no jokers or other special cards. a) What is the number of microstates? (4/13/52/cant tell) b) What is the number of macrostates?(4/13/52/cant tell) c) What is the degeneracy of macrostate spade? (4/13/52/cant tell) d) What is the...
  8. V

    Does energy level determine degeneracy in 3D systems?

    Hi everyone. I can not remember if, in 3D, the higher it is the energy level, the higher it is its degeneracy. With a cubic well and with a 3D harmonic oscillator it holds... Does anyone know if it is a general rule or not (and in the case it is, where does this rule come from)?
  9. R

    Hydrogen partition function and degeneracy

    So I'm trying to use the excitation law of Boltzmann and the ionisation law of Saha to calculate stuff about what percentage of a quantity of hydrogen in ionised and in what energy state. I have the temperature and energy levels values, so i still need the: -statistical weight (degeneracy)...
  10. J

    Degeneracy: Is it Necessary for Experiments?

    Sometimes it happens that hamiltonian has degenerate eigenvalue. This degeneracy can be removed by modifying the Hamiltonian. If the modification is additional term proportional to an adjustable parameter, the resulting difference between the eigenvalues can be made arbitrarily small. Is...
  11. H

    Set Up Eigenspace for Particle in 1st State: nx,ny,nz

    So what I do know about degeneracy is that it's the size of an eigenspace in a certain state. How would I go about setting up the eigenspace? Let's say for a particle in the first state with n values nx = 2, ny = 1, nz = 1
  12. U

    Addition of Angular Momenta - Degeneracy

    Homework Statement Show that when angular momenta ##j_1## and ##j_2## are combined, the total number of states is ##(2j_1 +1)(2j_1+2)## Homework Equations The Attempt at a Solution For the two gyros in the box, there are ##2j_1 +1## possible orientations of the first gyro...
  13. H

    Understanding the Degeneracy Discriminant in Classical and Quantum Regimes

    The degeneracy discriminant, that specifies the requirement for applicability of classical Maxwell-Blotzman statistics, is z<<1 where z is fugacity and z=exp^(\mu/kT). However when T\to \infty we would have z\to 1 which means we are in quantum regime while it is obvious that we are in...
  14. D

    Degeneracy of each of energies - Quantum

    Hello, I don't understand something in this exercice and i have another question:Homework Statement Use separation of variables in Cartesian coordinates to solve the infinite cubical well (or "particle in a box"): V (x, y, z) = { O,if x, y, z are all between 0 and a...
  15. S

    Ground State Degeneracy in ferromagnetic Heisenberg model

    I am reading the book "Lecture notes on Electron Correlation and Magnetism" by Patrik Fazekas. It says, "The ground state (of Heisenberg FM model) is not unique. We have just found that the system has the maximum value of the total spin Stot = LS. Sztot = LS state which is maximally...
  16. H

    How Do You Calculate Degeneracy in a 2D Particle in a Box?

    Question Particle in a box (2D) Determine the energy levels (degeneracy) of the lowest three I found that E = A (4a^2 + b^2) where A is a constant a and b are positive integers (principle quantum number) My steps I assume 4a^2 + b^2 = k where k is also a positive integer...
  17. ajayguhan

    Degeneracy Factor: Exploring Photon & Polarization

    What is degeneracy factor? What is the degeneracy factor of photon? What is the difference between degeneracy factor and polarization of light?
  18. Ken G

    Are there widespread misconceptions about degeneracy pressure?

    Two statements that are often made about degeneracy pressure are: 1) It is a new or special kind of pressure that requires quantum mechanics, in contrast with ideal gas pressure, which in effect involves somewhat mysterious forces that emerge from the Pauli exclusion principle, and 2) it...
  19. P

    Can Electron Degeneracy Pressure be Visualized?

    I have a question: is there any way to accurately "visualize" the phenomenon of electron degeneracy pressure? I understand that the main concept behind it is the Pauli Exclusion Principle. However, I was reading about the Chandrasekhar limit, and that it's derived from the fact that although a...
  20. T

    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...
  21. G

    Calculating Helmholtz energy (incl degeneracy)

    Hi I'm trying to calculate Helmholtz free energy from definition \frac{\partial A}{\partial T}=\frac{\partial}{\partial T}(kT\mathrm{ln}\; Z). First \langle E\rangle=\sum\limits_i p_i E_i=\sum\limits_i E_i\frac{g_i e^{-\beta E_i}}{Z}=\frac{1}{Z}\sum\limits_i E_i g_i e^{-\beta...
  22. K

    Can someone explain degeneracy pressure in simple terms?

    Sorry for the noob question, but is there a resource anyone can point me to for an easy to understand explanation of degeneracy pressure? I have no scientific background at all, so when I say easy I'm not kidding, but I am looking for a bit more than the pop sci bit of the Wikipedia entry...
  23. H

    Mistake in proof regarding degeneracy property

    Greetings, I was trying to prove a theorem regarding degeneracy, and I succeeded. However, I also proved the converse of the if-then part of the theorem (underlined below), which I know is wrong. I can't spot my mistake though. The theorem and my proof are written below - could someone...
  24. L

    Rotational invariance and degeneracy (quantum mechanics)

    Homework Statement Show that if a Hamiltonian H is invariant under all rotations, then the eigenstates of H are also eigenstates of L^{2} and they have a degeneracy of 2l+1. Homework Equations The professor told us to recall that J: \vec{L}=(L_x,L_y,L_z)...
  25. Z

    Degeneracy Parameter A: What is N, V, and k?

    When calculating whether Boltzmann, Bose-Einstein, or Fermi Dirac should be used there is an equation for Degeneracy Parameter A which can be initially used. Degeneracy Parameter A = (N/V) (h^2 / 2 TT m k T)^(3/2) What is represented by N and V and k? Is N the Number Density, V Volume and k...
  26. C

    Degeneracy for different energy states in Infinite cubic well

    Alright, I'm back with yet another question... So the prof was explaining that the energy in an infinite cubical well is E((h2∏2)/2ma2))(nx2+ny2+nz2) Which is all well and good, and he gave us the example of: ψ1,2,1 = E = 6((h2∏2)/2ma2)) And with little explanation mixed it up once...
  27. N

    Quantum harmoinic oscillator degeneracy

    Ok, I know this question will sound really stupid but I'm just not following the derivation for the formula of degeneracy's given by 1/2(n+1)(n+2) This is what I get n1+n2+n3=n so for a given n1, n2+n3=n-n1 Then this is the line I don't understand (and I'm sure its something simple I'm...
  28. T

    Is sixfold degeneracy the maximum degeneracy an angular momentum can have?

    If ψ is normalize-able and a function of nx, ny, nz, is the maximum energy degeneracy 6? I.E. There can be degeneracy at the same Energy with each state taking a different value of n, yet adding up to some (nx^2+ny^2+nz^2)=Same E, due to the linearity of the operators involved. I guess the...
  29. V

    Quantum degeneracy problem, electron on a ring

    Homework Statement Homework Equations Below The Attempt at a Solution So this is a lot like the infinite square well, except periodic. If S is an arc length, then S = \theta R so \frac{d^2}{dS^2} = \frac{1}{R^2}\frac{d^2}{d\theta^2}, which is more convenient to use in the hamiltonian. So...
  30. H

    Is Degeneracy in Quantum States Equally Probable in Thermodynamic Equilibrium?

    In Wikipedia this sentence is written about degeneracy; In physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, do you agree with the bold statement?
  31. G

    3-fold degeneracy: what is it?

    Hi everybody, I apologize for my trivial question, I'm reading a paper by Freedman and Kitaev and, when describing anyons and quantum Hall Effect with v=1/3, they say that "the ground state of electron liquid on the torus is 3-fold degenerate". What is the meaning? It means that there are 3...
  32. L

    Degeneracy of hydrogen energy levels

    Consider a universe where the intrinsic spin of the electron is S = 5/2, but all other parameters and Rules of Quantum Mechanics are the same. Find the degeneracy of the n=1 and n=2 levels of hydrogen. My understanding is that electrons in an atom have 4 quantum numbers n,l,ml,ms, and different...
  33. D

    Neutron stars not sustained by degeneracy force?

    I have just attended a talk, where the speaker (a professor in Hong Kong University) claims that neutron stars don't collapse due to "nuclear forces". He further explains that those nuclear forces are residual strong forces (i.e. exchange of pions). However, the mainstream saying (according to...
  34. J

    Did quantum degeneracy pressure drive inflation?

    Was cosmic inflation partly driven by the quantum degeneracy pressure of the quarks and electrons? Just after the Big Bang they would all be sitting on top of each other - just what fermions don't like to do!
  35. H

    Isotropic harmonic potential degeneracy of two identical particles

    Hello Everyone, I've come across the following question and can't seem to get the right answer out:Homework Statement Two identical particles are in an isotropic harmonic potential. Show that, if the particles do not interact and there are no spin-orbit forces the degeneracies of the three...
  36. Rapier

    What Is the Lowest Two-Fold Degenerate Energy Level in a 3D Quantum Box?

    Homework Statement A photon is confined by impenetrable barriers to a three-dimensional box of dimensions A = 300 nm by B = 400 nm by C = 500 nm, where A, B, and C are along the x, y, and z-axes, respectively. d) Let the triplet of integers (nx, ny, nz) denote an energy level. If two triplets...
  37. O

    Degeneracy of Hydrogen atomic orbitals with different l-values but same n-value

    I am terribly confused. I have always been hearing that in the hydrogen atom, 2s and 2p orbitals have the same energy. Similarly, the 3s, 3p and 3d orbitals have the same energies. This is also suggested by the hydrogen spectrum, my professor also believes the same, and I am unable to find...
  38. P

    How do you solve degeneracy for 2-D particle in a box?

    Homework Statement What is the degeneracy of the energy level E =65 E0 of the two dimensional particle in a box? Answer Homework Equations E=(h_^2/8mL^2)*(nx^2+ny^2)--> I think we use this eq. The Attempt at a Solution
  39. S

    Finding degeneracy of N Quantum Harmonic Oscillator

    Hey guys, For a particular problem I have to determine the total degeneracy across N 3-D Quantum Harmonic oscillators. Given that the degree of degeneracy for a 3-D harmonic oscillator is given by: (n+1)(n+2)/2 and the Total energy of N 3d quantum harmonic oscillators is given by...
  40. J

    Exploring the Role of Electron Degeneracy in White Dwarf Stars

    Yesterday i had a thought about when our sun runs out of energy and collapses into a white dwarf, it is my understanding when this happens a white dwarf is held up by electron degeneracy, all the electrons are under immense pressure but cannot fall to the lowest energy state therefore stopping...
  41. J

    How Does Polarization Relate to Photon Degeneracy?

    In photon statistics, g is defined as the internal degeneracy per particle, and the text gives the example that photon have two possible polarization states in three space dimensions, thus g=2. Why is the number of possible polarization equals the internal degeneracy of the particle?
  42. E

    Degeneracy in Linear Programming

    Homework Statement Consider the standard form polyhedron {x | Ax = b, x>=0} , and assume that the rows of the matrix A are linearly independent. (a) Suppose that two different bases lead to the same basic solution. Show that the basic solution is degenerate (has less than m non-zero...
  43. C

    Quantum mechanics, symmetry and degeneracy

    Homework Statement I'm struggling to understand the concept of symmetry in quantum mechanics. My notes state "In general if the probability density has lower symmetry than the hamiltonian, the wavefunction will be degenerate". I don't really get the connection with the hamiltonian. It...
  44. D

    Degeneracy in Wave Function Energy States

    Homework Statement What is meant by the term degenerate when referring to wave function energy states? Do the wavefunctions for degenerate states necessarily look the same? Explain. Homework Equations The Attempt at a Solution Degeneracy, in terms of wave function energy states...
  45. W

    Exploring "True" vs. Measured Degeneracy in Physical Systems

    Firstly hello, this is the first time I have posted here (although I have used the site to find info in the past). My query is best illustrated, I think, with an example. Suppose we have some physical system with corresponding state vector \left| \psi \right> = a \left| 0 \right> + b \left|...
  46. H

    Degeneracy in Quantum Mechanics

    Can anyone explain to me what Degeneracy is properly. I know its something to do with having different eigenvalues on the same energy level or something like that, but have not been able to find a good explanation in any textbooks or anywhere online. And how does something have infinite...
  47. W

    Level degeneracy of the transverse ising model

    the hamiltonian is H=-J \sum_{l=1}^N \sigma_l^z \sigma_{l+1}^z+ g \sigma_l^x here we assume periodic boundary condition my problem is, what is the highest possible degeneracy of the levels? initially i expected 2 but numerically i find that it is 4 i cannot understand it
  48. H

    Calculating Sodium Atom Ground State Degeneracy

    Hi guys, Just got a question I'm a little stuck on and would love a push in the right direction Q) Using Hartree's theory calculate the degeneracy of the ground state of the Sodium atom. Its a previous exam question and I'm struggling to find much descriptive information about the topic...
  49. Y

    4-Fold Degeneracy of 14th Energy Level in Cubic Well

    Homework Statement In reference to the particle in an infinite, cubic well I have been asked to calculate the degeneracy of the 14th energy level and comment on its special nature. 2. The attempt at a solution Notation: E_n = E (n_x , n_y , n_z) E_14 =...
  50. M

    Degeneracy of the Quantum Linear Oscillator

    So, today while doing my homework for statistical mechanics I was reading about the quantum linear oscillator in the textbook, "Classical and Statistical Thermodynamics" by Ashley H. Carter. In it, after discussing the quantized energy it says: "Note that the energies are equally spaced and...
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