page_title: Wave function
section_title: Finite potential barrier
hierarchical_section_title: Wave function / Non-relativistic examples / Finite potential barrier
caption_attribution_description: English: created by Bamse 05:12, 31 July 2006 (UTC)
context_page_description: A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ.
  The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state.
  For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform. Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin.
context_section_description: One of most prominent features of the wave mechanics is a possibility for a particle to reach a location with a prohibitive (in classical mechanics) force potential. A common model is the "potential barrier", the one-dimensional case has the potential
  and the steady-state solutions to the wave equation have the form (for some constants k, κ)
  Note that these wave functions are not normalized; see scattering theory for discussion.
  The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative x): setting Aᵣ = 1 corresponds to firing particles singly; the terms containing Aᵣ and Cᵣ signify motion to the right, while Aₗ and Cₗ – to the left. Under this beam interpretation, put Cₗ = 0 since no particles are coming from the right. By applying the continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above.
  In a semiconductor crystallite whose radius is smaller than the size of its exciton Bohr radius, the excitons are squeezed, leading to quantum confinement. The energy levels can then be modeled using the particle in a box model in which the energy of different states is dependent on the length of the box.
caption: Scattering at a finite potential barrier of height V0. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. E > V0 for this illustration.
exif: {}