What is Spin operator: Definition and 31 Discussions

Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. For photons, spin is the quantum-mechanical counterpart of the polarization of light; for electrons, the spin has no classical counterpart.The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The existence of the electron spin can also be inferred theoretically from spin–statistics theorem and from the Pauli exclusion principle—and vice versa, given the particular spin of the electron, one may derive the Pauli exclusion principle.
Spin is described mathematically as a vector for some particles such as photons, and as spinors and bispinors for other particles such as electrons. Spinors and bispinors behave similarly to vectors: they have definite magnitudes and change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change. These are indicated by assigning the particle a spin quantum number.The SI unit of spin is the same as classical angular momentum (i.e. N·m·s or kg·m2·s−1). In practice, spin is given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ, which has the same dimensions as angular momentum, although this is not the full computation of this value. Very often, the "spin quantum number" is simply called "spin". The fact that it is a quantum number is implicit.
Wolfgang Pauli in 1924 was the first to propose a doubling of the number of available electron states due to a two-valued non-classical "hidden rotation". In 1925, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the old quantum theory of Bohr and Sommerfeld. Ralph Kronig anticipated the Uhlenbeck–Goudsmit model in discussion with Hendrik Kramers several months earlier in Copenhagen, but did not publish. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it.

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  1. cianfa72

    I EPR in Bohm formulation

    Hi, I was reading about the EPR paradox in Bohm simplified formulation. From my understanding the paradox is that Bob is actually able to get a value for the positron's spin along both the ##z## and ##x## axes. Since electron and positron are entangled, he get the value of spin along ##z##...
  2. Rayan

    Sequential Stern Gerlech experiment

    So I thought that when the $m_l = 1$ beam passes through the second SG-magnet, it should split into 3 different beams with equal probability corresponding to $ m_l = -1 , 0 , 1 $ since the field here is aligned along z-axis and hence independent of the x-axis splitting. And I thought that the...
  3. cianfa72

    I Representation of Spin 1/2 quantum state

    Hi, I'm aware of the wave function ##\Psi## of a quantum system represents basically the "continuous components" of a quantum state (a point/vector in the infinite-dimension Hilbert space) in a basis. If we take the ##\delta(x - \bar x)## eigenfunctions as basis on Hilbert space then the wave...
  4. 4

    I Spin operator and spin quantum number give different values, why?

    Assume spin 1/2 particle So the spin operator gives +/- hbar/2 eg. S |n+> = +/- hbar/2 |n+> But S= s(s+1) hbar = sqrt(3)/2 hbar So I'm off by a factor of sqrt(3). I suspect I am missing something fundamental about my understanding of spin. My apologies and thanks in advance.
  5. P

    Spin matrix representation in any arbitrary direction

    I've tried to use the 1st equation as a matrix to determine, but it clearly isn't a diagonal matrix. My guess is that I need to find the spin matrix along the direction ##\hat{n}##, but do I need to find the eigenstates of ##\sigma \cdot \hat{n}## first and check if they form a diagonal matrix...
  6. W

    A Define spin operators for numerical groundstate obtained by ED

    Hi, I want to measure spin components of a ground state of some models. These ground states are obtained by ED. The states for constructing the Hamiltonian are integers representing spins in binary. As the ground state (and the other eigenvectors) are now not anymore in a suitable representation...
  7. Garlic

    I How does the expectation value of the spin operator evolve over time?

    Dear PF, As an excercise I am to find out how the expectation value of the spin operator evolves over time. There was a hint, stating that it is enough to show that $$ e^{i \frac{\phi ( \hat{n} \cdot \sigma )}{2}} \sigma_i e^{- i \frac{\phi ( \hat{n} \cdot \sigma )}{2}} = [R_{ \hat{n} }]_{ij}...
  8. schwarzg

    A Simultanious eigenstate of Hubbard Hamiltonian and Spin operator in tw

    Please see this page and give me an advice. https://physics.stackexchange.com/questions/499269/simultanious-eigenstate-of-hubbard-hamiltonian-and-spin-operator-in-two-site-mod Known fact 1. If two operators ##A## and ##B## commute, ##[A,B]=0##, they have simultaneous eigenstates. That means...
  9. Muthumanimaran

    B Time average value of Spin operator

    From the book Introduction to Quantum Mechanics by Griffiths,. In the section 6.4.1 (weak field zeeman effect) Griffiths tells that the time average value of S operator is just the projection of S onto J while finding the expectation value of J+S $$S_{avg}=\frac{(S.J)J}{J^2}$$ How to prove this?
  10. S

    Expectation Value and Probabilities of Spin Operator Sy

    Homework Statement (a) If a particle is in the spin state ## χ = 1/5 \begin{pmatrix} i \\ 3 \\ \end{pmatrix} ## , calculate the expectation value <Sy>(b) If you measured the observable Sy on the particle in spin state given in (a), what values might you get and what is the probability of...
  11. S

    Normalised eigenspinors and eigenvalues of the spin operator

    Homework Statement Find the normalised eigenspinors and eigenvalues of the spin operator Sy for a spin 1⁄2 particle If X+ and X- represent the normalised eigenspinors of the operator Sy, show that X+ and X- are orthogonal. Homework Equations det | Sy - λI | = 0 Sy = ## ħ/2 \begin{bmatrix} 0...
  12. BigDig123

    I Is the total Spin operator a vector

    Hello, I am learning about Excited states of Helium in my undergrad course. I was wondering if the total spin operator Ŝ is a vector quantity or not. Thanks for your help.
  13. X

    I Pauli Spin Operator Eigenvalues For Two Electron System

    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...
  14. chi_rho

    A Transforming Spin Matrices (Sx, Sy, Sz) to a Spherical Basis

    Say I have {S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\\ \end{array}\right)} Right now, this spin operator is in the Cartesian basis. I want to transform it into the spherical basis. Since, {\vec{S}} acts like a vector I think that I only need to...
  15. phys-student

    Spin system, quantum mechanics

    Homework Statement Consider a spin system with noninteracting spin 1/2 particles. The magnetic moment of the system is written as: μ = (ħq/2mc)σ Where σ = (σx, σy, σz) is the Pauli spin operator of the particle. A magnetic field of strength Bz is applied along the z direction and a second...
  16. J

    How do I find eigenstates and eigenvalues from a spin operator?

    Homework Statement I have a spin operator and have to find the eigenstates from it and then calculate the eigenvalues. I think I managed to get the eigenvalues but am not sure how to get the eigenstates.Homework Equations The Attempt at a Solution I think I managed to get the eigenvalues out...
  17. S

    Finding the eigenvalues of the spin operator

    1. What are the possible eigenvalues of the spin operator \vec{S} for a spin 1/2 particle? Homework Equations I think these are correct: \vec{S} = \frac{\hbar}{2} ( \sigma_x + \sigma_y + \sigma_z ) \sigma_x = \left(\begin{array}{cc}0 & 1\\1 & 0\end{array}\right),\quad...
  18. jk22

    Rotation of Spin Operator and Vector in 3D Space

    If we consider a spin 1/2 particle, then, the rotation of the spinor for each direction is given by a rotation matrix of half the angle let say theta Rspin=\left(\begin{array}{cc} cos(\theta/2) & -sin(\theta/2)\\sin(\theta/2) & cos(\theta/2)\end{array}\right) and the new component of the spin...
  19. G

    Spin-1 particles' spin operator

    The S_{z} operator for a spin-1 particle is S_{z}=\frac{h}{2\pi}[1 0 0//0 0 0//0 0 -1] I'm given the particle state |\phi>=[1 // i // -2] What are the probabilities of getting each one of the possible results? Now... we can say the possible measure results will be 1,0,-1 and the...
  20. R

    Eigenvalues/Eigenstates of Spin Operator S in xz Plane

    Homework Statement Find the eigenvalues and eigenstates of the spin operator S of an electron in the direction of a unit vector n; assume that n lies in the xz plane. Homework Equations S|m>= h m|m> The Attempt at a Solution This question is from Zettili QM and they have...
  21. S

    Expectation Value of Spin Operator

    Hey, I'm having trouble interpreting a question, as the solutions say something different... Anyways the question part d) below: So we want to determine the expectation value of the y-component of the electron spin on the eigenstate of Sx, now I would of thought this was given by...
  22. A

    Eigenvector of a spin operator

    Homework Statement Homework Equations The Attempt at a Solution I don't know what's wrong with my work. I can't obtain the eigenvector provided in the model answer. My work Model Answer
  23. F

    Eigenfunctions of spin operator

    What are the eigenfunctions of the spin operators? I know the spin operators are given by Pauli matricies (https://en.wikipedia.org/wiki/Spin_operator#Mathematical_formulation_of_spin), and I know what the eigenvalues are (and the eigenvectors), but I have no idea what the eigenfunctions of the...
  24. R

    Discover the Spin of an Electron Using Angular Momentum Operator and Eigenvalues

    I should Use the fact that in general the eigenvalues of the square of the angular momentum operator is J(J + 1)h and show the spin of the electron. I have J= L+S and J2 = L2+ S2 Homework Statement But how could i find the spin of the electron
  25. Q

    QM - Spin operator conjugate question

    Homework Statement Okay so I've got a question I really need answered first up! If I have a 2x1 matrix for Psi, is Psi* a 1x2 matrix with all the 'i's turned to '-i's? Now onto the actual question - http://imgur.com/3ucb4" - part b only Homework Equations http://imgur.com/bcEm3"...
  26. T

    Eigenvalues and Eigenstates of Spin Operator

    I'm not exactly looking for help finding the eigenvalues of the spin operator, I'm mainly wondering if there is a better technique to do it. Homework Statement Find the eigenvalues and corresponding eigenstates of a spin 1/2 particle in an arbitrary direction (θ,\phi) using the Pauli...
  27. U

    In the second quantization spin operator, what are Pauli spin vector indices?

    If you look up the second quantization spin operator, you'll notice that there are two indices on the pauli vector for two possible spins. The operator sums over these two indices. Since the pauli vector is an unchanging quantity what do these indices physically correspond to?
  28. N

    How is equation 4.61 derived from n dot s in the Arbitrary Spin Operator?

    http://www.tampa.phys.ucl.ac.uk/~tania/QM4226/SEC4B.pdf At the above link, I'm not quite sure how the instructor got to the matrix definition for Sn(equation 4.61 on page 4) from n dot s. Does someone know of a link that doesn't skip that step?
  29. J

    Quantum mechanics - spin operator eigenvalue probability?

    Hi, I have this problem on a past exam paper I am having some trouble with: "in the conventional basis of the eigenstates of the Sz operator, the spin state of a spin-1/2 particle is described by the vector: u = \left( \stackrel{cos a}{e^i^b sina} \right) where a and B are constants...
  30. N

    Exploring the Commutation of Spin Operator and Magnetic Field

    Homework Statement I need to show the commutation between the spin operator and a uniform magnetic field will produce the same result as the cross product between them. Does this make sense? I don't see how it can be possible. Homework Equations [s,B] (The s should also have a hat...
  31. I

    Proof of Dot and Cross Product of Arbitrary Vectors with Pauli Spin Operator

    I need to show: (\mathbf{\sigma} \cdot \mathbf{a})(\mathbf{\sigma} \cdot \mathbf{b})=\mathbf{a} \cdot \mathbf{b} I + i \mathbf{\sigma} \cdot (\mathbf{a} \times \mathbf{b}) where a and b are arbitrary vectors, sigma is the pauli spin operator. I was just wondering what the dot product...