# Does the spin has a determinate value in magnetic field?

Consider a 1/2 spin particle like proton in magnetic field $$B$$ in the $$z$$ direction with strength as high as in NMR appliances. Then the anomalous Zeeman Effect occurs. And the particle has two possible levels - one for spin up $$z$$ and one for spin down $$z$$. In NMR it can absorb a photon and go from spin down to spin up and then re-emit a photon going from up to down.

But QM says that the spin is projected on the $$z$$ axis. Then placing a 1/2 particle in $$B$$ is a measurement without anyone to observe it. As far as I am aware a measurement must include an observation from an experimenter (as is in Schrödinger cat)? Is this true?

On the other hand it is stated that the spin precesses around $$B$$. Now I am not sure if a precession and a definitive spin projection are mutually compatible. Maybe the spin revolves only about $$+z$$ (or alternatively only about $$-z$$ half axis)? Then it has $$+1/2$$ spin in $$z$$ (or $$-1/2$$ when rotation around $$–z$$) and is in superposition on $$x$$ and $$y$$ axis?

The quantum-mechanical spin state of a spin-1/2 particle is quite a complicated beast if you want to understand it in detail.

As a first approach, it's true that for any spin-1/2 system, given a quantization axis (say, the $$z$$ axis), there are only two fully-distinguishable states, $$|{\uparrow}\rangle$$ and $$|{\downarrow}\rangle$$ ("up" and "down"). However, it is also perfectly possible for the system to be in a superposition state of these two, with arbitrary complex-valued amplitudes, so the most general state has the form $$|\psi\rangle = \cos(\theta/2)|{\uparrow}\rangle + e^{i\phi}\sin(\theta/2)|{\downarrow}\rangle \tag 1$$ for two parameters $$\theta\in[0,\pi]$$ and $$\phi\in[0,2\pi)$$. (Why can it be written in this form? Because it must be normalized to unit norm, and because we don't care about global phases.)

Now, here's the thing: for any state of a spin-1/2 system, of the form $$(1)$$, there exists an axis of quantization for which said state is the "up" state along that axis of quantization; in other words, there always exists a unit direction vector $$\hat{\mathbf n}$$ such that $$|\psi\rangle$$ is an eigenstate of the spin component operator $$\hat{S}_\hat{\mathbf n} = \hat{\mathbf n}\cdot \hat{\mathbf S}$$ in the same way that $$|{\uparrow}\rangle$$ is an eigenstate of $$\hat{S}_z$$. Moreover, this unit direction vector $$\hat{\mathbf n}$$ has polar spherical coordinates $$(\theta,\phi)$$.

This means that we can think of the set of all spin states of a spin-1/2 system as a sphere, which we call the Bloch sphere. This is a crucial, core concept, and it is well worth reading up on it, both at its Wikipedia page as well as in textbooks. (If you want a specific suggestion, try Nielsen and Chuang.)

So, how does this interface with NMR?

• If all you do is add a magnetic field $$\mathbf B$$, then it is indeed correct that the spin will precess around $$\mathbf B$$, in the sense that the unit-vector Bloch-sphere representation $$\hat{\mathbf n}$$ of $$|\psi\rangle$$ will precess around the magnetic-field vector.
• On the other hand, if one performs a projective measurement of the spin component along any chosen quantization axis (be it the $$z$$ axis or something else), then the measurement outcomes will only ever be "up" or "down" along that chosen axis.

So, what constitutes a measurement? well, for starters, it requires an actual interaction with an external system (starting with fluorescent emission). But what really constitutes a measurement, you ask? That's one of the biggest open problems in the foundations of quantum mechanics.

It's important to note here that the spin precession sounds like it's a very likely candidate to the physical mechanism behind the emission, in the same way that a hydrogen atom placed in an electronic superposition state has a charge oscillation that matches the emission frequency, but that's basically a red herring. The EM field is a quantum entity, and the mechanism for the emission is quantized transitions in interaction with the quantized EM field.

Hopefully this is helpful and enough to get you started.

• Emilio, my concern is: is the superposition projected by the magnetic field on z axis or not? It turns out in Copenhagen interpretation that it is not. There is no splitting of energy levels, even no transition between them until not anyone (with high intelectual capacity) observed the photon with a special instrument he build for that purpose. Oct 24 '19 at 7:26

Consider a 1/2 spin particle like proton in magnetic field B... Then the anomalous Zeeman Effect occurs.

The Zeeman effect is observable for atoms, not for single subatomic particles. Only the "connection" of nucleus and electrons emits photons with defined peaks in relation to each other.

... the particle(s of an atom) has two possible levels - one for spin up z and one for spin down z. In NMR it can absorb a photon and go from spin down to spin up and then reemit a photon going from up to down.

Hardly the Zeeman effect is observable for single atoms. It is the interaction of atoms with each other that orientate the subatomic particles in different directions. The external magnetic field align particles with different orientations of their magnetic dipoles. During their relaxation they emit the photons which one see as the Zeeman broadening.

But QM says that the spin is projected on the z axis.

The spin is directed in any direction. It is the z axis which is “adjusted” to the spin as a normal. From this “sketch” it is now possible to describe the observed phenomena.

• Holger. tahnks. Maybe its a terminology error of mine and mostly Zeeman effect as a term is used for atoms. But splitting of energy levels in nuclei is the same thing. A quote from Wiki (Zeeman effect) - "The Zeeman effect is very important in applications such as nuclear magnetic resonance spectroscopy". H1 (hydrogen) is the most used in NMR so with electron or not does not make any difference. The spin must orient parallel/antiparallel to B and there is energy difference E=h.gama.B. That is exactly what happens in Zeeman too. I read somewhere that it is termed "nuclear Zeeman effect". Oct 16 '19 at 6:45
• Zeeman splitting is also observed in accelarators where the work is done with just protons. Oct 16 '19 at 11:47
• The main question is in fact: When placing a 1/2 particle in B does it have +1/2 value (or -1/2) on z objectively? And if yes - then is this a measurement without anyone to observe it. As far as I am aware a measurement must include an observation from an experimenter (as is in Schroedinger cat)? Oct 16 '19 at 12:02
• Or it is the particle in superposition of spin up and spin down on z and rotates between them? Oct 16 '19 at 12:12
• @Mercury I’m very interested to read about Zeeman effect from protons in particle accelerators. Could you give some source? Oct 17 '19 at 3:23