page_title: Probability amplitude
hierarchical_section_title: Probability amplitude
caption_attribution_description: English: Calculated 5d1 orbital of an electron's eigenstate in the Coulomb-field of a hydrogen nucleus. An eigenstate is a state which keeps it's shape except for a complex phase when the Hamilton operator is applied, thus being invariant in time while obeying the Schrödinger equation. The wavefunction is: &#x03C8; 5 , 2 , 1 ( r , &#x03D1; , &#x03C6; ) = ( 2 5 a 0 ) 3 5 5 &#x22C5; 7 ! &#x22C5; e &#x2212; r / ( 5 a 0 ) &#x22C5; ( 2 r 5 a 0 ) 2 &#x22C5; L 2 5 ( 2 r / ( 5 a 0 ) ) &#x22C5; Y 2 1 ( &#x03D1; , &#x03C6; ) {\displaystyle \psi _{5,2,1}(r,\vartheta ,\varphi )={\sqrt {{\left({\frac {2}{5\,a_{0}}}\right)}^{3}{\frac {5}{5\cdot 7!}}}}\cdot e^{\textstyle -r/(5\,a_{0})}\cdot \left({\frac {2\,r}{5\,a_{0}}}\right)^{2}\cdot L_{2}^{5}(2\,r/(5\,a_{0}))\cdot Y_{2}^{1}(\vartheta ,\varphi )} The state is an eigenstate of H, L² and Lz, which constitute a complete set of commuting observables. The quantum numbers mean that the following quantities have a sharp certain value: n = 5: Energy: E = &#x2212; 1 R y / n 2 = &#x2212; 13.6 e V / 25 {\displaystyle E=-1\,\mathrm {Ry} /n^{2}=-13.6\,\mathrm {eV} /25} l = 2: Angular momentum: | L | = l ( l + 1 ) &#x210F; = 6 &#x210F; {\displaystyle |L|={\sqrt {l\,(l+1)}}\,\hbar ={\sqrt {6}}\,\hbar } m = 1: Angular momentum in z-direction: L z = m &#x210F; = &#x210F; {\displaystyle L_{z}=m\,\hbar =\hbar } Since l=2, this is called a d-orbital. The orbital is aligned around the z-axis, but remains an eigenfunction of H and L² if rotated to any direction. The depicted rigid body is where the probability density exceeds a certain value. The color shows the complex phase of the wavefunction, where blue means real positive, red means imaginary positive, yellow means real negative and green means imaginary negative. The image is raytraced using modified Phong lighting. The fine structure is neglected.
context_page_description: In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.
  Probability amplitudes provide a relationship between the wave function of a system and the results of observations of that system, a link first proposed by Max Born. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein.
context_section_description: In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.
  Probability amplitudes provide a relationship between the wave function (or, more generally, of a quantum state vector) of a system and the results of observations of that system, a link first proposed by Max Born. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding (see References), and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein. It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.
caption: A wave function for a single electron on 5d atomic orbital of a hydrogen atom. The solid body shows the places where the electron's probability density is above a certain value (here 0.02 nm−3): this is calculated from the probability amplitude. The hue on the colored surface shows the complex phase of the wave function.
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