# 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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2. ### 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?
3. ### 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...
4. ### 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...
5. ### 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.
6. ### 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| +...

13. ### 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?
14. ### 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...
15. ### 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...
16. ### 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...
17. ### 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...

42. ### 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...
43. ### 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...
44. ### 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...
45. ### 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...
46. ### 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...
47. ### 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^+...
48. ### 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?
49. ### 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...
50. ### 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 ->...