Question

How are the quantum numbers $n$, $1$ and $m_1$ arrived at? Explain the significance of these quantum numbers.?

Answer 1

Quantum numbers are a way to describe the discrete states that a particle can hold. These can then be used for solutions to the Schrodinger equation(above).

Principal Quantum Number (n)—this designates the size of the orbital, also called the shell. The larger the value for $n$, the greater the average distance of an electron in the orbital from the nucleus and therefore the larger the orbital. (I could always remember this trend by relating it back to Coulomb’s Law of Attractive Forces.) From the equation that defines the line spectrum of Hydrogen, we can also know that the value for n represents energy levels.

Angular Momentum Quantum Number (l)—this describes the shape of the orbital, but again, you can relate it back to the periodic table. The values of $l$ are integers that depend on the value of the principal quantum number, $n$. For any given value of $n$, the possible range of values for $l$ go from $0$ to $n-1$. If $n=1$, there’s only one possible value of $l$; that is, $0$ ($n-1$ where $n=1$). If $n=2$, there’s two values for $l$ : $0$ and $1$. If $n=3$, $l$ has three values: $0$, $1$, and $2$.

Magnetic Quantum Number (ml)—this describes the orientation of the orbital in space, or the direction along the axes in which it faces. Within a sub-shell, the value of this depends on the value of $l$. For a certain value of $l$, there are $(2l+1)$ integral values, or $-l,\dots 0,\dots +l$