Which of the following sets of quantum numbers describe valid orbitals? Check all that apply.
[tex]$n=1, l=0, m=0$[/tex]
[tex]$n=2, l=1, m=3$[/tex]
[tex]$n=2, I=2, m=2$[/tex]
[tex]$n=3, l=0, m=0$[/tex]
[tex]$n=5, l=4, m=-3$[/tex]
[tex]$n=4, l=-2, m=2$[/tex]
Answer (2)
The valid sets of quantum numbers are n = 1 , l = 0 , m = 0 , n = 3 , l = 0 , m = 0 , and n = 5 , l = 4 , m = − 3 . The other sets provide invalid combinations based on the quantum number rules. Thus, the answer is that the valid sets are identified accordingly.
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Check if each set of quantum numbers satisfies the conditions: n is a positive integer, 0 ≤ l ≤ n − 1 , and − l ≤ m ≤ l .
The set n = 1 , l = 0 , m = 0 is valid.
The set n = 3 , l = 0 , m = 0 is valid.
The set n = 5 , l = 4 , m = − 3 is valid.
The other sets are invalid because they violate the conditions for l or m .
The valid sets are: n = 1 , l = 0 , m = 0 ; n = 3 , l = 0 , m = 0 ; n = 5 , l = 4 , m = − 3
Explanation
Understanding the Rules for Quantum Numbers We need to determine which of the given sets of quantum numbers are valid. To do this, we must check if each set satisfies the following rules:
The principal quantum number, n , must be a positive integer: n = 1 , 2 , 3 , ...
The angular momentum quantum number, l , must be a non-negative integer less than n : 0 ≤ l ≤ n − 1
The magnetic quantum number, m , must be an integer between − l and + l : − l ≤ m ≤ l
Checking Each Set of Quantum Numbers Let's analyze each set of quantum numbers:
Set 1: n = 1 , l = 0 , m = 0
n = 1 is a positive integer, so it's valid.
l = 0 satisfies 0 ≤ l ≤ n − 1 because 0 ≤ 0 ≤ 0 is true.
m = 0 satisfies − l ≤ m ≤ l because − 0 ≤ 0 ≤ 0 is true.
Therefore, this set is valid .
Set 2: n = 2 , l = 1 , m = 3
n = 2 is a positive integer, so it's valid.
l = 1 satisfies 0 ≤ l ≤ n − 1 because 0 ≤ 1 ≤ 1 is true.
m = 3 does not satisfy − l ≤ m ≤ l because − 1 ≤ 3 ≤ 1 is false.
Therefore, this set is invalid .
Set 3: n = 2 , l = 2 , m = 2
n = 2 is a positive integer, so it's valid.
l = 2 does not satisfy 0 ≤ l ≤ n − 1 because 0 ≤ 2 ≤ 1 is false.
Therefore, this set is invalid .
Set 4: n = 3 , l = 0 , m = 0
n = 3 is a positive integer, so it's valid.
l = 0 satisfies 0 ≤ l ≤ n − 1 because 0 ≤ 0 ≤ 2 is true.
m = 0 satisfies − l ≤ m ≤ l because − 0 ≤ 0 ≤ 0 is true.
Therefore, this set is valid .
Set 5: n = 5 , l = 4 , m = − 3
n = 5 is a positive integer, so it's valid.
l = 4 satisfies 0 ≤ l ≤ n − 1 because 0 ≤ 4 ≤ 4 is true.
m = − 3 satisfies − l ≤ m ≤ l because − 4 ≤ − 3 ≤ 4 is true.
Therefore, this set is valid .
Set 6: n = 4 , l = − 2 , m = 2
n = 4 is a positive integer, so it's valid.
l = − 2 does not satisfy 0 ≤ l ≤ n − 1 because 0 ≤ − 2 ≤ 3 is false.
Therefore, this set is invalid .
Identifying Valid Sets Based on our analysis, the valid sets of quantum numbers are:
n = 1 , l = 0 , m = 0
n = 3 , l = 0 , m = 0
n = 5 , l = 4 , m = − 3
Examples
Understanding quantum numbers is crucial in fields like material science and quantum computing. For instance, designing new materials with specific electronic properties requires precise control over the quantum states of electrons. Similarly, in quantum computing, manipulating qubits (quantum bits) relies on accurately defining and controlling their quantum states using quantum numbers.
