Answer :
To determine which sets of quantum numbers describe valid orbitals, we need to understand the rules and constraints governing quantum numbers in quantum mechanics:
1. Principal quantum number (n):
- Must be a positive integer. [tex]\( n = 1, 2, 3, \dots \)[/tex]
2. Azimuthal quantum number (l):
- Must be a non-negative integer, ranging from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex] for a given value of [tex]\( n \)[/tex].
- [tex]\( l = 0, 1, 2, \dots, (n-1) \)[/tex]
3. Magnetic quantum number (m):
- Must be an integer, ranging from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex] for a given value of [tex]\( l \)[/tex].
- [tex]\( m = -l, -(l-1), \dots, 0, \dots, (l-1), +l \)[/tex]
Let's examine each set of quantum numbers:
1. [tex]\( n=1, l=0, m=0 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 0 \)[/tex]. Hence, [tex]\( l = 0 \)[/tex] is valid.
- [tex]\( m \)[/tex] ranges from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], i.e., [tex]\( m = -0 \)[/tex] to [tex]\( 0 \)[/tex]. Hence, [tex]\( m = 0 \)[/tex] is valid.
- This set is valid.
2. [tex]\( n=2, l=1, m=3 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 1 \)[/tex]. Hence, [tex]\( l = 1 \)[/tex] is valid.
- [tex]\( m \)[/tex] ranges from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], i.e., [tex]\( m = -1 \)[/tex] to [tex]\( 1 \)[/tex]. Hence, [tex]\( m = 3 \)[/tex] is not valid.
- This set is not valid.
3. [tex]\( n=2, l=2, m=2 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 1 \)[/tex]. Hence, [tex]\( l = 2 \)[/tex] is not valid.
- This set is not valid.
4. [tex]\( n=3, l=0, m=0 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 2 \)[/tex]. Hence, [tex]\( l = 0 \)[/tex] is valid.
- [tex]\( m \)[/tex] ranges from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], i.e., [tex]\( m = -0 \)[/tex] to [tex]\( 0 \)[/tex]. Hence, [tex]\( m = 0 \)[/tex] is valid.
- This set is valid.
5. [tex]\( n=5, l=4, m=-3 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 4 \)[/tex]. Hence, [tex]\( l = 4 \)[/tex] is valid.
- [tex]\( m \)[/tex] ranges from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], i.e., [tex]\( m = -4 \)[/tex] to [tex]\( 4 \)[/tex]. Hence, [tex]\( m = -3 \)[/tex] is valid.
- This set is valid.
6. [tex]\( n=4, l=-2, m=2 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 3 \)[/tex]. Hence, [tex]\( l = -2 \)[/tex] is not valid.
- This set is not valid.
In summary, the valid sets of quantum numbers are:
- [tex]\( (1, 0, 0) \)[/tex]
- [tex]\( (3, 0, 0) \)[/tex]
- [tex]\( (5, 4, -3) \)[/tex]
1. Principal quantum number (n):
- Must be a positive integer. [tex]\( n = 1, 2, 3, \dots \)[/tex]
2. Azimuthal quantum number (l):
- Must be a non-negative integer, ranging from [tex]\( 0 \)[/tex] to [tex]\( n-1 \)[/tex] for a given value of [tex]\( n \)[/tex].
- [tex]\( l = 0, 1, 2, \dots, (n-1) \)[/tex]
3. Magnetic quantum number (m):
- Must be an integer, ranging from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex] for a given value of [tex]\( l \)[/tex].
- [tex]\( m = -l, -(l-1), \dots, 0, \dots, (l-1), +l \)[/tex]
Let's examine each set of quantum numbers:
1. [tex]\( n=1, l=0, m=0 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 0 \)[/tex]. Hence, [tex]\( l = 0 \)[/tex] is valid.
- [tex]\( m \)[/tex] ranges from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], i.e., [tex]\( m = -0 \)[/tex] to [tex]\( 0 \)[/tex]. Hence, [tex]\( m = 0 \)[/tex] is valid.
- This set is valid.
2. [tex]\( n=2, l=1, m=3 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 1 \)[/tex]. Hence, [tex]\( l = 1 \)[/tex] is valid.
- [tex]\( m \)[/tex] ranges from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], i.e., [tex]\( m = -1 \)[/tex] to [tex]\( 1 \)[/tex]. Hence, [tex]\( m = 3 \)[/tex] is not valid.
- This set is not valid.
3. [tex]\( n=2, l=2, m=2 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 1 \)[/tex]. Hence, [tex]\( l = 2 \)[/tex] is not valid.
- This set is not valid.
4. [tex]\( n=3, l=0, m=0 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 2 \)[/tex]. Hence, [tex]\( l = 0 \)[/tex] is valid.
- [tex]\( m \)[/tex] ranges from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], i.e., [tex]\( m = -0 \)[/tex] to [tex]\( 0 \)[/tex]. Hence, [tex]\( m = 0 \)[/tex] is valid.
- This set is valid.
5. [tex]\( n=5, l=4, m=-3 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 4 \)[/tex]. Hence, [tex]\( l = 4 \)[/tex] is valid.
- [tex]\( m \)[/tex] ranges from [tex]\( -l \)[/tex] to [tex]\( +l \)[/tex], i.e., [tex]\( m = -4 \)[/tex] to [tex]\( 4 \)[/tex]. Hence, [tex]\( m = -3 \)[/tex] is valid.
- This set is valid.
6. [tex]\( n=4, l=-2, m=2 \)[/tex]:
- [tex]\( n \)[/tex] is a positive integer.
- [tex]\( l \)[/tex] ranges from 0 to [tex]\( n-1 \)[/tex], i.e., [tex]\( l = 0 \)[/tex] to [tex]\( 3 \)[/tex]. Hence, [tex]\( l = -2 \)[/tex] is not valid.
- This set is not valid.
In summary, the valid sets of quantum numbers are:
- [tex]\( (1, 0, 0) \)[/tex]
- [tex]\( (3, 0, 0) \)[/tex]
- [tex]\( (5, 4, -3) \)[/tex]
