Chapter 39 – More About Matter Waves

39-1 Energies of a Trapped Electron

• The confinement principle states that confinement of any type of wave leads to

quantization, which means that the wave is restricted to discrete states with certain

energies. States with intermediate energies are not allowed within confinement.

o Electrons confined to an infinitely deep potential energy well or infinite

potential well can exist in only certain discrete states. If the well is one-

dimensional, the energies associated with these quantum state are expressed

as:

represents the length of the well.

represents the electron mass.

represents the quantum number.

• The lowest energy is given by 1, not 0.

The electron can jump from one quantum state to another state only if its

energy change is equal to the difference between the higher energy ( )

to the lower energy ( ) expressed as

• If change occurs by photon absorption or emission, the energy of

the photon must be equal to the change in the electron’s energy:

39-2 Wave Functions of a Trapped Electron

• The wave function for an electron in an infinite one-dimensional potential energy well

along an axis is expressed as

• The probability that an electron will be detected in the interval between the coordinates

and can be given by the product

• The correspondence principle states that at large enough quantum numbers, the

predictions of quantum physics will coincide smoothly with the predictions of classical

physics.

• If the probability density of an electron is integrated over the entire axis, the total

probability must be equal to 1; the aforementioned can be expressed as:

o Using this equation to evaluate the amplitude of a wave function is referred to as

normalizing the wave function. Thus, the equation is also referred to as the

normalization equation.

o Confined systems in quantum physics require a state of minimum energy

referred to as the zero-point energy that cannot be a quantum number of zero.

39-3 An Electron in a Finite Well

• The wave function for an electron in a finite, one-dimensional potential well extends into

the walls, where the wave function decreases exponentially with well depth ( ).

o In comparison to the sates in an infinite well of relatively the same size, the

states in a finite well have a limited number, longer de Broglie wavelengths, and

lower energies. The infinite depth potential well is an idealization.

39-4 Two- and Three-Dimensional Electron Traps

• Nanocrystallites are granules of a semiconducting powdered material that is an

effective potential well for the electrons trapped within it.

• Quantum dots are artificial atoms.

• Quantum corrals are artificially grouped atoms in the shape of a circle utilized to move

the atoms.

• The quantized energies for an electron trapped in a two-dimensional infinite potential

well that forms are rectangular corral are express by

o represents the quantum number for the respective well width ( )

• The wave functions for a matter wave/electron in a two-dimensional potential well are

given by the following expression:

39-5 The Hydrogen Atom

• The hydrogen atom is a fixed potential trap with the electron moving around inside it.

• The Bohr model is a planetary model in which the electron orbits the central proton with

an angular momentum that is limited to values expressed as

The quantum number in the Bohr model is incorrectly restricted to exclude 0. While the

Bohr model of the hydrogen atom successfully evaluated the energy levels for the atom

in order to explain the emission and absorption spectrum (i.e. the collection of

wavelength lines) of the atom, the model is incorrect in almost every other instance.

• Schrödinger’s equation gives the correct values of angular momentum and its quantized

energies:

o The atom (or electron) can change energy only by jumping between the allowed

energies.

If the jump is by photon absorption or emission, the restriction in energy is

expressed as

• (Rydberg constant)

• The radial probability density ( ) for a state of hydrogen atom is defined so that the

probability density is defined as the probability of the electron being detected

somewhere in the space between two spherical shells of the radii and that are

centered on the nucleus.

o Normalization of the statement requires the expression:

• The probability that the electron will be detected between any two given radii can be

expressed as

• The following are the respective quantum numbers of the Hydrogen atom:

o The orbital quantum number is a measure of the magnitude of the angular

momentum associated with the corresponding quantum state.

o The orbital magnetic quantum number is related to the orientation in space of

the angular momentum vector.

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