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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.
Consequently, the wave function also became a four-component function, governed by the Dirac equation that, in free space, read (+ (= )) =. This has again the form of the Schrödinger equation, with the time derivative of the wave function being given by a Hamiltonian operator acting upon the wave function.
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 ...
A so-called eigenmode is a solution that oscillates in time with a well-defined constant angular frequency ω, so that the temporal part of the wave function takes the form e −iωt = cos(ωt) − i sin(ωt), and the amplitude is a function f(x) of the spatial variable x, giving a separation of variables for the wave function: (,) = ().
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 ...
Wave functions are assumed to be elements of the Hilbert space L 2 of square-integrable functions, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set {|φ n } of wave functions is orthonormal if
The term "wave function" is typically used for a different mathematical representation of the quantum state, one that uses spatial coordinates also called the "position representation". [9]: 324 When the wave function representation is used, the "reduction" is called "wave function collapse".
The concept of universal wavefunction was introduced by Hugh Everett in his 1956 PhD thesis draft The Theory of the Universal Wave Function. [8] It later received investigation from James Hartle and Stephen Hawking [ 9 ] who derived the Hartle–Hawking solution to the Wheeler–DeWitt equation to explain the initial conditions of the Big Bang ...