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4

When labeling states of the hydrogen atom, one doesn't refer to the z component of the angular momentum, but rather to the total angular momentum. The total angular momentum is positive, but, as you've stated, there are two states for $J=\frac{1}{2}$ with $L=0$, and those are $J_z=\pm\frac{1}{2}$ (Or some linear combination of them) As to why this is, ...

4

The fundamental difference between an electron's spin and that of a baseball is that the electron is (as far as we know) a point particle. It therefore cannot rotate in the usual sense, where individual parts move relative to the center of mass; we say that its angular momentum is intrinsic. The magnitude $\lvert\vec{S}\rvert^2$ of a particle's intrinsic ...

3

You are looking for the Wigner d-functions. They relate angular momentum eigenstates through rotation. As you can see in the link the definition is $$d^{(j)}_{m,m'}(\theta) = \langle jm|e^{-i\theta J_y}|j m' \rangle$$ where $e^{-i\theta J_y}$ is a unitary rotation operator. We'll have two sets of states: $|jm;0\rangle$ for the original basis and ...

2

Think about the kinetic energy observable $T$ for a spin-1/2 particle in 3D space. The particle's Hilbert space is technically ${\mathfrak H} = L^2({\mathbb R}^3) \otimes {\mathbb C}^2$, yet the kinetic energy operator $T$ is first defined on $L^2({\mathbb R}^3)$, where its eigenfunctions $\Psi_{\bf p}({\bf x})$ are easily found. The extension of $T$ to ...

2

Things have intrinsic spin. There's no "class of objects" that has spin and another class of objects that doesn't. Everything is a quantum object with a quantum state, and spin is a number that tells you how the state of the object transforms under rotations. It is different from "classical" angular momentum in that spin is not the operator associated to ...

1

The three generators of right-handed spinor rotations are given by $\left\{- i\sigma_x,-i\sigma_y,-i\sigma_z\right\}$, see for instance Peskin & Schroeder page 44, and the rotation matrix for a spinor rotation over an angle $\phi$ around a unit vector $\hat{s}$ is given by: $R~=~ \exp\left(-i\frac{\phi}{2}~\hat{s}\cdot\vec{\sigma}\right) ~=~ ... 1 Entanglement is a quantum mechanical phenomenon. It is a shorthand to saying " aspects of the wavefunction for these particles are completely known", i.e. the particles are entangled by the wavefunction describing their probabilistic behavior. A laser beam emerges from zillions of coherent photons, i.e. their phases with respect to each other and the beam ... 1 The Stern-Gerlach experiment uses an inhomogeneous magnetic field on a particle with a spin that is proportional to the particle's magnetic moment. An inhomogeneous magnetic field does increase the speed of the particle, not merely bend it. The link you cited does not claim otherwise. 1 The hyperphysics site you mention states spin rate of some$10^{32}$radian/s would be required to match the observed angular momentum. Classical angular momentum is calculated as$I\omega$, where$I = \frac{2}{5}mr^2$for a sphere. The mass of an electron is$9.11\times10^{-31}$kg and the site mentions an upper limit of$10^{-3}$fermis or$10^{-18}\$ ...

1

If the spin is an actual magnetic moment, then its behavior under time reversal is simply similar to that of classical magnetization, which changes sign. Think of magnetic fields and dipoles as generated by electric currents. Under time reversal the currents reverse direction and so do the corresponding magnetic fields or dipoles. At quantum level, spin ...

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