About the Stern Gerlach Experiment

The experiment in question is an experiment first executed by Otto Stern and Walther Gerlach back in 1922.
It involves sending a beam of particles, originally silver atoms, radiated from an oven, through an inhomogeneous magnetic field, onto a hit detector, for observing their deflection.
With a bit of knowledge of classical electromagnetism, one would expect to see deflection in all directions, at least along the axis in which the major change in the field occurs. The law that quantifies the deflection can be writen as :

where F is the "body-force", meaning it acts on the whole particle and changes it's velocity according to [F = ma], and m is the magnetic moment of the particle, caused by the orbit of the electrons around the atom, creating a little "Current loop".
Therefore, if we assume that the particles leave the oven with a magnetic moment in an arbitrary direction (which is not at all a strong assumption, for the oven has no favored direction), each particle should feel a different body-force according to the angle between its moment and the gradient of the external field B in each direction. On the other hand, results show that the deflection is (roughly) only in the 2 main directions, one takes the particle upwards and one downwards, relative to it's initial height, where the field and its gradient are both directed up. No classical explanation was found, but rather a different approach was taken, that assumes that the particles possess an intrinsic angular-momentum-like property that takes only certain quantized values, and in this case (silver atoms) only 2, one along the direction of the field, corresponding to a deflection upwards, and the other to the opposite way, which corresponds to a deflection downwards.
This intrinsic property was called "Spin".

There are actually 2 things we neglected in the classical analysis, but could not contribute to the explanation, nor even be important in the pattern obtained on the screen.
The first, and obvious, is gravitation, which pulls all the particles downwards in the same way. We neglect it for 2 reasons : 1. for particles of that scale it's so small that it barely affects the path of the particle in the short time it spends between leaving the oven and hitting the screen. 2. since it acts axactly the same on all particles, it would merely cause a deviation of all the pattern a little downward, which is irrelevant.
The second is precession of the angular momentum around the field direction (Larmor Precession) which can actually be even more complex for inhomogeneous field, but, since the equation that describes the time development of the magnetic moment is :

where gamma is the giromagnetic ratio. it describes a precession, with a huge frequency, but keeps the body force constant.




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