For the [tex]$n=3$[/tex] electron shell, which of the following quantum numbers are valid? Check all that apply.
[tex]$I=3$[/tex]
[tex]$m=3$[/tex]
[tex]$I=0$[/tex]
[tex]$m=-2$[/tex]
[tex]$I=-1$[/tex]
[tex]$m=2$[/tex]
Answer (1)
The possible values for the azimuthal quantum number l are 0 , 1 , 2 .
The possible values for the magnetic quantum number m range from − l to + l .
l = 0 is valid.
m = − 2 is valid.
m = 2 is valid.
The valid quantum numbers are l = 0 , m = − 2 , and m = 2 .
The final answer is: l = 0 , m = − 2 , m = 2
Explanation
Problem Analysis We are given the principal quantum number n = 3 and asked to determine which of the given values for the azimuthal quantum number l and the magnetic quantum number m are valid.
Possible Values of l For a given principal quantum number n , the possible values of the azimuthal quantum number l are integers ranging from 0 to n − 1 . In this case, since n = 3 , the possible values for l are 0 , 1 , 2 .
Possible Values of m For a given azimuthal quantum number l , the possible values of the magnetic quantum number m are integers ranging from − l to + l .
Checking the Given Values Now, let's check the given values:
l = 3 : This is not a valid value for l because l must be less than n , and 3 ≮ 3 .
m = 3 : This is not a valid value for m . The maximum value for m is l , and the maximum value for l when n = 3 is 2 . Therefore, m cannot be 3 .
l = 0 : This is a valid value for l because 0 is between 0 and n − 1 = 2 .
m = − 2 : This is a valid value for m . If l = 2 , then m can be − 2 , − 1 , 0 , 1 , 2 .
l = − 1 : This is not a valid value for l because l must be a non-negative integer.
m = 2 : This is a valid value for m . If l = 2 , then m can be − 2 , − 1 , 0 , 1 , 2 .
Examples
Understanding quantum numbers is crucial in fields like material science and nanotechnology. For example, when designing new semiconductors or quantum computing devices, engineers need to precisely control the energy levels and electron configurations of atoms. The quantum numbers n , l , and m dictate these properties, allowing for the creation of materials with specific electrical and optical characteristics. By manipulating these numbers, scientists can tailor materials for advanced technological applications.
