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 = 3 and l = − 1 are invalid because 0 ≤ l ≤ n − 1 .
m = 3 is invalid because − l ≤ m ≤ l must hold. Valid options are l = 0 , m = − 2 , and m = 2 .
The valid quantum numbers are l = 0 , m = − 2 , and 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 for l The azimuthal quantum number l can take values from 0 to n − 1 . Since n = 3 , the possible values for l are 0 , 1 , 2 .
Possible Values for m The magnetic quantum number m can take values from − l to + l , i.e., − l , − l + 1 , ... , 0 , ... , l − 1 , l .
Checking the Given Values Now, let's check each of the given values:
l = 3 : This is not a valid value for l because l can only be 0 , 1 , 2 when n = 3 .
m = 3 : To determine if this is valid, we need to consider the possible values of l . The maximum value of l is 2, so the maximum value of m is 2. Therefore, m = 3 is not valid.
l = 0 : This is a valid value for l . When l = 0 , m can only be 0 . So, l = 0 is valid.
m = − 2 : To determine if this is valid, we need to consider the possible values of l . If l = 0 , m = 0 . If l = 1 , m = − 1 , 0 , 1 . If l = 2 , m = − 2 , − 1 , 0 , 1 , 2 . Since m = − 2 is possible when l = 2 , m = − 2 is valid.
l = − 1 : This is not a valid value for l because l must be a non-negative integer.
m = 2 : To determine if this is valid, we need to consider the possible values of l . If l = 0 , m = 0 . If l = 1 , m = − 1 , 0 , 1 . If l = 2 , m = − 2 , − 1 , 0 , 1 , 2 . Since m = 2 is possible when l = 2 , m = 2 is valid.
Final Answer Therefore, the valid quantum numbers are l = 0 , m = − 2 , and m = 2 .
Examples
Understanding quantum numbers is crucial in fields like material science and quantum computing. For instance, designing new semiconductor materials requires precise control over the electronic structure, which is governed by these quantum numbers. By manipulating these numbers, scientists can tailor the properties of materials for specific applications, such as creating more efficient solar cells or developing novel quantum devices.
