Answer :
To determine which set of quantum numbers is valid, we need to follow the rules for quantum numbers in quantum mechanics:
1. The principal quantum number, [tex]\( n \)[/tex], must be a positive integer ([tex]\( n > 0 \)[/tex]).
2. The azimuthal quantum number, [tex]\( l \)[/tex], must be an integer that ranges from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex] ([tex]\( 0 \leq l \leq n-1 \)[/tex]).
3. The magnetic quantum number, [tex]\( m \)[/tex], must be an integer that ranges from [tex]\(-l \)[/tex] to [tex]\( l \)[/tex] ([tex]\( -l \leq m \leq l \)[/tex]).
Let's examine each set:
1. [tex]\( n=4, l=4, m=4 \)[/tex]
- The principal quantum number [tex]\( n = 4 \)[/tex] is valid since it is a positive integer.
- The azimuthal quantum number [tex]\( l = 4 \)[/tex] is not valid because it must be in the range from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex], i.e., [tex]\( 0 \leq l \leq 3 \)[/tex] for [tex]\( n = 4 \)[/tex]. Since [tex]\( l = 4 \)[/tex] does not satisfy [tex]\( l \leq 3 \)[/tex], this set is invalid.
2. [tex]\( n=1, l=-2, m=0 \)[/tex]
- The principal quantum number [tex]\( n = 1 \)[/tex] is valid since it is a positive integer.
- The azimuthal quantum number [tex]\( l = -2 \)[/tex] is not valid because it must be in the range from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex], i.e., [tex]\( 0 \leq l \leq 0 \)[/tex] for [tex]\( n=1 \)[/tex]. Since [tex]\( l = -2 \)[/tex] does not satisfy [tex]\( l \geq 0 \)[/tex], this set is invalid.
3. [tex]\( n=-1, l=0, m=0 \)[/tex]
- The principal quantum number [tex]\( n = -1 \)[/tex] is not valid since it must be a positive integer. Therefore, this set is invalid regardless of the values of [tex]\( l \)[/tex] and [tex]\( m \)[/tex].
4. [tex]\( n=4, l=3, m=3 \)[/tex]
- The principal quantum number [tex]\( n = 4 \)[/tex] is valid since it is a positive integer.
- The azimuthal quantum number [tex]\( l = 3 \)[/tex] is valid because it is within the range [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex] (i.e., [tex]\( 0 \leq l \leq 3 \)[/tex]).
- The magnetic quantum number [tex]\( m = 3 \)[/tex] is valid because it is within the range [tex]\(-l \)[/tex] to [tex]\( l \)[/tex] (i.e., [tex]\(-3 \leq m \leq 3\)[/tex]).
Therefore, the set of valid quantum numbers is:
[tex]\[ n=4, l=3, m=3 \][/tex]
Thus, option [tex]\( n=4, l=3, m=3 \)[/tex] is the correct and valid set of quantum numbers.
1. The principal quantum number, [tex]\( n \)[/tex], must be a positive integer ([tex]\( n > 0 \)[/tex]).
2. The azimuthal quantum number, [tex]\( l \)[/tex], must be an integer that ranges from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex] ([tex]\( 0 \leq l \leq n-1 \)[/tex]).
3. The magnetic quantum number, [tex]\( m \)[/tex], must be an integer that ranges from [tex]\(-l \)[/tex] to [tex]\( l \)[/tex] ([tex]\( -l \leq m \leq l \)[/tex]).
Let's examine each set:
1. [tex]\( n=4, l=4, m=4 \)[/tex]
- The principal quantum number [tex]\( n = 4 \)[/tex] is valid since it is a positive integer.
- The azimuthal quantum number [tex]\( l = 4 \)[/tex] is not valid because it must be in the range from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex], i.e., [tex]\( 0 \leq l \leq 3 \)[/tex] for [tex]\( n = 4 \)[/tex]. Since [tex]\( l = 4 \)[/tex] does not satisfy [tex]\( l \leq 3 \)[/tex], this set is invalid.
2. [tex]\( n=1, l=-2, m=0 \)[/tex]
- The principal quantum number [tex]\( n = 1 \)[/tex] is valid since it is a positive integer.
- The azimuthal quantum number [tex]\( l = -2 \)[/tex] is not valid because it must be in the range from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex], i.e., [tex]\( 0 \leq l \leq 0 \)[/tex] for [tex]\( n=1 \)[/tex]. Since [tex]\( l = -2 \)[/tex] does not satisfy [tex]\( l \geq 0 \)[/tex], this set is invalid.
3. [tex]\( n=-1, l=0, m=0 \)[/tex]
- The principal quantum number [tex]\( n = -1 \)[/tex] is not valid since it must be a positive integer. Therefore, this set is invalid regardless of the values of [tex]\( l \)[/tex] and [tex]\( m \)[/tex].
4. [tex]\( n=4, l=3, m=3 \)[/tex]
- The principal quantum number [tex]\( n = 4 \)[/tex] is valid since it is a positive integer.
- The azimuthal quantum number [tex]\( l = 3 \)[/tex] is valid because it is within the range [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex] (i.e., [tex]\( 0 \leq l \leq 3 \)[/tex]).
- The magnetic quantum number [tex]\( m = 3 \)[/tex] is valid because it is within the range [tex]\(-l \)[/tex] to [tex]\( l \)[/tex] (i.e., [tex]\(-3 \leq m \leq 3\)[/tex]).
Therefore, the set of valid quantum numbers is:
[tex]\[ n=4, l=3, m=3 \][/tex]
Thus, option [tex]\( n=4, l=3, m=3 \)[/tex] is the correct and valid set of quantum numbers.
