Glossary Collection

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A

Angular momentum

In quantum mechanics, the orbital angular momentum of an electron in an atom is quantized and characterized by the quantum number \( l \), with magnitude \( \sqrt{l(l+1)} \hbar \)

Associated Laguerre polynomials

A set of polynomials denoted by \( L_{k}^{m}(x) \), which are solutions to the associated Laguerre differential equation and are used in the radial wave functions of the hydrogen atom for states with angular momentum quantum number \( l \).

Asymptotic solution for ρ equation

Refers to the behavior of the radial wave function in terms of a scaled radial coordinate \( \rho \) (often \( \rho = \frac{2Z r}{n a_0} \)) at large or small values, typically showing exponential decay for bound states as \( \rho \to \infty \).

Atomic units

A system of units where the electron mass \( m_e \), elementary charge \( e \), reduced Planck’s constant \( \hbar \), and Coulomb’s constant \( k_e = \frac{1}{4\pi \epsilon_0} \) are set to 1, simplifying equations in atomic physics.

C

Centrifugal barrier

The term \( \frac{\hbar^2 l(l+1)}{2m r^2} \) in the effective potential of the radial Schrödinger equation, which acts as a repulsive potential for \( l > 0 \), affecting the electron’s radial distribution.

E

Effective potential

In the radial Schrödinger equation, it is the sum of the actual potential (e.g., Coulomb potential) and the centrifugal barrier term, determining the radial motion of the electron.

Energy levels

The discrete values of energy that an electron can have in an atom, determined by the principal quantum number \( n \) in the hydrogen atom, with \( E_n = -\frac{13.6}{n^2} \) eV.

I

Interdependency of l and m

The magnetic quantum number \( m \) can take integer values from \( -l \) to \( +l \), inclusive, where \( l \) is the azimuthal quantum number, indicating that for each \( l \), there are \( 2l + 1 \) possible values of \( m \).

Interdependency of l and n

In the hydrogen atom, the azimuthal quantum number \( l \) can range from 0 to \( n-1 \), where \( n \) is the principal quantum number, meaning that for a given energy level \( n \), there are \( n \) possible values of \( l \).

L

Laguerre polynomials

A set of orthogonal polynomials \( L_k(x) \) that are solutions to Laguerre’s differential equation and appear in the radial wave functions of the hydrogen atom for s-states (\( l = 0 \)).

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