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
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
• 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
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