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8/21/03 |
Unit 4Three-Dimensional Quantum Mechanics |
Most of the conceptual aspects of three-dimensional quantum mechanics are very similar to those of one-dimensional quantum mechanics. However, the properties of three-dimensional space do introduce several new and fascinating predictions of quantum mechanics that have been verified through experimental measurements.
Schrödinger’s Equation is easily generalized into two or three dimensions simply by including terms for the y- and/or z-coordinates that are of the same form as for the x-coordinate. In the case of a particle in a box, this equation is easily solved by separating the variables into four equations involving x, y, z, and t. The quantum solution to this simple problem shows how the boundary conditions in three-dimensional space produce three quantum numbers that affect the quantized energy levels of the particle. Furthermore, it shows why degenerate energy levels occur under certain conditions and how those degeneracies can be removed by changing the conditions only slightly.
For more complicated symmetries than those of a rectangular
box, namely, for the spherical symmetry of central forces, the solution
to Schrödinger’s equation shows why angular momentum is quantized. In fact, not
only is the magnitude of the angular momentum quantized according to the
orbital quantum number 
, but also is the component of the angular momentum quantized
according to the magnetic quantum number 
. The third quantum number required for a bounded particle in
a three-dimensional space is called the principle quantum number n. It
determines the quantized energy levels of the particle.
An extremely important example of the central force problem
is the quantum solution to the hydrogen
atom. This solution gives exactly the same energy levels as the
Bohr atom without the ad hoc assumption that the electron moves in a circular
orbit with its angular momentum quantized in units of 
. Instead, the quantum model of the atom shows that the
position of an electron is described by a wave function that behaves like a
cloud of probability surrounding the atom. In the ground state, the electron
has zero angular momentum with a position distribution function that simply
resides around the nucleus. At higher energies, the electron may or may not
have angular momentum, and its electron cloud can assume various interesting
shapes.
If the quantum model of the atom is placed in a magnetic field, the orbital angular momentum of the electron produces an orbital magnetism that interacts with the field. The energy of this interaction removes the degeneracy of the energy levels of the atom and produces the Zeeman effect observed as a fine structure in the atomic spectrum of the atom. Careful analysis of the experimental fine structure of the spectrum of an atom in a magnetic field shows that even when the electron has zero orbital angular momentum, it still has an intrinsic angular momentum called spin. The total angular momentum of the electron is simply the vector sum of its spin and its orbital angular momentum.
According to quantum mechanics, all fundamental particles of a given type are physically indistinguishable from one another. In other words, all electrons are identical to one another. Therefore, if two electrons are exchanged with one another the resulting state must be physically indistinguishable from the original. This inability to distinguish between the exchanging of two particles is called exchange symmetry. In the case of electrons, this exchange symmetry results in the Pauli exclusion principle that asserts that no two electrons can simultaneously exist in exactly the same quantum state. In other words, no two electrons can exist in the same place at the same time doing the same thing. Quantum mechanically, this means that no two electrons can have exactly the same quantum numbers.
Because of the exclusion principle, multi-electron atoms must have all their electrons in different states. Therefore, the ground states of the various atoms in the periodic table must consist of electrons arranged in increasingly higher energy levels. By building the atoms upward from the hydrogen atom, adding one electron at a time, quantum mechanics is able to predict the whole periodic table of the elements. However, the detailed spectra of multi-electron atoms are just as difficult to predict in quantum mechanics as the motion of three-bodied in classical mechanics. This is because the interactions of more than two bodies complicates the situation so much that computer simulations are often necessary to obtain accurate results. Nevertheless, the quantum model of the atom has been amazingly successful in predicting the observed properties of atoms and molecules. Therefore, quantum mechanics provides an accurate description of the microscopic world even though that description is sometimes difficult to comprehend.
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