Science Visualized: Why Electrons have ½ Spin

http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html

https://en.wikipedia.org/wiki/Spinors_in_three_dimensions
https://commons.wikimedia.org/wiki/File:Spinor_interpretation.svg
https://en.wikipedia.org/wiki/Spinor
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydcol.html#c2
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c1
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schrcn.html#c1
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/orbmag.html#c2
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html#c1
http://hyperphysics.phy-astr.gsu.edu/hbase/hydwf.html#c1
https://users.physics.ox.ac.uk/~Steane/teaching/rel_C_spinors.pdf
http://www.sjsu.edu/faculty/watkins/spinor.htm
http://users.aber.ac.uk/ruw/teach/237/hatom.php
http://faculty.wcas.northwestern.edu/~infocom/The%20Website/plates/Plate%201.html
https://www.youtube.com/watch?v=rLikRJOESbU
https://www.youtube.com/watch?v=BMIvWz-7GmU
https://www.youtube.com/watch?v=DvRzdCnsiYw
https://www.youtube.com/watch?v=imdFhDbWDyM
https://www.youtube.com/watch?v=Ei8CFin00PY
https://www.youtube.com/watch?v=lMFgfqRZYoc
https://www.youtube.com/watch?v=TWpyhsPAK14
https://www.youtube.com/watch?v=Uk5DUtHY7LM
{An ongoing modification process to various universities, Wikipedia and among other's explanation attempts}

A single point in space can spin continuously without becoming tangled. Notice that after a 360 degree rotation, the spiral flips between clockwise and counterclockwise orientations. It returns to its original configuration after spinning a full 720 degrees.

The intrinsic magnetic moment μ of a spin-1/2 particle with charge q, mass m, and spin angular momentum S:

___________________________________________________________________

The value of Bohr magneton

system of units value unit

SI^[1] 9.27400968(20)×10⁻²⁴ J·T⁻¹

CGS^[2] 9.27400968(20)×10⁻²¹ Erg·G⁻¹

eV^[3] 5.7883818066(38)×10⁻⁵ eV·T⁻¹

atomic units ¹⁄₂ $\frac{e \hbar}{m_\mathrm{e}}$

In atomic physics, the Bohr magneton (symbol μ_B) is a physical constant and the natural unit for expressing themagnetic moment of an electron caused by either its orbital or spin angular momentum.^[4]^[5]

The Bohr magneton is defined in SI units by

$\mu_\mathrm{B} = \frac{e \hbar}{2 m_\mathrm{e}}$

and in Gaussian CGS units by

$\mu_\mathrm{B} = \frac{e \hbar}{2 m_\mathrm{e} c}$

where

e is the elementary charge,
ħ is the reduced Planck constant,
m_e is the electron rest mass and
c is the speed of light.

The electron magnetic moment, which is the electron's intrinsic spin magnetic moment, is approximately one Bohr magneton.^[6]

The value of Bohr magneton
system of units	value	unit
SI^[1]	9.27400968(20)×10⁻²⁴	J·T⁻¹
CGS^[2]	9.27400968(20)×10⁻²¹	Erg·G⁻¹
eV^[3]	5.7883818066(38)×10⁻⁵	eV·T⁻¹
atomic units	¹⁄₂	$\frac{e \hbar}{m_\mathrm{e}}$

Angular momentum in quantum mechanics

Main article: Angular momentum operator

Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics. In relativistic quantum mechanics, it differs even more, in which the above relativistic definition becomes a tensorial operator.

Spin, orbital, and total angular momentum[edit]

Main article: Spin (physics)

Angular momenta of a classical object.

Left: "spin" angular momentum S is really orbital angular momentum of the object at every point,

right: extrinsic orbital angular momentum L about an axis,

top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω),^[26]

bottom: momentum p and its radial position r from the axis.The total angular momentum (spin plus orbital) is J. For a quantumparticle the interpretations are different; particle spin does nothave the above interpretation.

The classical definition of angular momentum as

\mathbf{L}=\mathbf{r}\times\mathbf{p}

can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator

and p as the quantum momentum operator.

L is then an operator, specifically called the orbital angular momentum operator.Specifically, L is a vector operator, meaning

\mathbf{L}=(L_x,L_y,L_z)

, where L_x, L_y, L_z are three different operators.

However, in quantum physics, there is another type of angular momentum, called spin angular momentum, represented by the spin operator S. Almost all elementary particles have spin. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. Allelementary particles have a characteristic spin, for example electrons always have "spin 1/2" (this actually means "spin ħ/2") while photons always have "spin 1" (this actually means "spin ħ").

Finally, there is total angular momentum J, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, J = L + S.) Conservation of angular momentum applies to J, but not to L or S; for example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total remaining constant.

Quantization

In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where

\hbar

is the reduced Planck constant and

\hat{n}

is any direction vector such as x, y, or z:

If you measure...	The result can be...
$L_{\hat{n}}$	$\ldots, -2\hbar, -\hbar, 0, \hbar, 2\hbar, \ldots$
$S_{\hat{n}}$ or $J_{\hat{n}}$	$\ldots, -\frac{3}{2}\hbar, -\hbar, -\frac{1}{2}\hbar, 0, \frac{1}{2}\hbar, \hbar, \frac{3}{2}\hbar, \ldots$
$L^2$ ( $= L_x^2+L_y^2+L_z^2$ )	$(\hbar^2 n(n+1))$ , where $n=0,1,2,\ldots$
$S^2$ or $J^2$	$(\hbar^2 n(n+1))$ , where $n=0,\frac{1}{2},1,\frac{3}{2}, \ldots$

In this standing wave on a circular string, the circle is broken into exactly 8wavelengths. A standing wave like this can have 0,1,2, or any integer number of wavelengths around the circle, but itcannot have a non-integer number of wavelengths like 8.3. In quantum mechanics, angular momentum is quantized for a similar reason.

(There are additional restrictions as well, see angular momentum operator for details.)

The reduced Planck constant

\hbar

is tiny by everyday standards, about 10⁻³⁴ J s, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure ofelectron shells and subshells in chemistry is significantly affected by the quantization of angular momentum.

Quantization of angular momentum was first postulated by Niels Bohr in his Bohr model of the atom and was later predicted by Erwin Schrödinger in his Schrödinger equation.

Uncertainty

In the definition

\mathbf{L}=\mathbf{r}\times\mathbf{p}

, six operators are involved: The position operators

r_x

r_y

r_z

, and the momentum operators

p_x

p_y

p_z

. However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.

The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example

L_xL_y \neq L_yL_x

. (For the precise commutation relations, see angular momentum operator.)

Total angular momentum as generator of rotations

As mentioned above, orbital angular momentum L is defined as in classical mechanics:

\mathbf{L}=\mathbf{r}\times\mathbf{p}

, but total angular momentum J is defined in a different, more basic way: Jis defined as the "generator of rotations".^[27] More specifically, J is defined so that the operator

R(\hat{n},\phi) \equiv \exp\left(-\frac{i}{\hbar}\phi\, \mathbf{J}\cdot \hat{\mathbf{n}}\right)

is the rotation operator that takes any system and rotates it by angle

\phi

about the axis

\hat{\mathbf{n}}

. (The "exp" in the formula refers to operator exponential)

The relationship between the angular momentum operator and the rotation operators is the same as the relationship between lie algebras and lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in Noether's theorem that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.

Complex Phase

quantum mechanical spin is described by a complex-valued vector with two components called a spinor

http://www.sjsu.edu/faculty/watkins/spinor.htm

Notice that after a 360 degree rotation, the spiral flips between clockwise and counterclockwise orientations.

Magnetic Quantum Number, m

In atomic physics, the magnetic quantum number is the third of a set of quantum numbers (the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number) which describe the unique quantum state of an electron and is designated by the letter m. The magnetic quantum number denotes the energy levels available within a subshell. There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers n, ℓ, m, and s specify the complete and unique quantum state of a single electron in an atom called its wavefunction or orbital. The wavefunction of the Schrödinger wave equation

reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first 3 quantum numbers are all interrelated. The magnetic quantum number arose in the solution of the azimuthal part of the wave equation as shown below.

Separating the SchrÖdinger Equation

The magnetic quantum number associated with the quantum state is designated as m. The quantum number m refers, loosely, to the direction of the angular momentum vector. The magnetic quantum number m only affects the electron's energy if it is in a magnetic field because in the absence of one, all spherical harmonics corresponding to the different arbitrary values of m are equivalent. "M" also affects the probability cloud.

Given a particular ℓ, m is entitled to be any integer from -ℓ up to ℓ. More precisely, for a given orbital momentum quantum number ℓ (representing the azimuthal quantum number associated with angular momentum), there are 2ℓ+1 integral magnetic quantum numbers m ranging from -ℓ to ℓ, which restrict the fraction of the total angular momentum along the quantization axis so that they are limited to the values m. This phenomenon is known as space quantization.^[1] It was first demonstrated by two German physicists, Otto Stern and Walther Gerlach. Since each electronic orbit has a magnetic moment in a magnetic field the electronic orbit will be subject to a torque which tends to make the vector

\mathbf{L}

parallel to the field. The precession of the electronic orbit in a magnetic field is called the Larmor precession.

To describe the magnetic quantum number m you begin with an atomic electron's angular momentum, L, which is related to its quantum number ℓ by the following equation:

\mathbf{L} = \hbar\sqrt{\ell(\ell+1)}

where

\hbar = h/2\pi

is the reduced Planck constant. The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels.^[1]

To show that only certain discrete amounts of angular momentum are allowed, ℓ has to be an integer. The quantum number m refers to the projection of the angular momentum for any given direction, conventionally called the z direction. L_z, the component of angular momentum in the z direction, is given by the formula:^[1]

\mathbf{L_z} = m\hbar.

Another way of stating the formula for the magnetic quantum number

(m_l = -\ell,-\ell + 1,..., 0, ..., \ell - 1, \ell)

is the eigenvalue, J_z=m_ℓh/2π.

Where the quantum number ℓ is the subshell, the magnetic number m represents the number of possible values for available energy levels of that subshell as shown in the table below.^[1]

Relationship between Quantum Numbers
Orbital	Values	Number of Values for m
s	$\ell=0,\quad m=0$	1
p	$\ell=1,\quad m=-1,0,+1$	3
d	$\ell=2,\quad m=-2,-1,0,+1,+2$	5
f	$\ell=3,\quad m = -3,-2,-1,0,+1,+2,+3$	7
g	$\ell=4,\quad m = -4,-3,-2,-1,0,+1,+2,+3,+4$	9

The magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field, hence the name magnetic quantum number (Zeeman effect).

However, the actual magnetic dipole moment of an electron in an atomic orbital arrives not only from the electron angular momentum, but also from the electron spin, expressed in the spin quantum number.

So, this all results from a nice derivation taken from the link above for separating the equation