To determine whether a set of quantum numbers is valid, we need to check if they satisfy certain conditions associated with quantum mechanical principles. Specifically, we must validate that the quantum numbers follow these rules:
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 satisfies [tex]\( 0 \leq l < n \)[/tex].
3. The magnetic quantum number [tex]\( m \)[/tex] must be an integer that falls within the range [tex]\( -l \leq m \leq l \)[/tex].
Let’s verify each set of quantum numbers one by one:
1. For the set [tex]\( n=2, l=1, m=0 \)[/tex]:
- The principal quantum number [tex]\( n \)[/tex] is 2, which is a positive integer ([tex]\( n > 0 \)[/tex]).
- The azimuthal quantum number [tex]\( l \)[/tex] is 1, which satisfies [tex]\( 0 \leq l < n \)[/tex] because [tex]\( 0 \leq 1 < 2 \)[/tex].
- The magnetic quantum number [tex]\( m \)[/tex] is 0, which falls within the range [tex]\( -l \leq m \leq l \)[/tex] because [tex]\( -1 \leq 0 \leq 1 \)[/tex].
This set is valid.
2. For the set [tex]\( n=1, l=0, m=0 \)[/tex]:
- The principal quantum number [tex]\( n \)[/tex] is 1, which is a positive integer ([tex]\( n > 0 \)[/tex]).
- The azimuthal quantum number [tex]\( l \)[/tex] is 0, which satisfies [tex]\( 0 \leq l < n \)[/tex] because [tex]\( 0 \leq 0 < 1 \)[/tex].
- The magnetic quantum number [tex]\( m \)[/tex] is 0, which falls within the range [tex]\( -l \leq m \leq l \)[/tex] because [tex]\( -0 \leq 0 \leq 0 \)[/tex].
This set is valid.
3. For the set [tex]\( n=3, l=3, m=3 \)[/tex]:
- The principal quantum number [tex]\( n \)[/tex] is 3, which is a positive integer ([tex]\( n > 0 \)[/tex]).
- The azimuthal quantum number [tex]\( l \)[/tex] is 3. However, it must satisfy [tex]\( 0 \leq l < n \)[/tex]. For [tex]\( n=3 \)[/tex], [tex]\( l \)[/tex] must be less than 3. Thus, [tex]\( l = 3 \)[/tex] does not satisfy [tex]\( l < n \)[/tex].
- Even if we checked the magnetic quantum number [tex]\( m \)[/tex], it also is 3, and it falls within the range [tex]\( -l \leq m \leq l \)[/tex] because [tex]\( -3 \leq 3 \leq 3 \)[/tex]. But this doesn't matter since the [tex]\( l \)[/tex] value is already invalid.
This set is not valid.
Therefore, the set of quantum numbers [tex]\( n = 3, l = 3, m = 3 \)[/tex] is not a valid set, satisfying the given answer ([tex]\( \text{True, True, False} \)[/tex]).
