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The Quantum Theory of Magnetism
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Human Geography. Research and Information. Social Work. Warfare and Defence. Magnetism: A Very Short Introduction. Google Preview. We might put it this way. On the other hand, there are situations, such as in a plasma or a region of space with many free electrons, where the electrons do obey the laws of classical mechanics.
And in those circumstances, some of the theorems from classical magnetism are worthwhile.
Also, the classical arguments are of some value for historical reasons. The first few times that people were able to guess at the meaning and behavior of magnetic materials, they used classical arguments. Finally, as we have already illustrated, classical mechanics can give us some useful guesses as to what might happen—even though the really honest way to study this subject would be to learn quantum mechanics first and then to understand the magnetism in terms of quantum mechanics.
We will have to lean on the classical mechanics as kind of half showing what happens, realizing, however, that the arguments are really not correct. We therefore make a series of theorems about classical magnetism that will confuse you because they will prove different things. Except for the last theorem, every one of them will be wrong. Furthermore, they will all be wrong as a description of the physical world, because quantum mechanics is left out.
The first theorem we want to prove from classical mechanics is the following: If an electron is moving in a circular orbit for example, revolving around a nucleus under the influence of a central force , there is a definite ratio between the magnetic moment and the angular momentum. The magnitude of the angular momentum is the mass of the electron times the velocity times the radius. See Fig. It is directed perpendicular to the plane of the orbit. The magnetic moment of the same orbit is the current times the area. However, if you keep going with the classical physics, you find other places where it gives the wrong answers, and it is a great game to try to remember which things are right and which things are wrong.
We might as well give you immediately what is true in general in quantum mechanics.
First, Eq. The electron also has a spin rotation about its own axis something like the earth rotating on its axis , and as a result of that spin it has both an angular momentum and a magnetic moment. In any atom there are, generally speaking, several electrons and some combination of spin and orbit rotations which builds up a total angular momentum and a total magnetic moment.
Although there is no classical reason why it should be so, it is always true in quantum mechanics that for an isolated atom the direction of the magnetic moment is exactly opposite to the direction of the angular momentum. This formula does not, of course, tell us very much. It says that the magnetic moment is parallel to the angular momentum, but can have any magnitude. The form of Eq. You might also be interested in what happens in nuclei. In nuclei there are protons and neutrons which may move around in some kind of orbit and at the same time, like an electron, have an intrinsic spin.
Again the magnetic moment is parallel to the angular momentum. It is like a little magnet, and it has the kind of magnetic moment that a rotating negative charge would have. One of the consequences of having the magnetic moment proportional to the angular momentum is that an atomic magnet placed in a magnetic field will precess. First we will argue classically. Therefore the torque due to the magnetic field will not cause the magnet to line up. The angular momentum—and with it the magnetic moment—precesses about an axis parallel to the magnetic field.
Scientists Switch On and Off Magnetism Using Quantum Mechanics
According to the classical theory, then, the electron orbits—and spins—in an atom should precess in a magnetic field. Is it also true quantum-mechanically? Next we want to look at dia magnetism from the classical point of view. It can be worked out in several ways, but one of the nice ways is the following. Suppose that we slowly turn on a magnetic field in the vicinity of an atom. As the magnetic field changes an electric field is generated by magnetic induction. We would like to write Eq. It is therefore usually more convenient to write Eq.
This is diamagnetism of matter. It is this magnetic effect that is responsible for the small force on a piece of bismuth in a nonuniform magnetic field. You could compute the force by working out the energy of the induced moments in the field and seeing how the energy changes as the material is moved into or out of the high-field region.
Classical mechanics cannot supply an answer. We must go back and start over with quantum mechanics. In an atom we cannot really say where an electron is, but only know the probability that it will be at some place. This equation, of course, is the moment for one electron. The total moment is given by the sum over all the electrons in the atom. The surprising thing is that the classical argument and quantum mechanics give the same answer, although, as we shall see, the classical argument that gives Eq.
The same diamagnetic effect occurs even when an atom already has a permanent moment. Then the system will precess in the magnetic field. As the whole atom precesses, it takes up an additional small angular velocity, and that slow turning gives a small current which represents a correction to the magnetic moment.
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This is just the diamagnetic effect represented in another way. That has already been included in the diamagnetic term. We can already conclude something from our results so far. Thus, according to the classical theory, all systems of electrons would precess with the same angular velocity.