What is Degeneracy: Definition and 147 Discussions

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. kamui1

    Solving for degeneracy electron cloud temperature

    When I try P_rel_e = P_ideal I couldn't get a single number that is close to the given T_Max. It might be that I used the wrong equations but I am not sure. Can anyone give me some guidence on this question?
  2. E

    I Understanding No Energy Degeneracy in Sakurai's Quantum Mechanics

    Hello, I'm hoping someone can help me understand a statement in Sakurai Modern Quantum Mechanics (3rd edition). In particular, in the section that describes free particle in infinite spherical well (page 198, section 3.7.2), after the text has shown that for a given ##l## value, the energy...
  3. H

    Degeneracy of the energy level

    Energy of the One-dimensional box: ground state: En = (n^2*h^2) / (8mL^2), where n=1 twice the ground state: 2* En = 2 [(1^2*h^2) / (8mL^2)] Energy of the Three-dimensional box: En = (nx^2 + ny^2 + nz^2) *h^2 / (8mL^2) = 2 (1^2*h^2) / (8mL^2) As stated, twice the ground state energy of one...
  4. L

    I Double delta potential -- Degeneracy of bound states in one dimension?

    I have a question from the youtube lecture That part starts after 42 minutes and 47 seconds. Balakrishnan said that if delta barriers are very distant (largely separated) then we have degeneracy. I do not understand how this is possible when in 1d problems there is no degeneracy for bond states.
  5. mcas

    Perturbation theory and the secualr equation for double degeneration

    I've been assigned to do a problem from Landau which you can read below: I have no problem with finding the energy. Then I write down the equations: \begin{equation*} \begin{cases} (V_{11}-E^{(1)})|c_1|e^{i\alpha_1} + V_{21}e^{i\alpha_2}|c_2| = 0\\ V_{12}e^{i\alpha_1}|c_1| +...
  6. M

    Degenerate perturbation theory

    $$ W_{n,n} = \int_0^{2 \pi} \frac{1}{\sqrt{2 \pi}} e^{-inx} V_0 \cos(x) \frac{1}{\sqrt{2 \pi}} e^{inx} dx $$ $$ = 0 $$ $$ W_{n, -n} = \int_0^{2 \pi} \frac{1}{\sqrt{2 \pi}} e^{-inx} V_0 \cos(x) \frac{1}{\sqrt{2 \pi}} e^{-inx} dx $$ $$ = \frac{a n ( \sin(4 \pi n) + i \cos( 4 \pi n) - i...
  7. docnet

    Energy degeneracy in a cubical box

    For one-dimensional binding potential, a unique energy corresponds to a unique quantum state of the bound particle. In contrast, a particle of unique energy bound in a three-dimensional potential may be in one of several different quantum states. For example, suppose that the three-dimensional...
  8. G

    B Degeneracy Pressure: Pauli Principle & Neutron Stars

    Is degeneracy pressure created due to Pauli exclusion principle able to create some waves? Also at neutron star stage similarly are there waves created may be of higher energy? Can we talk of some harmonic motions in these stages?
  9. S

    Calculating degeneracy of the energy levels of a 2D harmonic oscillator

    Too dim for this kind of combinatorics. Could anyone refer me to/ explain a general way of approaching these without having to think :D. Thanks.
  10. Kayla Martin

    Why is the degeneracy criterion the same for all stars?

    Is it anything to do with the equation being independent of mass etc.. of the star? or to do with the Pauli exclusion principle? Any help will be much appreciated.
  11. J

    I Accuracy of the Density of States

    I'm trying to understand the detailed concept of why the density of states formula is accurate enough to calculate the number of quantum states of an energy level, including degeneracy, within a small energy interval of ##dE##. The discrete energie levels are calculated by $$E = \frac{h^2 \cdot...
  12. tanaygupta2000

    Degeneracy pressure of a White Dwarf Star

    So far, I am provided with all the required values for calculation, except N. If N = total number of electrons in star, then using N = mass of star/ mass of an electron should be no problem. Am I right?
  13. Noki Lee

    I Valley degeneracy in tunneling current

    I'm trying to understand the valley degeneracy to calculate the tunneling current. Here is the equation of tunneling current.$$I_T=q\frac {g_sg_v} {L} \sum_{k} v_g(k)(f_v-f_c)T$$ ##g_v## is valley degeneracy. I thought it comes from the symmetry of structures, depending on a certain point in...
  14. S

    Degeneracy of hydrogen energy levels

    I'm considering a hydrogen atom placed in an infinite potential on one side of the nucleus, i.e. ##V(x) = +\infty## for ##x < 0##. I require the wavefunctions to be odd in order to satisfy the boundry condition at ##x=0##. By parity of the spherical harmonics only states with ##l## odd are...
  15. binbagsss

    A Fractional Quantum Hall Effect- degeneracy of ground state (Tong's notes)

    Hi , I'm looking at the argument in David Tongs notes (http://www.damtp.cam.ac.uk/user/tong/qhe/three.pdf) for ground state degeneracy on depending on the topology of the manifold (page 97, section 3.2.4). I follow up to getting equation 3.31 but I'm stuck on the comment after : ' But such an...
  16. V

    Solid State Textbook on crystal field theory and degeneracy breaking

    In particular I would like to have a resource for the relation between group theory, crystal field symmetries and breaking of degeneracies of orbitals. I've taken a graduate condensed matter course and graduate quantum mechanics courses. I have some basic knowledge of group theory but can learn...
  17. J

    I Value of beta in Boltzmann Statistics taking degeneracy into account

    Hello, The relationship between entropy ##S##, the total number of particles ##N##, the total energy ##U(β)##, the partition function ##Z(β## and a yet to be defined constant ##β## is: $$S(\beta)=k_BN \cdot \ln(Z(\beta)) - \beta k_B \cdot U(\beta)$$ Which leads to: $$\frac{dS}{d\beta} =...
  18. P

    I Neutron vs electron degeneracy pressure

    How is neutron degeneracy pressure able to support a much higher density object such as a neutron star where electron degeneracy pressure only supports a comparatively less dense object such as a white dwarf. Conceptually I would think electron degeneracy pressure to be stronger due to the...
  19. S

    A Dipole moment of an isolated quantum system

    How to prove the dipole moment of an isolated quantum system in isotropic space is identically equal to zero, unless there exists an accidental degeneracy. Thanks in advance
  20. BlackPowder

    I Is the degeneracy of N2+ different from N2?

    The calculation of degeneracy of diatomic molecules can be easily found. However, there is no detail introduction of ions. Not sure if the electronic, vibrational, rotational, and nuclear spin statistical weights are differ from N2+ to N2. Please help. Thanks.
  21. mertcan

    I Linear programming -- Bland rule degeneracy

    hi, my question is related to a proof involves bland rule for avoiding the degeneracy. Initially I emphasized some sentences which have importance in attachment/file with yellow color. At the beginning, it says xs is entering variable and when it enters objective value does not change( because...
  22. LarryS

    I Probabilities for degenerate eigenvalues?

    In non-relativistic QM, given a wave function that has a degenerate eigenvalue for some observable, say energy. There is a whole subspace of eigenvectors associated with that single degenerate eigenvalue. How is the measurement probability for that degenerate eigenvalue computed from the...
  23. H

    A How to calculate the degeneracy of an energy band?

    Could somebody write the guide for calculate the degeneracy of energy band by group theory? For instance, the valence band of Si and Ge in Gamma point. Thanks a lot!
  24. P

    B Why is there no proton degeneracy pressure?

    Hello. I usually heard about electron degeneracy pressure and neutron degeneracy pressure. But I´ve never heard about a proton degeneracy pressure. Why is this?
  25. S

    B Angular momentum, degeneracy pressure, and cosmic inflation

    Considering the angular momentum of a collapsing star preventing it from resulting in a black hole by degeneracy pressure, are there ekpyrotic universe models that include angular momentum and degeneracy pressure as key factors of cosmic inflation?
  26. N

    Addition of spin and degeneracy

    Homework Statement S=1/2 I1=I2=1/2 (nuclear spin) I=I1+I2 J=S+I Homework Equations H= a S . I1 + a S.I2 The Attempt at a Solution A) Find the values of I, the eigenvalues of I2, their degree of degeneracy and show that [H,I2]=0Using momentum addition I get I = 0 or 1 I2 has for...
  27. D

    I Is the particle in a 1-D infinite well doubly degenerate?

    Hi. I'm confused about degeneracy in I-D As far as I understand it ( and please tell me if I'm wrong ) ; a free particle is doubly degenerate with a continuous energy spectrum with eigenfunctions eikx and e-ikx. A particle on a ring in I-D is doubly degenerate but this time the energy is...
  28. Ranku

    I Can negative pions replace electrons in atoms?

    Is Pauli Exclusion Principle alone responsible for electron degeneracy pressure?
  29. J

    B Neutron, White Dwarfs and Degeneracy Pressure

    Hi, I have been trying to get some physics behind the cause of the degeneracy pressure but have some confusion with the stuff I have found. Apparently the cause of degeneracy pressure can be explained through the uncertainty principle. If you trap electrons in a smaller and smaller space the...
  30. G

    I Common eigenfuctions for degeneracy

    Hello. I read the textbook and found that common eigenfunctions are even possible for degenerate eigenvalues. Let's say operators A and B commutes and eigenvalue a of operator A is N-fold degenerate, means that there are N linearly independent eigenfunctions having same eigenvalue a. These...
  31. FallenApple

    I Electron Degeneracy Pressure not Fundamental?

    So for what I understand, when a star collapses, the electrons do not like to overlap their quantum states because of the pauli exlusion principle. Is this different from an E&M force? If so, then why isn't it a fundamental interaction? All forces are made of a combination of the 4 fundamental...
  32. stevendaryl

    I Partition Function and Degeneracy

    There doesn't seem to be a forum that is specifically about statistical mechanics, so I'm posting this question here. I apologize for the long-winded introduction, but I think it's needed to provide context for my question: If you have a discrete collection of single-particle energy levels...
  33. Kara386

    Degeneracy of Fermionic gas

    Homework Statement The question asked me to show that the entropy of a fermionic gas is ##S = -k_B \Sigma_i (1-f_i)\ln(1-f_i) +f_i\ln(f_i)## Using the Fermi-Dirac distribution so ##f_i = \frac{n_i}{g_i}##. Homework EquationsThe Attempt at a Solution The number of microstates ##\Omega## is...
  34. P

    I Determining if an operator is degenerate

    Hi, I was wondering how you can formally determine whether a given operator is degenerate. I undertand you can produce the 'usual equation' det(Q-##\lambda ##)=0 and solve for ##\lambda ##, where Q is our operator. But if Q is a differential (for example ##\frac{p^2}{2m}= - \bar{h}...
  35. weezy

    I Degeneracy of energy levels greater than no. of particles?

    From statistical mechanics in zeemansky's book . He states that it's easy to see that for a closed system the no. Of degenerate states ##g_i## for energy level ##E_i## is greater than the number of particles ##N_i##occupying that energy state. I can't find a mathematical proof for it. Can I...
  36. ErikZorkin

    I Eigenvalue degeneracy in real physical systems

    I understand this question is rather marginal, but still think I might get some help here. I previously asked a question regarding the so-called computable Universe hypothesis which, roughly speaking, states that a universe, such as ours, may be (JUST IN PRINCIPLE) simulated on a large enough...
  37. M

    I Measuring entangled state of degenerate eigenstates

    Let's suppose that we have an entangled state of two systems ##A## and ##B##: $$ \frac{1}{2}\left(|\psi_1 \phi_1\rangle+|\psi_2 \phi_2\rangle \right) $$ where ##|\psi \rangle## and ##|\phi \rangle## are energy eigenstates of ##A## and ##B## respectively. However the eigenstates##|\phi_1\rangle##...
  38. A

    I Degeneracy in quantum statistics

    degeneracy,this word appears in my textbook many times,but i could not understand what it means in quantum statistics.also in my textbook it is said in bose-einstein statistics that " the deviation from perfect gas behaviour exhibited by bose-einstein gas is called gas degeneracy".but i can't...
  39. B

    Degeneracy pressure and stellar collapse

    I have a question. In a massive star (more than say 5 times the mass of the sun), the electron/neutron degeneracy pressure is unable to prevent the gravitational collapse. Does this imply that the Pauli's exclusion principle breaks down and two or more electrons/neutrons collapse to the same...
  40. 1

    Question on the degeneracies of a thermodynamic system

    Homework Statement A system possesses three energy levels $$E_1=\varepsilon$$ $$E_2=2\varepsilon$$ $$E_3=3\varepsilon$$ with degeneracies $$g(E_1)=g(E_3)=1$$ $$g(E_2)=2$$. Find the heat capacity of the system. Homework Equations $$\beta=\frac{1}{kT}$$ $$Z=\sum_i g_ie^{-\beta \varepsilon_i} \...
  41. Y

    Nonlinear electric susceptibility and degenerate frequencies

    Hi there, I'm having a little trouble understanding the "distinguishability" of frequencies in the nonlinear electric susceptibility tensor. As far as I understand, if we have a SHG process with two collinear beams of the same polarization and frequency ω, there is only one susceptibility...
  42. H

    Degeneracy removed when commuting observables are specified?

    The following is a proof that two commuting operators ##A## and ##B## possesses a complete set of common eigenfunctions. The issue I have with the proof is the claim that the eigenvalue ##a_n## together with the eigenvalue ##b_n## completely specify a particular simultaneous eigenfunction...
  43. S

    Counting States of Degeneracy in 3-D Particles in a Box

    So I'm trying to figure out the best way to count the degeneracy of states for a 3-d particle in a box. The problem breaks into the following: we have three integers greater than or equal to unity whose values are allowed to vary independently, and the sum of whose squares equals a fourth...
  44. P

    Rotational Energy and Degeneracy

    I have a question regarding the rotational energy equation, derived based on the rigid rotor assumption: ##\epsilon = k\theta_r J (J+1)## where k = Boltzmann constant, and J is the rotational quantum number. The degeneracy is 2J+1. Let's assume the constant quantity ##k\theta_r## = 1 and...
  45. T

    Prove that the nth energy level of atom has degeneracy n^2

    Homework Statement Prove that the nth energy level of an atom has degeneracy equal to n^2. Homework EquationsThe Attempt at a Solution I was thinking of using the sum from n=1 to N of n = N(N+1)/2 but my professor said i needed to change that equation up a bit to be able to show this. I was...
  46. S

    Degeneracy of a 2-dimensional isotropic Harmonic Oscillator

    Homework Statement The Hamiltonian is given by: H = \frac{1}{2} \sum_{i=1,2}[p_i^2 + q_i^2] We define the following operators: J = \frac{1}{2} (a_1^+ a_1 + a_2^+ a_2) J_1 = \frac{1}{2} (a_2^+ a_1 + a_1^+ a_2) J = \frac{i}{2} (a_2^+ a_1 - a_1^+ a_2) J = \frac{1}{2} (a_1^+ a_1 - a_2^+...
  47. Helios

    Collapsars vrs. QM degeneracy pressure

    The QM degeneracy pressure puts up a fight but the immensity of the star wins out. Why is this? Is the Pauli Exclusion Principle really a principle? Why does it surrender in this case?
  48. J

    What is the role of degeneracies in the condition for optical gain?

    I'm a novice studying laser physics and I came a across the condition for optical gain: \frac{N_2}{g_2} > \frac{N_1}{g_1} This is a basic set up where N_1 is the number of atoms in the lower energy state and N_2 is the number of atoms in the higher energy state. g_1, and g_2 are the...
  49. R

    Saha equation partition function for Argon?

    This question is in regards to the degeneracy of states for an Argon atom with just one missing electron. For hydrogen the problem of finding the partition function depends on finding the the ionized state of hydrogen divided by the non-ionized state... (please see Saha equation ->...