Larmor Procession

Magnetic moment (mu) :

this is a magnet by angular momentum of charge or spin. its value is:

mu = gamma S ( in case of spin ) = gamma L ( in case of angular momentum )

where gamma is (g-factor) x ( Bohr magneton ) / ( h_bar [ in case that the S or L is not using natural unit ] )

the g-factor is a dimensionless number, which reflect the environment of the spin, for orbital angular momentum, g = 1.

Bohr magneton = electron charge x h_bar / ( 2 mass), since different particle has different mass, their Bohr magneton value are different. electron is the lightest particle, so, it has largest value on Bohr magneton.

J-coupling ( i am not sure it is correct to say so)

since the spin and angular momentum are mathematical equal. they give the same effect when react with magnetic field, thus, we use a new term:

J = L + S

mu = gamma J

in this case, since the g-factor of L and S are different, and when we combine with different particles, the Bohr magnetons are different, in that case, special treatment is needed. but it does not change the principle.

Larmor frequency:

When applied a magnetic field on a magnetic moment, the field will cause the moment process around the axis of the field. the precession frequency is called Larmor frequency.

the procession can be understood in classical way or QM way.

Classical way:

the change of angular momentum is equal to the applied torque. and the torque is equal to the magnetic moment  cross product with the magnetic field.

d J / d t = mu x B = gamma J x B

solving gives the procession frequency is :

w = - gamma B

the minus sign is very important, it indicated that the J is precessing by left hand rule.

QM way:


The Schrödinger equation is :

i  d/dt |Psi > = H |Psi >

H is the Hamiltonian = - mu . B = - gamma J.B = - gamma B Jz = w Jz

Jz is an operator.

the solution is for |Psi> is

|Psi ( t ) > = Exp[ - i w Jz ] |Psi ( t = 0 ) >

for up state coefficient a1 and down state coefficient a2,

solving a1 and a2 gives:

a1 = Exp [ - i w t ]
a2 = Exp [ + i w  t ]

Thus, in QM point of view, the state does not change but only the phase.

For not using natural unit, we knew, w h_bar = Energy.