How to represent a spin 1/2

The above simulation shows the state of a spin-1/2 particle. The state is described through a unit vector, shown in yellow, pointing in the direction of the average value of the spin vector.

You can change the frame orientation by dragging the bounding sphere with the mouse. You can also change the spin state, either by entering new q and j values with the keyboard, or by dragging the yellow arrow with the mouse, the Ctrl key being pushed down.

Note that j and j +2p yield the same unit vector while they correspond to opposite states in the Hilbert space. In other words, you need to turn 4p for recovering the same state. As in the case of a propagating wavepacket, it should be mentioned that in quantum mechanics it is impossible to directly observe the entire state of the system as the above simulation might lead to believe. Indeed, it is impossible to simulatenously measure the different cartesian components of the spin vector. Therefore, you should take this as the simulation of a mathematical object - the system state - which cannot be directly observed for a single spin. However, if you have a large assembly of spins all prepared in the same state, you can measure each cartesian component for a third of the ensemble and hence measure the average spin vector. This general remark also holds for the spin simulations shown hereafter.

Note that this version is obsolete. Please see the new version.