REVIEW & SUMMARY Conduction Of Electricity In Solid
The Confinement Principle The confinement principle applies to waves of all kinds. including waves on a string and the
mailer waves of quantum physics. It states that confinement leads t o quantization-that is. to the existence of discrete states with discrete energies.
An Electron in an Infinite Potential Well
An infinite potential well is a device for confining an electron. From the confinement principle we expect that the mailer wave representing a trapped electron can exist only in a set of discrete states. For a one infinite potential well. the energies associated with these quantum states are.
in which L is the width of the well and n is a quantum number. If the electron is to change from one state to another. its energy must change by the amount.
where Ehi~h is the higher energy and blow is the lower energy. If the change is done by photon absorption or emission. the energy of the photon must be.
The probability density “‘;(.r) for an allowed state has the physical meaning that “‘;(.1′) dx is the probability that the electron will be detected in the interval between x and x + dx. For an electron in an infinite well, the probability densities are .
At high quantum numbers n, the electron tends toward classical behavior in that it tends to occupy all pans of the well with ual probability. This fact leads to the correspondence principle: At large enough quantum numbers. the predictions of quantum physics merge smoothly with those of classical physics.
Normalization and ZercrPoiRI Energy The amplitude-A? in 0Eq 40-12 can be found from the normalizing equation,
4 we see that the lowest permitted energy for the electron is not zero but the energy that corresponds to n = I. This lowest energy is called the zero-point energy of the electron-well system.
An Electron in a Finite Potential Well
A finite potential well is one for which the potential energy of an electron inside the well is less than that for one outside the well by a finite amount Uo. The wave function for an electron trapped in such a well extends into the walls of the well.
Two- and Three-Dimensional Electron Traps
tized energies for an electron trapped in a two-dimensional infinite potential well that forms a rectangular corral are
where II, is a quantum number for which the electron’s mailer wave fits in well width L, and 11, is a quantum number for which the electron’s mailer wave fish in well width L,.. Similarly. the energies for an electron trapped in a three-dimensional infinite potential well that forms a rectangular box are.
Here n is a third quantum number. for which the mailer wave fits in well width L
The Hydrogen Atom atom is . The potential energy function for the hy dragon atom.
The energies of the quantum states of the hydrogen atom are found from the three-dimensional form of Schrodinger equation to be
in which n is the principal quantum number. The hydrogen atom requires three quantum numbers for its complete description; their names and allowed values are shown in Table 40-2. _The radial probability density P(r) for a state of the hydrogen atom is defined so that P(r) dr is the probability that the electron will be detected between two concentric shells, centered on the atom’s nucleus. whose radii are rand r + dr. For the hydrogen atom’s ground state.