Quantum Numbers: Meaning & Significance

[rgshankar76]

What is the physical significance of the various quantum numbers like (l,m,s,j, etc)that are used to describe any system.

 

[RedX]

Well l is usually orbital angular momentum, m is orbital angular momentum along an axis, s is spin, and j is total angular momentum.

Actually l is a number which represents orbital angular momentum. The real value would be sqrt [l(l+1)h^2] I think, where h is Planck's constant divided by 2pi. Upon further consideration, yeah, you have m going from -l to l. Similarly for spin and total angular momentum.

Anyways, physically these quantum numbers will tell you what you will observe if you make a measurement of something you can observe. If a value of some observable is not one of these quantum numbers, then you won't observe it. For example if you take a particle where s=1/2, then if you measure the spin, then you'll get sqrt(1/2(1/2+1)h^2) because that's the only allowed value when you specify s=1/2.

Also all these quantum numbers may not have definite values simultaneously. You'll have to consider whether all the operators commute or not and can be simultaneously diagnoalized by a unitary change of basis. Some things always commute like the total angular momentum operator and one of the total angular momentum operators about an axis. Others don't like the total angular momentum operators about an axis with each other.

 

[R0man]

Quantum numbers play a crucial role in describing the physical properties of any system at the microscopic level. These numbers are used to characterize the energy levels, orbital shapes, and orientations of electrons in an atom or molecule.

The principal quantum number (n) determines the energy level of an electron, with higher values representing higher energy levels. This number also indirectly indicates the distance of the electron from the nucleus.

The azimuthal quantum number (l) indicates the shape of the electron's orbital, with different values representing different shapes such as s, p, d, and f orbitals.

The magnetic quantum number (m) specifies the orientation of the orbital in space, with different values representing different spatial orientations.

The spin quantum number (s) indicates the spin of the electron, with values of +1/2 or -1/2 representing the two possible spin states.

The total angular momentum quantum number (j) combines the values of l and s to represent the total angular momentum of the electron.

These quantum numbers have a direct physical significance as they determine the energy states and spatial distribution of electrons, which in turn affect the chemical and physical properties of atoms and molecules. For example, the different shapes of orbitals determine the electron density and bonding behavior of atoms in molecules.

Moreover, the quantum numbers also help in predicting and understanding the behavior of particles at the quantum level, where classical physics fails to provide an accurate description.

In summary, the various quantum numbers used to describe a system have significant physical significance as they provide a detailed understanding of the microscopic properties and behavior of particles, which ultimately affects the macroscopic properties of matter.