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The more general description of matter waves corresponding to a single particle type (e.g. a single electron or neutron only) would have a form similar to = (,) (() /) where now there is an additional spatial term (,) in the front, and the energy has been written more generally as a function of the wave vector. The various terms given ...
The figure can serve to illustrate some further properties of the function spaces of wave functions. In this case, the wave functions are square integrable. One can initially take the function space as the space of square integrable functions, usually denoted L 2. The displayed functions are solutions to the Schrödinger equation.
Here, (,) is a wave function, a function that assigns a complex number to each point at each time . The parameter m {\displaystyle m} is the mass of the particle, and V ( x , t ) {\displaystyle V(x,t)} is the potential that represents the environment in which the particle exists.
where is position, is the wave function, is a periodic function with the same periodicity as the crystal, the wave vector is the crystal momentum vector, is Euler's number, and is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states , and serve as a suitable basis for the wave functions or states of electrons ...
For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital . Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the ...
Below, a number of drum membrane vibration modes and the respective wave functions of the hydrogen atom are shown. A correspondence can be considered where the wave functions of a vibrating drum head are for a two-coordinate system ψ(r, θ) and the wave functions for a vibrating sphere are three-coordinate ψ(r, θ, φ).
Waves of the same type are often superposed and encountered simultaneously at a given point in space and time. The properties at that point are the sum of the properties of each component wave at that point. In general, the velocities are not the same, so the wave form will change over time and space.
Davisson and Germer's accidental discovery of the diffraction of electrons was the first direct evidence confirming de Broglie's hypothesis that particles can have wave properties as well. Davisson's attention to detail, his resources for conducting basic research, the expertise of colleagues, and luck all contributed to the experimental success.