Experiment: Spin Rotation Operator

[arpon]

How do we experimentally apply the operator $\exp{\left(-i\phi\frac{ S_z}{\hbar}\right)}$ on a quantum mechanical system? (Here $S_z$ is the spin angular momentum operator along the z-axis)
For example, on a beam of electrons?

 

[blue_leaf77]

Operation on a state by an operator is a mathematical concept, you cannot always implement this operation in an experiment. What an experiment can do is, as far as I know, to measure the state (e.g. by placing a detector in the particle's flight path) and to disturb the Hamiltonian of the system (e.g. by applying external fields). From these actions on the system, in the end you will typically try to measure various observable physical quantities which are, mathematically, expressed as a Hermitian operator. As for the rotation operator you have there which is not hermitian, this is one type of unitary operator or transformation whose action is just to change the reference frame of the observer. In practice it just amounts to a trivial spin coordinate transform, in this case a rotation around a chosen z axis.

 

[mikeyork]

Operation on a state by an operator is a mathematical concept, you cannot always implement this operation in an experiment. What an experiment can do is, as far as I know, to measure the state (e.g. by placing a detector in the particle's flight path) and to disturb the Hamiltonian of the system (e.g. by applying external fields). From these actions on the system, in the end you will typically try to measure various observable physical quantities which are, mathematically, expressed as a Hermitian operator. As for the rotation operator you have there which is not hermitian, this is one type of unitary operator or transformation whose action is just to change the reference frame of the observer. In practice it just amounts to a trivial coordinate transform, in this case a rotation around a chosen z axis. You simply rotate you ruler with which you hypothetically use to measure the particle's position.

Not quite. The operator given rotates only the spin reference frame. The position frame (or equivalently the orbital frame) is unchanged.

 

[blue_leaf77]

Not quite. The operator given rotates only the spin reference frame. The position frame (or equivalently the orbital frame) is unchanged.

You are right, my explanation was rather loose on the last part. Editing to post #2 has been done.

 

[DrClaude]

In the particular case of $\hat{S}_z$, you can do it by applying a magnetic field along $z$. To see why that is the case, consider the Hamiltonian $-\hat{\mu} \cdot \mathbf{B}$, where $\hat{\mu}$ is the magnetic moment and $\mathbf{B}$ the magnetic field, and write down the time evolution operator.