Light has the dual characteristics of particles (photons) and electromagnetic
waves. The photon has an energy of 𝐸 = ℎ𝜈 (𝜈: frequency, h: Planck constant) and the
momentum of 𝑝⃗ = ℎ𝑘⃗⃗ (𝑘⃗⃗: wavenumber and 1|𝑘⃗⃗| is the wavelength). The photon density is
proportional to the square of the amplitude of the electromagnetic waves. The
fundamental aspect of quantum mechanics is that these characteristics apply to all
matters. The properties of matters are described by wave functions. The probability of
the existence of the matter is proportional to the square of the associated wave function.
When a matter is localized in a limited region, it can only assume discrete values of
energy because the wavelength of the matter wave must be an integral division of the
region. The phase of the wavefunction has uncertainty on order 1/2 radians; therefore,
position and momentum (time and energy) cannot be simultaneously determined. As the
size of the localization area of the wavefunction becomes smaller, the minimum kinetic
energy becomes larger because of the smaller wavelength (larger momentum
uncertainty).
The Schroedinger equation was derived based on the idea that the relationship between
the frequency and the wavenumber corresponds to that between energy and momentum
given by classical mechanics, which makes it possible to obtain the wave functions of
matters in the energy eigenstates. Several examples of solutions to the Schroedinger
equation are introduced. The mixture between different energy eigenstates and the shift
in the energy eigenvalues are induced by electromagnetic fields. The temporal change of
the wave function (transition between different energy states) is also obtained using the
Schroedinger equation.
Keywords: Adiabatic rapid passage, Backbody radiation, Bohr radius, Boson, Eigenfunction, Eigenvalue, Electric induced transparency (EIT), Fermion, Operator, Particle-wave duality, Photoelectronic effect, Rabi oscillation, Schroedinger equation, Stark shift, Uncertainty principle, Zeeman shift.
