Precession of spin in a magnetic field

[docnet]

TL;DR Summary

what happens during the precession of spin in a magnetic field

In this lecture Lenny Susskind describes a spin in a magnetic field precesses around the axis of the direction of the magnetic field. This description is also frequently found in NMR theory which is a semi-classical theory.

Lenny says if the magnetic field ##B_o## is applied in the ##z## direction and the spin was originally pointing along the ##x## direction, then the spin will precesses around the ##z## axis. Then Lenny describes around 2:02:00 what sounds contradictory. The spin is not actually in the x axis. The spin is in a superposition of states up and down, which are aligned with the ##z## axis. Only the average of the spin is in the perpendicular direction. The how can there be a measurable component of spin perpendicular to z which is precessing??

The description seems to contradict itself. The spin in the magnetic field is represented with an arrow pointing in an oblique direction, yet what is really there is a superposition of states up and down, along Z.

In NMR spectroscopy someone puts a collection of spins in a magnetic field ##B_z,##. Spins exist in the up (lower energy) and down (higher energy) states with some probability corresponding to the Boltsman distribution. The individual spins are described as precessing around the z axis, and the precessions of individual spins not in phase with each other. Then someone applies a radio frequency pulse to the spins, tipping the bulk magnetization (is this the same thing as spin?) into the x-axis and they start to precess around the z axis in phase with each other. what is really happening when a spin "precesses", and what is the difference between these two precessions in NMR?link:

 

[PeterDonis]

The spin is not actually in the ##x## axis. The spin is in a superposition of states up and down, which are aligned with the ##z## axis.

No. These are not two different states of spin; they are just two different descriptions of the same state of spin. If you measured this spin about the ##x## axis, you would not get "up" or "down" each with some probability; you would always get "up". That's why it can be described as a spin "about the ##x## axis" with no superposition involved. What Susskind is doing is just switching between these two descriptions when it is convenient to do so.

 

[docnet]

No. These are not two different states of spin; they are just two different descriptions of the same state of spin. If you measured this spin about the x axis, you would not get "up" or "down" each with some probability; you would always get "up". That's why it can be described as a spin "about the x axis" with no superposition involved. What Susskind is doing is just switching between these two descriptions when it is convenient to do so.

Thank you for your insight.. I sometimes become confused about quantum mechanics. I forgot classical logic often fails in quantum mechanics.

 

[vanhees71]

The classical meaning of spin precession in a (for simplicity homogeneous) magnetic field in ##z## direction in non-relativistic quantum theory (i.e., using the Pauli equation) can be most easily derived in the Heisenberg picture. The relevant part of the Hamiltonian is
$$\hat{H}=-\frac{g q B}{2m} \hat{s}_z.$$
In the Heisenberg picture the entire time evolution of the observable operators is due to the full Hamiltonian (the statistical operator, representing the state is time-independent).

The time dependence of the spin components is then given by
$$\dot{\hat{\sigma}}_j=\frac{1}{\mathrm{i} \hbar} [\hat{s}_j,\hat{H}].$$
This leads to
$$\dot{\hat{\sigma}}_x=-\frac{\omega_L}{\mathrm{i} \hbar} [\hat{\sigma}_x,\hat{\sigma}_z] =\omega_L \hat{\sigma}_y,$$
$$\dot{\hat{\sigma}}_y=-\frac{\omega_L}{\mathrm{i} \hbar} [\hat{\sigma}_y,\hat{\sigma}_z] =-\omega_L \hat{\sigma}_x,$$
$$\dot{\hat{\sigma}}_z=-\frac{\omega_L}{\mathrm{i} \hbar} [\hat{\sigma}_z,\hat{\sigma}_z] =0.$$
Or in vector notation
$$\dot{\hat{\vec{\sigma}}}=-\omega_L \vec{e}_3 \times \hat{\vec{\sigma}}.$$
Here
$$\omega_L=\frac{g q B}{2m}$$
is the Larmor frequency.

This means that the (expectation value of the) spin precesses (for positively charged particles in mathematical negative direction) around the magnetic field. That's known as Larmor precession.

One of the most important applications is nuclear magnetic resonance (NMR), which in the public is known as magnetic resonance imaging (MRI) to avoid the evil "n word" ;-)).

 

[docnet]

This means that the (expectation value of the) spin precesses (for positively charged particles in mathematical negative direction) around the magnetic field. That's known as Larmor precession.

Thank you for explaining with equations! you confirmed professor Susskind's lecture with details.

Okay, so a question that I was trying to ask in my first post, but was unable to find the words, is the following.

It is established that a spin situated along a ##90## degree angle from the ##B_z## axis precesses around ##z##. Now, this spin has a probability to emit a photon and align itself with the ##z## axis. (In NMR terms, the collection of spins ini ##B_z## undergo longitudinal relaxation and loses energy to the lattice, reaching equilibrium magnetization along ##z##. Because NMR deals with a system of identical spins, this ##z## equilibration is described as happening gradually. The average of the collection of spins is an arrow that moves gradually towards ##z## while precessing around ##z## with the larmor frequency. Equilibration is reached when the bulk spin magnetic moment is aligned along ##z##)

Lenny describes this for an individual spin. In ##B_z##, the spin "up" state in the direction of ##z## has energy like ##\frac{-w}{2}## and the corresponding spin "down" has energy like ##\frac{w}{2}##. Given a fixed ##|B_z|## and a fixed magnetic moment of the nucleus, a spin "down" along ##z## state will transition to "up" while emitting a photon with energy like ##w##. The energy of the emitted photon is the same for a spin in the ##x## axis. It will not emit a photon with energy ##\frac{w}{2}## because it is now halfway between ##z## up and ##z## down. This spin only emits a photon with energy ##w##, but with a probability ##.5## because a spin in the x-axis is in a superposition of spin up and spin down along ##z##.

What confuses me is the spin has a definite direction in ##x## but either emits a photon with energy ##w## or ##0##, with probability of ##.5## in each case. Is this because the magnetic field itself is a measuring apparatus?

 

[PeterDonis]

Is this because the magnetic field itself is a measuring apparatus?

More precisely, the NMR device, which detects the emitted photons, is a measuring apparatus. And since the magnetic field produced by the device is aligned along the ##z## axis, the measurement being realized--the detections of photons--will depend on the ##z## spin states.

 

[docnet]

Frankly I do not know much about NMRI, but I know a bit about NMR because my undergraduate research project uses solid state NMR spectroscopy. (apparently I do not have to know much about NMR theory to use NMR spectroscopy to resolve protein stucture).

NMR spectrometers use magnetic fields which can be homogeneous or gradient applied, and elaborate radio frequency pulses are designed to induce and transfer "labeled" magnetizations between populations of nuclei in a sample. NMR spectrometers have a detector coil that detects the Larmor frequencies of different collections of phase-aligned nuclei, and Fourier transforms the waves into signals. The numbers of different NMR spectroscopy experiments and the types of data generated are numerous.

The photons which are emitted by nuclei during the down ##z## to up ##z## quantum state transitions are lost to the lattice (described in NMR text vaguely as the degrees of freedom available to the nuclei) during relaxation, and are converted to heat, but I could be wrong about the last part.

 

[hutchphd]

Lenny describes this for an individual spin. In Bz, the spin "up" state in the direction of z has energy like −w2 and the corresponding spin "down" has energy like w2. Given a fixed |Bz| and a fixed magnetic moment of the nucleus, a spin "down" along z state will transition to "up" while emitting a photon with energy like w. The energy of the emitted photon is the same for a spin in the x axis. It will not emit a photon with energy w2 because it is now halfway between z up and z down. This spin only emits a photon with energy w, but with a probability .5 because a spin in the x-axis is in a superposition of spin up and spin down along z.

This is a bit confused. Absent the external field there is no energy difference between the various spin states and hence no useful measurement possible.
The presence of the external field means that there is now an axis defined which splits the energy eigenstates according to spin along that axis. We usually call it the z axis. You can still talk about x and y spin states but they will not be energy eigenstates. Further the transition between the up/down z spin states is well characterized and easily measured by the quantum of energy that supplied by a correctly circularly polarized EM field quantum.
It is interesting that the classical theory of magnetic resonance works out just fine, although the quantum one is actually easier to use somehow.

 

[docnet]

This is a bit confused. Absent the external field there is no energy difference between the various spin states and hence no useful measurement possible.
The presence of the external field means that there is now an axis defined which splits the energy eigenstates according to spin along that axis. We usually call it the z axis. You can still talk about x and y spin states but they will not be energy eigenstates. Further the transition between the up/down z spin states is well characterized and easily measured by the quantum of energy that supplied by a correctly circularly polarized EM field quantum.
It is interesting that the classical theory of magnetic resonance works out just fine, although the quantum one is actually easier to use somehow.

That is true. In NMR theory the splitting of the energy eigenstates is called Zeeman splitting, and for spin ##I=\frac{1}{2}## nuclei there are two energy states. In ##|B_o|=0##, the two energy eigenstates are degenerate. As the strength of ##B_o## increases, the difference in energy levels increases linearly with ##|B_o|## and with ##|γ|## in the first order approximation. ##γ## is the gyromagnetic ratio intrinsic to nuclei and contains the spin magnetic momentum.

The system Lenny's lecture describes the ##x##-aligned spin at the moment a magnetic field ##B_z## is turned on. The precession happens before the nucleus (or electron) releases a photon and becomes aligned with the ##z## axis. In semi-classical NMR the spins are allowed to fully align with the ##z## axis before a radio frequency pulse is applied to the sample. This pulse tips the individual spins into the ##x## axis, whose precession is then measured.

 

[hutchphd]

I don't see where he describes "turning on" the field at all. Where in the timeline is this.
This too is very garbled I fear.

 

[docnet]

I don't see where he describes "turning on" the field at all. Where in the timeline is this.
This too is very garbled I fear.

If something that I wrote seems "garbled", please explain directly because I am willing to learn. I am new to quantum mechanics and by no means a know-it-all. That said, I think how the system was prepared is irrelevant. I don't think Lenny mentions how the spin was prepared because that is besides the point. Turning on the field is how I imagined the spin was prepared, so maybe that caused your confusion. If you think semi-classical NMR theory is "easy to use" like you said, please take a look at the paper I attach. It is a recent paper from our group on developing radio frequency pulse sequences for SSNMR spectroscopy :)Article: Scaled Recoupling of Chemical Shift Anisotropies at High Magnetic Fields under MAS with Interspersed C -elements
link:https://www.researchgate.net/publication/342723893_Scaled_Recoupling_of_Chemical_Shift_Anisotropies_at_High_Magnetic_Fields_under_MAS_with_Interspersed_C_-elements

edited for a link to article