Pauli Definition and 229 Threads

Wolfgang Pauli's matrices are $$\sigma_x=\begin{bmatrix}0& 1\\1 & 0\end{bmatrix},\quad \sigma_y=\begin{bmatrix}0& -i\\i & 0\end{bmatrix},\quad \sigma_z=\begin{bmatrix}1& 0\\0 & -1\end{bmatrix}$$ He introduces these equations as "the equations of motion" of the spin in a magnetic field. $$...

In the fermi gas model, there is assumption that there is a 3D potential well, but there is "energetic degeneration" for each three index "nx, ny, nz". Now the problem is with that image, if there is degeration, for some level En there may be 10 distinctive state with same energy, so there is 20...

My guess would be to do an integral of the form $$\frac{\int d^4k}{(2\pi)^4}k^2(\frac{1}{(k^2-m^2+i\epsilon)}-\frac{1}{k^2-\Lambda_1^2+i\epsilon})(\frac{1}{(k^2-m^2+i\epsilon)}-\frac{1}{k^2-\Lambda_2^2+i\epsilon})$$ before Wick otating and integrating. Any help is appreciated. Thanks.

Electromagnetism in the atoms is why we can't pass through a bank vault. But supposed electromagnetism were canceled for an object, what would happen to the residual or remaining Pauli Exclusion principle? Would it still cause resistance to passing through the vault? On a second scenerio, what...

How did Pauli determine his exclusion principle? Was it based on how he posited electron shells filled? Is the fact that fermions are antisymmetic a mathematical solution to make the principle work with quantum theory?

http://nautil.us/issue/93/forerunners/the-synchronicity-of-wolfgang-pauli-and-carl-jung

Is the imaginary number i "necessary" in the pauli matrices simply because of the condition of having 3 mutually orthogonal axi? If space were two dimensional we wouldn't need the i imaginary number?

For high temperature superconductivity, people usually say two quasifree electrons are pairing, one is spin up and the other one is spin down. So, if that is the case, each two electrons will have zero spin angular momentum. Since the superconductivity is the magnetic properties and spin is the...

I have a question about this note: https://ocw.mit.edu/courses/physics/8-06-quantum-physics-iii-spring-2018/lecture-notes/MIT8_06S18ch2.pdf I don't understand the expression (2.2.15). The complete relation would be $$ \pi_i \pi_j = \frac{1}{2}\left(\left[\pi_i, \pi_j\right] + \left\{\pi_i...

How do the Pauli spin matrices transform under an inversion ? I think I mean to say the 3 dimensional improper rotation which is just in 3 dimensional matrix notation minus the identity - so exactly how are the 2 dimensional Pauli spin matrices changed. And under a 180 rotation do the 'y' and...

We know that S2 commutes with Sz and so they share their eigenspace. Now since S2 also commutes with Sx, as per my understanding, the eigenvectors of S2 and Sz should also be the eigenvectors of Sx. But since the paulic matrices σx and σy are not diagonlized in the eigenbasis of S2, it is clear...

Hello! I am a bit confused about the mechanism behind the Pauli exclusion principle. From what I read, it is motivated based on QFT arguments (for example if you don't impose antisymmetry of the fermionic wavefunction you get non-locality, or infinitely negative energies etc.) so mathematically...

The exlcusion principle seems intuitive enough to me when the states being considered are eigenstates, however how does it work exactly with general states? It seems to me that if we're allowed to consider general quantum states then the principle breaks down, since we can always find states...

http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html "Postulate 2. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics. " "Postulate 6. The total wavefunction must be antisymmetric with respect to the interchange of all...

Hi :) I have several questions about the Pauli matrices, I have seen them when the lecturer showed us Stern-Gerlach experiment , and we did some really weird assumptions on what we think they should be. 1- why did we assume that all of those matrices should satisfy σ2 = I (the identity...

This question is more a question I'd ask in a chat rather than formally on paper/forum. If we take the free electron model, the electrons are considered as non interacting. It is essentially a 1 particle problem where the potential is constant through space. The electrons are not perturbed at...

electrons are repelled using the Pauli exclusion principle?

Consider the pairing term in Weizsäcker formula. Here https://en.wikipedia.org/wiki/Semi-empirical_mass_formula#Pairing_term it is claimed that: I don't understand how Pauli exclusion principle should be the cause of this. This term comes from spin-spin interaction (or "coupling"), but I do not...

Hello! I am a bit confused about the Pauli exclusion principle. Let's say I have 3 electrons. Due to energy considerations the first 2 go to the ground state, and they can be only 2 electrons there, because the position wavefunction has only one option ##\psi_{100}## (and again due to energy...

Say you have two particles a and b with respective positions ##x_a## and ##x_b##. Particle a is in the state ##\psi_a##, and particle b is in the state ##\psi_b##. If they are distinguishable, the wavefunction is $$\psi=\psi_a(x_a)\psi_b(x_b)$$ However, if they are identical fermions, the...

Homework Statement This is not a homework problem. It's an example in a textbook. 3 electrons. For ##S=3/2##, we have that $$ m_{s_1} = m_{s_2} = m_{s_3} = 1/2 $$ Therefore by the Pauli Exclusion principle, $$ m_{l_1} \neq m_{l_2} \neq m_{l_3} $$ and they take the values ##-1,0,1##...

Hi guys, Do virtual particles, when they are fermions, obey Pauli exclusion principle as real fermions do? More specifically, what I am wondering is the following: Fermion fields would have some energy at every point in spacetime due to the uncertainty principle. Now, is it possible for the...

I'm studying for a qualifying exam and I see something very strange in the answer key to one of the problems from a past qualifying exam. It appears the sigma^2 for a two electron system has eigenvalues according to the picture below of 4s(s+1) while from my understand of Sakurai it should have...

So I have been studying the case of spin 1/2 and I have understood how the formulations work through to find the spin matrices. However I do not get an intuitive understanding of what they mean and why they are formulated the way they are. I follow Griffith's book and in it as he begins to solve...

Homework Statement [/B] I know the pauli matrices in terms of the z-basis, but can't find them in terms of the other bases. I would like to know what they are. Homework Equations The book says they are cyclic, via the relations XY=iZ, but this doesn't seem to apply when I use this to find the...

I met with a little conflict between Pauli and Einstein? Can you please help. Its a thought experiment. Consider a single crystal which is 1km long. During its formation, due to Pauli’s exclusion principle, no two electron will have same quantum state. Now consider two electron, one with E and...

Suppose that somehow we could artificially bypass Pauli exclusion principle, and make electrons or any fermions for that matter occupy more than one state at the same time? What consequences in nature will we see? what phenomenons will occur? Suppose this mechanism for bypassing is limited in...

Hey guys, Hope all is well. I am trying to understand the process that takes us from the Pauli equation to the Dirac equation. Whilst I understand the motivation is to have a lorrentz covariant equation I don't really understand A.) how this was done B.) what the physical result...

This is not part of my coursework but a question from a past paper (that we don't have solutions to). 1. Homework Statement Construct the matrix ##\sigma_{-} = \sigma_{x} - i\sigma_{y}## and show that the states resulting from ##\sigma_{-}## acting on the eigenstates of ##\sigma_{z} ## are...

What is the explanation for the case that fermion's can't occupy the same quantum state? Is there some logic behind this or is it just observation?

Homework Statement Hey :-) I just need some help for a short calculation. I have to show, that (\sigma \cdot a)(\sigma \cdot b) = (a \cdot b) + i \sigma \cdot (a \times b) The Attempt at a Solution I am quiet sure, that my mistake is on the right side, so I will show you my...

The event horizon of a black hole appears to be plastered with 'afterimages' of everything that ever fell into it. (Because gravitational time dilation makes every such object appear to stop at the event horizon.) Now, suppose an event horizon is 'full' as defined by the Pauli exclusion...

I'm really interested in quantum theory and would like to learn all that I can about it. I'm looking books for learning quantum physics that contains derivation of Heisenberg uncertainty principle, dirac notation, pauli matrices, quantum operators, hawking radiation, etc. What are good books to...

Say I have ##n_{a}## bosons in some state ##a##, then the transition rate from some state ##b## to state ##a##, ##W^{boson}_{b\rightarrow a}##, is enhanced by a factor of ##n_{a}+1## compared to the corresponding transition probability for distinguishable particles, ##W_{b\rightarrow a}##, i.e...

A nice discussion of the density operator for a qubit can be found here: http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere-rotations.pdf

Dear all, The Hamiltonian for a spin-orbit coupling is given by: \mathcal{H}_1 = -\frac{\hbar^2\nabla^2}{2m}+\frac{\alpha}{2i}(\boldsymbol \sigma \cdot \nabla + \nabla \cdot \boldsymbol \sigma) Where \boldsymbol \sigma = (\sigma_x, \sigma_y, \sigma_z) are the Pauli-matrices. I have to...

Homework Statement I am watching a course on Relativistic Quantum Mechanics to freshen up, and I have found to have some issues regarding simple operator algebra. This particular issue on the Pauli Equation (generalization of the Schrodinger equation that includes spin corrections) in an...

Homework Statement Suppose the vector ##\phi## transforms under SU(2) as: $$\phi' = (\exp(-i \alpha \cdot t))_{ij}\phi_j,$$ where ## (t_j)_{kl} = −i \epsilon_{jkl}## and ##j, k, l \in \left\{1, 2, 3\right\}.## Based on ##\phi,## we define the ##2 \times 2## matrix ##\sigma = \tau \cdot...

I was reading about the Lennard-Jones potential, and I believe I understand the derivation of the 6th power dependency of dipole dipole interactions (Van der Waal forces) well. The most I have been able to find about the 12th power dependency of Pauli repulsion is that is has no theoretical...

How do we get (6.265)? Shouldn't we have ##exp(-i\frac{\alpha}{2}\hat{n}.\sigma)=\cos(\frac{\alpha}{2}\hat{n}.\sigma)-i\sin(\frac{\alpha}{2}\hat{n}.\sigma)##?

How does the Pauli-exclusion principle explain ionization energy trends? Is it just that as you move down the periodic table, the electrons experience repulsion between each other, and thus the atoms get bigger?

Lepton Universality and Pauli Exclusion Put in a possibly oversimplified way, lepton universality says that electrons, muons, and taus all behave in the same way except for mass effects. The question is “Does this apply to Pauli exclusion?” Due to the Pauli exclusion principle, only two...

I'm not sure I have the right approach here: Using the three 2 X 2 Pauli spin matrices, let $ \vec{\sigma} = \hat{x} \sigma_1 + \hat{y} \sigma_2 +\hat{z} \sigma_3 $ and $\vec{a}, \vec{b}$ are ordinary vectors, Show that $ \left( \vec{\sigma} \cdot \vec{a} \right) \left( \vec{\sigma} \cdot...

I have had no problem while finding the eigen vectors for the x and y components of pauli matrix. However, while solving for the z- component, I got stuck. The eigen values are 1 and -1. While solving for the eigen vector corresponding to the eigen value 1 using (\sigma _z-\lambda I)X=0, I got...

Hi everybody, a teacher of mine has told me that any complex, self adjoint matrix 2*2 which trace is zero can be written as a linear combination of the pauli matrices. I want to prove that, but I haven't been able to. Please, could somebody point me a book where it is proven, or tell me how to...

This is one of those question you won't find the answer in any book. From Wikipedia: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers (n, ℓ, mℓ and ms). But how can an electron know the state (the quantum numbers) of the other...

Homework Statement Look at the matrix: A = sin t sin p s_x + sin t sin p s_y +cos t s_z where s_i are the pauli matrices a) Find the eigenvalues and normalized eigenvectors (are they orthogonal)? b) Write the eigenvector of s_x with positive eigenvalue as a linear combination of the...

The principle states that two electrons cannot have the same quantum numbers. And I've read that this applies to "fermions"- protons, neutrons, 1/2 spin particles. But how exactly does this apply to, say, a proton? Sorry if I sound stupid...I've got all my knowledge about this through the...

Homework Statement I'm looking at solving a simple theory of massive gravity for Scwarzchild type solutions. I've attatched the paper that I'm working with. I've tried to add the 3 possible cubic terms to L_mass parametrized by constants. It doesn't seem possible to solve for B as a function of...

Hi, Can someone please explain this to me? "The axis of rotation for a non-quantum-mechanical object can point any way it likes. The Earth could rotate around an axis ninety degrees from the current one, so that the North Pole always faces the sun and the South Pole always faces away from it...