Question

The orbital angular momentum of an electron corresponding to n=4 and m= -3 is:
A. 0
B. $\dfrac{h}{{\sqrt 2 \pi }}$
C. $\dfrac{{\sqrt 6 h}}{{2\pi }}$
D. $\dfrac{{\sqrt 3 h}}{\pi }$

Answer 1

Hint: Hint: Angular momentum is a physical quantity defined for rotating motion. The angular momentum of the electron is determined by the angular quantum number by using the formula:
$mvr = \dfrac{h}{{2\pi }}\sqrt {l(l + 1)} $
Where m is mass, V is velocity, r is the radius, h is planck's constant having value =$6.626 \times {10^{ - 34}}{\text{jH}}{{\text{z}}^{{\text{ - 1}}}}$
And l is the angular quantum number.

Complete step by step answer:
Given, n is the principal quantum number and m is the magnetic quantum number. Since the value of the azimuthal quantum number depends upon the principal quantum number. For a given value of the principal quantum number (n), azimuthal quantum number (l) can have the value from zero to (n-1).
Here, the value of the principal quantum number (n) is given = 4
The value of the azimuthal quantum number (l) will be 0 to (4-1) which is 0,1,2,3.
The value of m range from –l to +l and the maximum value of m is 2m+1
Here, m = -3 hence the value of l corresponding to 3. Hence the angular momentum will be:
$
  L = \dfrac{h}{{2\pi }}\sqrt {3(3 + 1)} \\
   \Rightarrow L = \dfrac{h}{{2\pi }}\sqrt {3 \times 4} \\
   \Rightarrow L = \dfrac{h}{{2\pi }}\sqrt {12} \\
   \Rightarrow L = \dfrac{{\sqrt 3 h}}{\pi } \\
$

Hence the correct answer is option D .

Note:
Quantum number is the index number which is used to specify the position and energy of an electron in an atom. There are four types of quantum numbers.
1.Principal quantum number (n): It tells about the number of major energy levels. It has any positive whole number value except zero.
2.Angular quantum number (l): It is also known as the azimuthal quantum number. It tells about the number of subshells in a shell. it can be calculated as l=0 to (n-1)
3.Magnetic quantum number (m): It tells about the number of orbitals in a subshell. It can be calculated as m= -l to +l.
4.Spin quantum number(s): It tells about the spin of electrons. If the spin of the electron is clockwise its value is $\dfrac{1}{2}$ and in case of anticlockwise direction, it has a value $\dfrac{-1}{2}$.