Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
