- #1
Tainty
- 27
- 1
I have some trouble understanding the concept of spin precession in an external field from a quantum mechanics viewpoint. Hopefully someone will be willing to enlighten me.
Consider a single spin ½ particle. In the absence of an external field, the projection of the spin angular momentum on any arbitrarily defined quantization axis can only take on two specific values, i.e. -½ (spin down) and +½ (spin up). Let us define a quantization axis z and further suppose we have prepared this particle in a particular eigenstate, say the +½ state. In other words, this is equivalent to saying that the particle has a definite projection angular momentum on the z-axis equal to ħ/2.
We now turn on or place this particle in an external and constant magnetic field B which is aligned with our chosen quantization axis (z-axis). In certain (for e.g. nuclear magnetic resonance (NMR)) material that I've come across online, it is usually said the particle will now begin to precess about the B field, i.e. the z-axis.
My questions are as follows:
1. Will a particle prepared in the +½ Sz eigenstate precess?
When referring to material on the quantum mechanics of precession, one usually defines the spin ½ particle with an arbitrary orientation in space (see for e.g.)
In summary, it is something typically along the lines of defining the particle in a quantum state:
Ψ = cosθ|1> + sinθ|2> where θ is the angle between the spin angular momentum vector and the z-axis that the B field is applied along.
It can then be shown that the expectation value of the various projections of the magnetic dipole moment resembles that of precession about the z-axis at a precession frequency equivalent to the Larmor frequency.
My reasoning is that a particle prepared in +½ Sz eigenstate has a projection that is aligned with the z-axis, so θ=0 and therefore no precession should occur. Is this correct?
In the absence of an external field, the energies for the pair of eigenstates (spin up and spin down) are degenerate and so the time dependency within the Schrödinger equation only amounts to a constant phase factor. However, upon application of the magnetic field along the z-axis, the phase factors for each eigenstate are different. My second question is:
2. For the same conditions as described above: (i.e. a particle prepared in the +½ Sz eigenstate, with a B field applied along the z-axis), how does this spin ½ particle evolve with time? Since it is in a stationary state, does it ever flip-flop between eigenstates at the Larmor frequency?
Any help would be most appreciated. Thanks!
Consider a single spin ½ particle. In the absence of an external field, the projection of the spin angular momentum on any arbitrarily defined quantization axis can only take on two specific values, i.e. -½ (spin down) and +½ (spin up). Let us define a quantization axis z and further suppose we have prepared this particle in a particular eigenstate, say the +½ state. In other words, this is equivalent to saying that the particle has a definite projection angular momentum on the z-axis equal to ħ/2.
We now turn on or place this particle in an external and constant magnetic field B which is aligned with our chosen quantization axis (z-axis). In certain (for e.g. nuclear magnetic resonance (NMR)) material that I've come across online, it is usually said the particle will now begin to precess about the B field, i.e. the z-axis.
My questions are as follows:
1. Will a particle prepared in the +½ Sz eigenstate precess?
When referring to material on the quantum mechanics of precession, one usually defines the spin ½ particle with an arbitrary orientation in space (see for e.g.)
In summary, it is something typically along the lines of defining the particle in a quantum state:
Ψ = cosθ|1> + sinθ|2> where θ is the angle between the spin angular momentum vector and the z-axis that the B field is applied along.
It can then be shown that the expectation value of the various projections of the magnetic dipole moment resembles that of precession about the z-axis at a precession frequency equivalent to the Larmor frequency.
My reasoning is that a particle prepared in +½ Sz eigenstate has a projection that is aligned with the z-axis, so θ=0 and therefore no precession should occur. Is this correct?
In the absence of an external field, the energies for the pair of eigenstates (spin up and spin down) are degenerate and so the time dependency within the Schrödinger equation only amounts to a constant phase factor. However, upon application of the magnetic field along the z-axis, the phase factors for each eigenstate are different. My second question is:
2. For the same conditions as described above: (i.e. a particle prepared in the +½ Sz eigenstate, with a B field applied along the z-axis), how does this spin ½ particle evolve with time? Since it is in a stationary state, does it ever flip-flop between eigenstates at the Larmor frequency?
Any help would be most appreciated. Thanks!