Bohr’s model of electrons orbiting in atoms had already invoked the idea of angular momentum within the atom. Remember that angular momentum is the measure of an orbiting body’s tendency to keep orbiting. In Bohr’s model, an electron’s angular momentum is determined by its mass, angular speed, and distance from the nucleus. The semi-classical picture of electrons orbiting around nuclei of atoms relied on the electrons’ motion (and angular momentum) to resist the attractive force from the positively charged nucleus and remain in stable orbits.
The concept of angular momentum persists in quantum wave mechanics, and the earlier postulate that energy is quantized comes naturally from the math of the Schroedinger equation in three dimensions.
With wave mechanics, we no longer have the idea that electrons move in circular orbits. Instead, the allowed solutions are standing waves around the nucleus. Even seasoned quantum physicists have a hard time visualizing the angular momentum of an electron orbital, in which the particle does not actually follow any path in space, so don’t feel bad if you also find it difficult.
The important idea is that in any three-dimensional bound system, angular momentum is real and it must be quantized. Even more than that, angular momentum is a vector, which has both a magnitude and a direction. According to Schroedinger’s equation, both the total amount of angular momentum and its direction are quantized. The quantization of the amount of the angular momentum was expressed as the quantum number. For the quantization of the direction we used the letter m, which labeled possible values for the projection of the angular momentum vector onto any fixed axis.
Recall that the quantum works of Planck, Einstein, and Bohr gained acceptance only when guys in lab coats were able to observe predicted effects stemming from energy quantization. Scientists were naturally eager to uncover any direct experimental evidence of the angular momentum quantization, too. Otto Stern, one of Max Born’s assistants in Germany, proposed a way to do this in 1921, but it took a year or so and the help of Walter Gerlach to actually get it done.
Here is the gist: let’s say that the quantum number l represents the orbital angular momentum of an electron. Since the electron has an electric charge, an orbiting electron should feel a force whenever the atom moves through a nonuniform (i.e., inhomogeneous) magnetic field. If the direction of the field points up, that force will be either up or down depending on the direction of the electron’s orbital angular momentum. Stern and Gerlach therefore decided to shoot a beam of neutral atoms through a strong, inhomogeneous magnetic field and examine the deflection of that beam.
For atoms with zero angular momentum, Stern and Gerlach expected no deflection. For non-zero angular momentum, there would be a whole range of deflections if the vector could be oriented randomly in any direction. But if the direction were truly quantized, then only certain deflections would occur, and they would see distinct spots or bands when they measured the beam’s position after it passed through the magnetic field.
I cloud take any whole number value from zero on up (limited only by the principal quantum number, n). The possible values for m were positive or negative whole numbers, ranging from –l to +l. As a result, there is only one possible projection for l = 0, three for l = 1, five for l = 2, etc. The number of projections would always be an odd number, regardless of the quantum state the electrons are in. This is why Stern and Gerlach were so befuddled by what they actually saw when they conducted the experiment.
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