To determine which set of quantum numbers is valid, we need to examine the following 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 angular momentum quantum number [tex]\( l \)[/tex] must be an integer such that [tex]\( 0 \le l < n \)[/tex].
3. The magnetic quantum number [tex]\( m \)[/tex] must be an integer such that [tex]\( -l \le m \le l \)[/tex].
Let's check each set of quantum numbers against these rules:
### Set 1: [tex]\( n=4, l=4, m=4 \)[/tex]
- [tex]\( n = 4 \)[/tex]: Positive integer, so this is valid.
- [tex]\( l = 4 \)[/tex]: Must satisfy [tex]\( 0 \le l < n \)[/tex]. Here, [tex]\( l = 4 \)[/tex] is not less than [tex]\( n = 4 \)[/tex]. Therefore, this is invalid.
### Set 2: [tex]\( n=1, l=-2, m=0 \)[/tex]
- [tex]\( n = 1 \)[/tex]: Positive integer, so this is valid.
- [tex]\( l = -2 \)[/tex]: Must satisfy [tex]\( 0 \le l < n \)[/tex]. Here, [tex]\( l = -2 \)[/tex] is less than 0, which is invalid.
### Set 3: [tex]\( n=-1, l=0, m=0 \)[/tex]
- [tex]\( n = -1 \)[/tex]: Must be a positive integer. Here, [tex]\( n = -1 \)[/tex] is not positive, so this is invalid.
### Set 4: [tex]\( n=4, l=3, m=3 \)[/tex]
- [tex]\( n = 4 \)[/tex]: Positive integer, so this is valid.
- [tex]\( l = 3 \)[/tex]: Must satisfy [tex]\( 0 \le l < n \)[/tex]. Here, [tex]\( l = 3 \)[/tex] is within the range [tex]\( 0 \le l < 4 \)[/tex]. This is valid.
- [tex]\( m = 3 \)[/tex]: Must satisfy [tex]\( -l \le m \le l \)[/tex]. Here, [tex]\( -3 \le 3 \le 3 \)[/tex]. This is valid.
Therefore, the only set of quantum numbers that adheres to all the rules is:
[tex]\[ n=4, l=3, m=3 \][/tex]
This is option 4.
