reduced Planck constant

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1: 18.39 Physical Applications

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18.39.1 $$\left(\frac{-{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}+V(x)\right)\psi (x,t)=\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial t}\psi (x,t),$$

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Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $m$: nonnegative integer, $x$: real variable, $V(x)$: potential energy and $\psi (x,t)$: wavefunction

Permalink:

http://dlmf.nist.gov/18.39.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(i), §18.39 and Ch.18

►where $\mathrm{\hslash }$ is the reduced Planck’s constant. …

2: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. The reduced mass is $m=m_1{m}_2/(m_1+m_2)$, and at energy of relative motion $E$ with relative orbital angular momentum $\mathrm{\ell }\mathrm{\hslash }$, the Schrödinger equation for the radial wave function $w(s)$ is given by … ►For $Z_1{Z}_2=-1$ and $m=m_{\mathrm{e}}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_0=\mathrm{\hslash }/(m_{\mathrm{e}}c\alpha )$, and to a multiple of the Rydberg constant, …

3: 14.30 Spherical and Spheroidal Harmonics

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14.30.11 $$L^2{Y}_{l,m}={\mathrm{\hslash }}^{2}l(l+1)Y_{l,m},$$ $l=0,1,2,\mathrm{\dots }$,

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Symbols:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer, $l$: nonnegative integer and $L$: angular momentum operator

Permalink:

http://dlmf.nist.gov/14.30.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

… ►where $\mathrm{\hslash }$ is the reduced Planck’s constant. …