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Chapter 7: Problem 69

Which of the following sets of quantum numbers are not allowed in the hydrogen atom? For the sets of quantum numbers that are incorrect, state what is wrong in each set. a. \(n=3, \ell=2, m_{\ell}=2\) b. \(n=4, \ell=3, m_{\ell}=4\) c. \(n=0, \ell=0, m_{\ell}=0\) d. \(n=2, \ell=-1, m_{\ell}=1\)

Short Answer

Expert verified
The sets of quantum numbers that are not allowed in the hydrogen atom are: b. $n=4, \ell=3, m_{\ell}=4$ - The magnetic quantum number is incorrect, as it should range from -3 to 3. c. $n=0, \ell=0, m_{\ell}=0$ - The principal quantum number is incorrect, as it must be a positive integer. d. $n=2, \ell=-1, m_{\ell}=1$ - The azimuthal quantum number is incorrect, as it must be a non-negative integer.

Step by step solution

01

Case a: \(n=3, \ell=2, m_{\ell}=2\)#

Let's check if these quantum numbers satisfy the rules mentioned above for hydrogen atom: 1. The principal quantum number: \(n = 3\) is a positive integer. 2. The azimuthal quantum number: \(l = 2\) is a non-negative integer, and less than n (3). 3. The magnetic quantum number: \(m_{l} = 2\) is an integer ranging from -2 to 2. All the conditions are satisfied, so the set of quantum numbers is allowed.
02

Case b: \(n=4, \ell=3, m_{\ell}=4\)#

Let's check if these quantum numbers satisfy the rules mentioned above for hydrogen atom: 1. The principal quantum number: \(n = 4\) is a positive integer. 2. The azimuthal quantum number: \(l = 3\) is a non-negative integer, and less than n (4). 3. The magnetic quantum number: \(m_{l} = 4\) is not in the allowed range of -3 to 3. The magnetic quantum number does not satisfy the condition, so this set of quantum numbers is not allowed.
03

Case c: \(n=0, \ell=0, m_{\ell}=0\)#

Let's check if these quantum numbers satisfy the rules mentioned above for hydrogen atom: 1. The principal quantum number: \(n = 0\) is not a positive integer. The principal quantum number does not satisfy the condition, so this set of quantum numbers is not allowed.
04

Case d: \(n=2, \ell=-1, m_{\ell}=1\)#

Let's check if these quantum numbers satisfy the rules mentioned above for hydrogen atom: 1. The principal quantum number: \(n = 2\) is a positive integer. 2. The azimuthal quantum number: \(l = -1\) is not a non-negative integer. The azimuthal quantum number does not satisfy the condition, so this set of quantum numbers is not allowed. In conclusion, only set (a) has allowed quantum numbers for the hydrogen atom. Set (b) has an incorrect magnetic quantum number, set (c) has an incorrect principal quantum number, and set (d) has an incorrect azimuthal quantum number.

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