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What are the possible values of the principal quantum number [tex]n[/tex]?

A. all the non-negative integers: [tex]0,1,2,3[/tex], etc.
B. all the positive integers: [tex]1,2,3[/tex], etc.
C. represented by the formula: [tex]n=2 k+1[/tex], where [tex]k[/tex] is a positive integer
D. all the integers: [tex]-3,-2,-1,0,1,2,3[/tex], etc.
E. represented by the formula: [tex]k![/tex], where [tex]k[/tex] is a positive integer

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GinnyAnsweridn
The principal quantum number [tex]\( n \)[/tex] is a fundamental quantum number that primarily determines the energy level of an electron in an atom. It can only take specific values, and these values are crucial in quantum mechanics and atomic theory. To determine the possible values of the principal quantum number [tex]\( n \)[/tex], let's analyze the given options:

1. All the non-negative integers: [tex]\( 0, 1, 2, 3, \ldots \)[/tex]

If [tex]\( n \)[/tex] included zero, we would consider the series to begin with 0, which is not usually acceptable for the principal quantum number in the context of atomic orbitals. Generally, the principal quantum number starts from 1.

2. All the positive integers: [tex]\( 1, 2, 3, \ldots \)[/tex]

This is a sequence that starts from 1 and continues indefinitely with all the values being positive integers. This fits with our understanding that the principal quantum number [tex]\( n \)[/tex] signifies distinct energy levels which are counted starting from 1.

3. Represented by the formula: [tex]\( n = 2k+1 \)[/tex], where [tex]\( k \)[/tex] is a positive integer

This formula represents a specific set of integers where each term is an odd number starting from 3 (when [tex]\( k = 1 \)[/tex]). For example, when [tex]\( k = 1 \)[/tex], [tex]\( n = 3 \)[/tex]; when [tex]\( k = 2 \)[/tex], [tex]\( n = 5 \)[/tex]. This does not include [tex]\( 1 \)[/tex] and is not consistent with the definition of principal quantum numbers.

4. All the integers: [tex]\( -3, -2, -1, 0, 1, 2, 3, \ldots \)[/tex]

Including negative integers and zero is not appropriate for the principal quantum number because it represents possible energy levels that are positive and sequential, starting from 1.

5. Represented by the formula: [tex]\( k! \)[/tex], where [tex]\( k \)[/tex] is a positive integer

Factorials grow very rapidly and start from [tex]\( 1! = 1 \)[/tex], [tex]\( 2! = 2 \)[/tex], [tex]\( 3! = 6 \)[/tex]. This set of numbers is not sequential and regular like the principal quantum numbers.

Given all this analysis, the correct understanding and acceptable set of values are:

- All the positive integers: [tex]\( 1, 2, 3, \ldots \)[/tex]

Therefore, the principal quantum number [tex]\( n \)[/tex] can take all the positive integers.
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