Download Take silver atoms with an electron that has a moment of µz = −g e(e

Take silver atoms with an electron that has a moment of
µz = −ge (eh̄/2me )ms
If the electrons have two different kinds of spin directions, atoms with those electrons
should have different total spins and then they should respond differently to magnetic
fields.
In the Stern-Gerlach experiment they send these atoms in the x direction between magnets
one above the other, where z is the up/down direction and the magnets have length l (not
to be mixed up with the quantum number).
Although in the area between the magnets in the Stern-Gerlach experiment the magnetic
~ is homogeneous, near the end of the passage thru the magnets, it becomes inhomofield, B,
geneous. The field gives a potential energy V = −µ.B to the atom so if its inhomogeneous
it gives a force on the atom of
~
Fz = −∂V /∂z = µ.(∂ B/∂z).
This force acts on the spins to change the direction of their momentum, p by an angle
∆θ or θ. The new momentum is the old one plus the change in the perpendicular direction
(perp=perpendicular) ∆pperp .
∆θ ≈ sin(∆θ) = ∆pperp /p.
Now p = mA v, where mA is the mass of the atom and v is its velocity. t is the time the
atom spends beteen the magnets so t = l/v and
R
∆pperp = F dt (the impulse) is equal to
−µz .(∂B/∂z)l)/mavv.
We replace µz with its expression above and ma vv with ma v 2 which is twice the kinetic
energy (Ek ) and get:
∆θ = θ = (ge (eh̄/2me )(∂B/∂z)lms )/(2Ek ).
We can cancel the ge with the factor 2 and get
∆θ = θ = ((eh̄/2me )(∂B/∂z)lms )/(Ek ).
1