Appendix 12 - 1

Appendix 12: Magnetic Precession in Static and Oscillating Magnetic Fields:The Rotating Coordinate System

Mara PrentissRonald WalsworthSeptember 2002

Before learning the full quantum mechanical treatment of a spin 1/2 particle in amagnetic field, it is useful to consider the more familiar and intuitive problem of the evolution ofa classical particle with magnetic moment

†

r M . The quantum mechanical problem will turn out to

be analogous.

The unperturbed system treated here is a single object or particle with magnetic moment

†

r M in the presence of a strong static magnetic field in the z direction. The perturbation will be aweak rotating magnetic field in the x-y plane (which is largely equivalent to an oscillating fieldin the x direction).

Unperturbed System

The energy of the unperturbed system depends on the angle q between the magneticmoment and the magnetic field.

†

r B = B0 ˆ z

†

U = -r

M ⋅r B = -

r M B0 cosq

For a negatively charged particle

†

r M = -g

r L

where g is called the gyromagnetic ratio, and

†

r L is the angular momentum of the particle (which

for an electron is also the spin).

The torque on the magnetic moment is given by

†

r t =

dr L

dt=

r M ¥

r B

Hence

†

dr

M dt

= -g dr L

dt= -g

r M ¥

r B = g2

r L ¥

r B .

Note that the magnitude of the magnetic moment does not change since

Appendix 12 - 2

†

dr L

2

dt=

dr L ⋅

r L ( )

dt= 2

r L ⋅ d

r L

dt= 2

r L ⋅

r M ¥

r B ( ) = 2

r L ⋅ -g

r L ¥

r B ( ) = 0

Similarly the angle between

†

r M and

†

r B (i.e., the projection of

†

r M along the z axis) does not

change, since

†

dr B ⋅

r L ( )

dt= 0 +

r B ⋅ d

r L

dt=

r B ⋅

r M ¥

r B ( ) = -g

r B ⋅

r L ¥

r B ( ) = 0

In the quantum mechanical problem of a particle with spin there will be only two possibleorientations of the spin, up or down. Both spin up and spin down will be eigenstates, just as thez component of the spin is a constant in the classical case.

The equations of motion for the magnetic moment become

†

dMx

dt= -w0My

dMy

dt= w0Mx

dMz

dt= 0

where

†

w0 = gB0 is known as the Larmor frequency.

With the initial condition

†

r M (0) = Mx (0) ˆ x + Mz (0)ˆ z

i.e.,

†

q ≠ 0, The solutions are

†

Mx (t) = Mx (0)cosw0tMy (t) = Mx (0)sinw0t

That is, the magnetic moment precesses around the z-axis at the Larmor frequency.

Rotating Frame

Now consider the problem in the reference frame rotating at the Larmor frequency

†

w0. Inthis reference frame,

†

r M is a constant, which makes the time evolution trivial. The time

evolution in the lab frame can then be obtained by transforming from the rotating frame back tothe lab frame.

Appendix 12 - 3

Consider in general the transformation to a reference frame rotating at a frequency w. The rotation matrix for the transformation between the two frames is

†

R =

coswt sinwt 0-sinwt coswt 0

0 0 1

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

The magnetic moment in the new reference frame is given by

†

M ¢ x (t) = Mx coswt + My sinwtM ¢ y (t) = -Mx sinwt + My coswtM ¢ z (t) = Mz

The equations or motion in the rotating frame are

†

dM ¢ x

dt= w -w0( )M ¢ y

dM ¢ y

dt= - w -w0( )M ¢ x

dM ¢ z

dt= 0

These equations can be recombined into a vector equation

†

d ¢ r

M dt

= -g ¢ r

M ¥r B eff

where

†

r B eff = B0 -

wg

Ê

Ë Á

ˆ

¯ ˜ ̂ z = -

Dg

ˆ z

and

†

D = w -w0 .

Note that at resonance, one has

†

w = w0 and thus

†

r B eff = 0, so there is no precession of the

magnetic moment in the rotating frame. For any other frequency there is a precession because therotating frame is out of synch with the Larmor frequency. The equations above can betransformed back into the lab frame:

Appendix 12 - 4

†

r M Æ

r ¢ M

dr

M dt

Æd

r ¢ M

dt+

r w ¥

r ¢ M

r B eff Æ

r B -

r w g

Perturbation by a Rotating Magnetic Field

Next, consider an applied B field, which precesses in the x-y plane at frequency w:

†

r B pert = B1 coswt ˆ x + sinwt ˆ y ( )

[Note: in the laboratory it is often impractical to create a rotating magnetic field, so onecommonly employs a field that oscillates along a particular direction, e.g., the x axis. Thisoscillating field is equivalent to the sum of two counter-rotating fields: one field rotates with thesame helicity as the magnetic moment's rotation, while the other field rotates with oppositehelicity and can usually be ignored because it is effectively very far off resonance (at

†

w @ 2w0).]

With the addition of

†

r B pert the equations of motion become

†

dr

M dt

= -gr

M ¥r B = -g

r M ¥ B0 ˆ z + B1 coswt ˆ x + sinwt ˆ y ( )[ ]

In the frame rotating at w this becomes

†

dr

¢ M dt

= -gr

¢ M ¥r B eff = -g

r ¢ M ¥ B0 -

wg

Ê

Ë Á

ˆ

¯ ˜ ̂ z + B1 ˆ x

È

Î Í

˘

˚ ˙

In component form one has

†

d ¢ M xdt

= DMy

d ¢ M ydt

= -DMx - gB1Mz

d ¢ M zdt

= gB1My

On resonance in the rotating frame, i.e., at

†

w = w0, the magnetic field lies ONLY alongthe x direction, so the magnetic moment simply precesses around x. This means that Mx isconstant and the component of

†

r M perpendicular to x rotates in the y-z plane. Thus the

component of

†

r M which is along the z-axis at time t=0 will be along the -z axis at

†

t = p gB1 . In

other words, all of the projection of

†

r M along the positive z-axis will be transferred to a

Appendix 12 - 5

projection along the negative z-axis. Classically and geometrically, this is a consequence of theaxis of rotation being perpendicular to the y-z plane. Quantum mechanically, the appliedrotating field allows a coupling between the spin up and spin down states. In either picture, thisinversion effect is known as resonant Rabi oscillation.

Mathematically, the solutions are

†

¢ M x (t) = ¢ M x (0)¢ M y (t) = - ¢ M z(0)singB1t + ¢ M y (0)cosgB1t¢ M z(t) = - ¢ M y (0)singB1t + ¢ M z (0)cosgB1t

Thus the magnetic moment rotates around the x-axis at the Rabi frequency

†

W = gB1.

If

†

r B pert is tuned off resonance, such that

†

w @ w0, then the effective magnetic field will

have some z component as well as an x component. Thus the component of

†

r M in the positive z

direction will NEVER rotate completely into the negative z direction (and visa versa). As thedetuning of w from w0 gets larger and larger, the effective field approaches the z direction andthe rotations between the positive and negative components of the magnetic moment approacheszero.

Mathematically, in the rotating frame

†

r ¢ M precesses about

†

r B at a frequency

†

Weff = gr B eff

2= D2 + (gB1)

2 = D2 + W2

This is the off-resonant Rabi oscillation frequency.

The effect of the field is thus to set up a correlation between the spin up and spin downstates since something, which is in spin up at one time, will rotate into spin down at another time.In the stationary frame, the moment precesses around the moving magnetic field vector.

Mathematically, in the rotating frame, the time evolution of the components becomes

†

¢ M x (t) = ¢ M x (0)gB1( )2

+ D2 cosWeff tWeff

2 + ¢ M y (0) DWeff

sinWeff t + ¢ M z (0) DWWeff

2 cosWeff t - 1( )

†

¢ M y (t) = - ¢ M x (0) DWeff

sinWeff t + ¢ M y (0)cosWeff t - ¢ M z(0) WWeff

2 sinWeff t

†

¢ M z(t) = ¢ M x (0) DWWeff

2 cosWeff t - 1( ) + ¢ M y (0) WWeff

2 sinWeff t + ¢ M z(0) 1+W

Weff

Ê

Ë Á Á

ˆ

¯ ˜ ˜

2

cosWeff t - 1( )È

Î

Í Í

˘

˚

˙ ˙

Appendix 12 - 6