## Splitting the Eigenstates with Stern-Gerlach

A beam of atoms can be split into the eigenstates of with a Stern-Gerlach apparatus. A magnetic moment is associated with angular momentum.

This magnetic moment interacts with an external field, adding a term to the Hamiltonian.

If the magnetic field has a gradient in the z direction, there is a force exerted (classically).

A magnet with a strong gradient to the field is shown below.

Lets assume the field gradient is in the z direction.

In the Stern-Gerlach experiment, a beam of atoms (assume ) is sent into a magnet with a strong field gradient. The atoms come from an oven through some collimator to form a beam. The beam is said to be unpolarized since the three m states are equally likely no particular state has been prepared. An unpolarized, beam of atoms will be split into the three beams (of equal intensity) corresponding to the different eigenvalues of .

The atoms in the top beam are in the state. If we put them through another Stern-Gerlach apparatus, they will all go into the top beam again. Similarly for the middle beam in the state and the lower beam in the state.

We can make a fancy Stern-Gerlach apparatus which puts the beam back together as shown below.

We can represent the apparatus by the symbol to the right.

We can use this apparatus to prepare an eigenstate. The apparatus below picks out the state

again representing the apparatus by the symbol at the right. We could also represent our apparatus plus blocking by an operator

where we are writing the states according to the value, either +, -, or 0. This is a projection operator onto the + state.

An apparatus which blocks both the + and - beams

would be represented by the projection operator

Similarly, an apparatus with no blocking could be written as the sum of the three projection operators.

If we block only the beam, the apparatus would be represented by

* Example: A series of Stern-Gerlachs.*

Jim Branson 2013-04-22