Write an relation between the angular momentum quantum number (1)
and magnetic quantum number (m) with suitable example
Answers
The magnetic quantum number (symbol ml) is one of four quantum numbers in atomic physics. The set is: principal quantum number, azimuthal quantum number, magnetic quantum number, and spin quantum number. Together, they describe the unique quantum state of an electron. The magnetic quantum number distinguishes the orbitals available within a subshell, and is used to calculate the azimuthal component of the orientation of orbital in space. Electrons in a particular subshell (such as s, p, d, or f) are defined by values of ℓ (0, 1, 2, or 3). The value of ml can range from -ℓ to +ℓ, inclusive of zero. Thus the s, p, d, and f subshells contain 1, 3, 5, and 7 orbitals each, with values of m within the ranges 0, ±1, ±2, ±3 respectively. Each of these orbitals can accommodate up to two electrons (with opposite spins), forming the basis of the periodic table.
Derivation Edit
These orbitals have magnetic quantum numbers {\displaystyle m=-\ell ,\ldots ,\ell } {\displaystyle m=-\ell ,\ldots ,\ell } from left to right in ascending order. The {\displaystyle e^{mi\phi }} {\displaystyle e^{mi\phi }} dependence of the azimuthal component can be seen as a color gradient repeating m times around the vertical axis.
There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers {\displaystyle n} n, {\displaystyle \ell } \ell , {\displaystyle m_{l}} m_{l}, and {\displaystyle s} s[dubious – discuss] specify the complete and unique quantum state of a single electron in an atom called its wavefunction or orbital. The Schrödinger equation for the wavefunction of an atom with one electron is a separable partial differential equation. (This is not the case for the helium atom or other atoms with mutually interacting electrons, which require more sophisticated methods for solution[1]) This means that the wavefunction as expressed in spherical coordinates can be broken down into the product of three functions of the radius, colatitude (or polar) angle, and azimuth:[2]
{\displaystyle \psi (r,\theta ,\phi )=R(r)P(\theta )F(\phi )} {\displaystyle \psi (r,\theta ,\phi )=R(r)P(\theta )F(\phi )}
The differential equation for {\displaystyle F} F can be solved in the form {\displaystyle F(\phi )=Ae^{\lambda \phi }} {\displaystyle F(\phi )=Ae^{\lambda \phi }}. Because values of the azimuth angle {\displaystyle \phi } \phi differing by 2 {\displaystyle \pi } \pi (360 degrees in radians) represent the same position in space, and the overall magnitude of {\displaystyle F} F does not grow with arbitrarily large {\displaystyle \phi } \phi as it would for a real exponent, the coefficient {\displaystyle \lambda } \lambda must be quantized to integer multiples of {\displaystyle i} i, producing an imaginary exponent: {\displaystyle \lambda =im_{l}} {\displaystyle \lambda =im_{l}}.[3] These integers are the magnetic quantum numbers. The same constant appears in the colatitude equation, where larger values of {\displaystyle m_{l}} m_{l}2 tend to decrease the magnitude of {\displaystyle P(\theta )} P(\theta ), and values of {\displaystyle m_{l}} m_{l} greater than the azimuthal quantum number {\displaystyle \ell } \ell do not permit any solution for {\displaystyle P(\theta )} P(\theta ).
If plthe two waves are of different energies, then there is a net angular momentum, and the electron has a net movement in one direction or the other. That gives rise to a magnetic moment, and the larger the difference in energies, the larger the magnetic moment, and the larger the magnetic moment quantum number .