Will a spin up particle stay spin up?

Quantum mechanics is weird, if we put a particle in a specific position it will never stay there. Very quickly its position gets randomized, blurred. Does this happen with other properties, for example spin? Will a spin up particle stay spin up?

Cliff notes: a spin up particle won't in general stay spin up, but in certain situations it will (for example, if it's free).

The standard way to describe a physical quantity (observable) whose value stays the same is to say that "it's conserved". The question can then be generalized as follows:

Which observables are conserved?

The answer is: whether or not any particular observable is conserved doesn't just depend on the observable itself, but on the physical situation, i.e. the Hamiltonian.

As an easy example, in classical mechanics for a particle in free space the position won't be conserved but the momentum will. For a particle in a gravitational field neither the position nor the momentum is conserved.

In quantum mechanics, it can be shown that an observable is conserved if and only if it commutes with the Hamiltonian. 

With regard to the spin, if for example you have a particle in a magnetic field, the field will interact with the spin and can change its value, kind of like the gravitational field in the above example will interact with the particle’s momentum and will change its value. From the mathematical perspective, the Hamiltonian will have a term involving the magnetic field, which will cause it not to commute with the spin. For a free particle there is no such term, and the spin, as well as the momentum, will be conserved.

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