Redshift Academy


Orbital Angular Momentum
-------------------------





Orbiting electron with mass, m, and velocity, v, can be viewed as a 
current I around a loop with area A.  The magnetic moment, μ is
given by,

μ = IA and is a vector perpendicular to the plane of the loop.

The angular momentum of a particle is L = mvr (a vector)





I = e/T = ev/2πr

This can be rewritten as,

I = emvr/2πmr²

μ = IA = eL/2m since L = mvr and A = πr²

Solutions to the Schrodinger Equation for the Hydrogen atom show
that |L| is quantized.

|μ| = (e/2m)√l(l + 1)h where l is the ORBITAL QUANTUM NUMBER

    = √l(l + 1)μ_B where μ_B is the BOHR MAGNETON = eh/2m

Also, from the analysis of the Hydrogen atom we get the MAGNETIC 
(AZIMUTHAL) QUANTUM NUMBER, m_l, which corresponds to the 
quantization of the z-component of angular momentum.  m_l can take 
the values of,

        -l, -l+1,...., l-1, l 

Thus, for l = 2 we get,

        -2, -1, 0, 1, 2

This can be represented as:





The magnitude of z-component of the magnetic moment is given by

|μ_z| = m_lμ_B

Thus, the projection of orbital angular momentun, L, onto the z axis 
is quantized in units of h.

Electron Spin
-------------

Stern-Gerlach showed experimentally that electrons possessed an intrinsic 
angular momentum and a magnetic moment. Classically this could occur if 
the electron were a spinning ball of charge, and this property was called 
electron spin. Quantization of angular momentum had already arisen for 
orbital angular momentum, and if this electron spin behaved the same way, 
an angular momentum quantum number s = 1/2 was required to give just 
two states. 

Electrons have intrinsic angular momentum characterized by quantum number 
1/2. In the pattern of other quantized angular momenta, this gives total 
angular momentum of: 

S²  = S(S + 1)h²

   = (1/2)(1/2 + 1)h²

   = (3/4)h²

|S| = √(3)/2h 

The resulting fine structure which is observed corresponds to two 
possibilities for the z-component of the angular momentum.

S_z = +/-(1/2)h



          

Combining Orbital and Spin Angular Momentum
-------------------------------------------

This diagram can be seen as describing a single electron,  or multiple 
electrons for which the spin and orbital angular momenta have been 
combined to produce composite angular momenta, S and L respectively. 

The combination is a special kind of vector addition as is illustrated for 
the single electron case l=1 and s=1/2.  As in the case of the orbital 
angular momentum alone, the projection of the total angular 
momentum along a direction in space is quantized to values differeing 
by one unit of angular momentum. 





Angular Momentum in a Magnetic Field
------------------------------------

Once you have combined orbital and spin angular momenta according 
to the vector model, the resulting total angular momentum can be 
visualized as precessing about any externally applied magnetic field 
as shown in the following diagram.





Larmor Frequency 
-------------------

When you have a magnetic moment directed at some finite angle 
with respect to the magnetic field direction, the field will 
exert a torque on the magnetic moment. This causes it to precess 
about the magnetic field direction. This is analogous to the 
precession of a spinning top around the gravity field.  
Classically, the frequency of precession is given by 

ω_L = qB/2m

An isolated electron has an angular momentum and a magnetic 
moment resulting from its spin. While an electron's spin is 
sometimes visualized as a literal rotation about an axis, it 
is in fact a fundamentally different, quantum-mechanical 
phenomenon with no true analogue in classical physics. 
Consequently, there is no reason to expect the above classical 
relation to hold. In fact it does not, giving the wrong result 
by a dimensionless factor called the electron g-factor, denoted 
by g_e where g_e = 2(1 + α/2π + ...) and α is the fine-structure 
constant (= 1/137) - a coupling constant that characterizes 
the strength of the electromagnetic interaction between 
elementary charged particles.  Thus,

ω_L = qg_eB/2m = g_eμ_BB/h 

μ_B = qh/2m ... The Bohr Magneton

The magnetic moment is usually expressed as a multiple of the 
Bohr magneton. 

The result represents the angular frequency associated with the 
transition between the two possible spin states.  In turn, this 
can be expressed as the absorption or emission a photon with 
energy according to the Planck relationship.

E = hω_L

  = g_eμ_BB

  = 2μ_BB

Quantum Mechanical Treatment
----------------------------

In general H = (μ/2)σ.B

For the z direction:

H_z = (μ/2)σ_z.B
.
σ_z = (ih)[H,σ_z] = 0

Thus, there is no change in energy in the z direction (energy 
is conserved).
.
σ_x = (iμBh)[σ_z,σ_x] = 0 since [σ_z,σ_x] = 2iσ_y

  = -μBσ₂/2

Likewise,
.
σ_y = μBσ_x/2

These equations have the following solutions:

σ_y = sinμBt/h

σ_x = cosμBt/h

This represents precession about the axis of B.