J. Heisenberg Uncertainty and the 2D spin
The 2D structured spin gives an example which provides a basic understanding of how Heisenberg Uncertainty Principle works. The mathematical basis for this Principle follows because observables in quantum mechanics sometimes do not commute. Examples are position-momentum; energy-time; difference orthogonal components of the angular momentum vector. For spin, the Pauli spin operators do not commute.
Immediately it should jump out that the two components of the 2D spin are orthogonal and cannot simultaneously be measured. Each carries a magnetic moment and hence each has a spin operator which must be orthogonal. It follows from Heisenberg that only one of the two components can be measured in one experiment. One is always missed.
This point becomes important in analyzing EPR experiments. It also means that one cannot properly characterize spin by measuring it because the √2 spin states cannot be observed.
Here, my plan is to give a visualization of how only one of the two axes can be measured and the other one not.
Laboratory and Body Fixed Frame
In spectroscopy, it is common to define a body fixed frame for a molecule. In that frame its structure is more easily expressed. Then, using rotation matrices, the body frame is transformed to the laboratory frame, where experiments are done. The same is done here for the 2D spin. See the figure,
On the left one 2D spin is show in its body fixed frame. Each spin generally has a different frame, and all are related to the Lab frame by two angles. Note also in the body frame, the two spin orientations are shown but only one exists at any instant. Here space is isotropic, so the √2 spin is shown bisecting the quadrants.
Let us now apply an external magnetic field along the laboratory Z axis. Recall that a spin carries a magnetic moment that is so tiny that it is swamped by any external measuring field, so it lines up with that field,
Space is now no longer isotropic in the presence of a measuring probe, and so the √2 spin cannot form, nor can the mirror states. Since the spin is oriented some way, one axis is going to be closer to the applied field than the other. That one lines up while the other axis spins in the plane perpendicular to the applied field,
Then it looks just like a usual spin ½ (shown in the middle above) with the same magnitude of magnetic moment, and is described by a single axis of quantization. The 2D spin looks like a point particle when observed.
But it isn’t. The application of a measuring probe makes it impossible to measure the √2 spin and the results are identical had the experiment assumed the usual point particle spin.
I know of only one quantum calculation of the Stern-Gerlach experiment, which the above depicts. That is by Scully et al in 1969*. I did the same calculation using the 2D spin and the results confirm the conclusion of Scully et al: that the linear momentum and the spin angular momentum couple so the particle is deflected in the direction of the applied field. The orthogonal states are averaged away.
*M. O. Scully, W. E. Lamb Jr., and A. Barut, On the Theory of the Stern-Gerlach Apparatus, Foundations of Physics, 1987, 17, 575.
There is a mathematical “trick” that is often used to simplify things. If one looks at an object which is spinning around some axis, it is possible to transform the equations so that they are all spinning at the same rate. In other words we change coordinate frames to the rotating or spinning frame. In that frame, the 2D spin as not spinning anymore and the two orthogonal components are frozen as in the figure below,
The orientation of the applied field is “a” which could very well be the laboratory Z axis. The experimental result, which the 2D spin confirms, is that the Z component displays two pure states which are the usual spin ½ states of |±>Z. Experimentally, in a Stern-Gerlach experiment, the deflected spin leaves two spots on a photographic state, one for the “+” state and the other for the “-“ state.
The same results would be observed for the 2D spin.
Note in the figure in the spinning frame that the two orthogonal components are frozen along directions “a’ and “d”. We have already stated that the “a” component lies along the laboratory frame and gives the two states |±>a. Experimenters refer to the detection of different states as “channels”. That is, for this experiment, there are two channels in the “a” direction and every time a “+” state is measured, it is added to the “+” channel bin. A bin is simply a place (a computer file) which gives the number of events that occur in that channel. Likewise there is a “-“ channel and bin. Creating an inhomogeneous magnetic field and setting up the channels and bins is the job of the experimentalist. He counts “clicks” which are the responses of events from detectors. This is the experimental set up.
Now let us look at the “d” direction. In the spinning frame it looks the same as the “a” direction, so what the experimentalist must do to detect along the “d” direction is to build the same sort of apparatus but which is spinning at the same speed as the spin is spinning. Although this is an impossible task, we can think about it and perform a gedenken experiment.
If it were possible to build an apparatus like the above, then we could measure simultaneously along both the “a” and the ”d” axes. Then we would have four distinct channels and four distinct bins. Two bins would count clicks along the laboratory axis and two bins would count clicks in the spinning frame.
This means that the 2D spin displays twice as many clicks as does the usual spin which has only one axis.
In reality, knowing that spins spin at megahertz frequencies, it seems an impossible task to be able to create such an apparatus (although I am quick to add that experimentalists are extremely creative and may be able to pull something off like this somehow.)
The conclusion is that although four clicks are theoretically possible for the 2D spin, only two can ever be measured. Coming back to the Heisenberg Uncertainty Principle, it is clear why it is impossible to simultaneously measure along two orthogonal axes of spin quantization: we cannot build such an apparatus.
This makes a difference for the EPR coincidence experiments because in a given run, N coincidences are detected. However there are actually 2N coincidences possible but only half can ever be detected in a given experiment.
The reality is that it is unlikely that such an apparatus can be built and so those spinning orthogonal components simply phase randomize and are not detected.
This also underlines something that is quite unexpected. Physics is an experimental science. However for spin, if it turns out the 2D version is viable, then there are some states that can never be observed.