The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a Fourier transform. Wave functions can be functions of variables other than position, such as momentum. One has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. For example, a wave function might assign a complex number to each point in a region of space. Wave functions are composed of complex numbers. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). In quantum physics, a wave function (or wavefunction), is a mathematical description of the quantum state of an isolated quantum system. The wave function of an initially very localized free particle. Panels (G–H) further show two different wave functions that are solutions of the Schrödinger equation but not standing waves. Panels (C–F) show four different standing-wave solutions of the Schrödinger equation. Rather, it is represented as a wave here, the vertical axis shows the real part (blue) and imaginary part (red) of the wave function. The quantum process (C–H) has no such trajectory. The classical process (A–B) is represented as the motion of a particle along a trajectory. Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle.
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