How Does the Stern-Gerlach Experiment Reveal Quantum Angular Momentum?

[Tony11235]

In a certain Stern-Gerlach experiment a bean of potassium atoms (mass: M=6.4673x10^-26 kg) emerges from an oven at a temperature of 150 C. That means that each potassium atom has a kinetic energy of 8.76x10^-21 J. The atoms pass through an inhomogeneous magnetic field, whose gradient is 1.2x10^2 T/cm for a distance of 2.0 cm, and continue through a free-field space of 10.0 cm before being deposited on a collector plate. What is the maximum distance between the two lines at the detector?

I'm not sure what is involved. I think my main problem is the picture. What two lines at the detector is the problem referring two? Right now I just can't focus I guess.

 

[Doc Al]

The potassium atoms have a magnetic moment (due to the electron spin), thus they will feel a net force in an inhomgeneous magnetic field. Classically, the component of the magnetic moment in the direction of the field gradient would take on a continuum of values. But quantum mechanically, the component of the magnetic moment is quantized to only two values (spin up or spin down); thus the beam of atoms is split in two, forming two "lines" on the plate.

 

[kyle8921]

The Stern-Gerlach experiment is a classic experiment in quantum mechanics that demonstrated the quantization of angular momentum. In this experiment, a beam of atoms is passed through an inhomogeneous magnetic field, causing the atoms to split into two distinct paths. The distance between these two paths, also known as the separation of the two lines, is a direct result of the magnetic moment of the atoms.

In order to determine the maximum distance between the two lines at the detector, we need to calculate the deflection of the potassium atoms in the magnetic field. This can be done using the formula for the deflection of a charged particle in a magnetic field, which is given by F=qvB, where F is the force, q is the charge, v is the velocity, and B is the magnetic field strength.

In this case, the atoms are neutral, so we must use their magnetic moment (μ) instead of charge. The magnetic moment is given by μ = γL, where γ is the gyromagnetic ratio and L is the angular momentum. The gyromagnetic ratio for potassium is known to be 1.4x10^7 T^-1s^-1.

The angular momentum (L) can be calculated using the formula L = Iω, where I is the moment of inertia and ω is the angular velocity. In this case, the atoms are moving in a straight line, so ω = 0 and therefore L = 0. This means that the magnetic moment for the potassium atoms is also equal to 0.

Substituting this into the formula for the deflection, we get F = 0, meaning there is no deflection of the atoms in the magnetic field. Therefore, the maximum distance between the two lines at the detector is also equal to 0.

In conclusion, in this particular Stern-Gerlach experiment, the maximum distance between the two lines at the detector is 0, as the atoms do not experience any deflection in the magnetic field. This result is in accordance with the theory of quantum mechanics, which predicts that neutral particles with zero magnetic moment will not experience any deflection in a magnetic field.