Question

Which of the following sets of quantum numbers are not allowed? For each incorrect set, state why it is incorrect. a. \(n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2}\) b. \(n=4, \ell=3, m_{\ell}=2, m_{s}=-\frac{1}{2}\) c. \(n=4, \ell=1, m_{\ell}=1, m_{s}=+\frac{1}{2}\) d. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-1\) e. \(n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2}\) f. \(n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2}\)

Short answer

Sets a, d, e, and f are not allowed due to their quantum numbers being outside the allowed range. Set a is incorrect because \(\ell\) is not within the allowed range. Set d is incorrect because \(m_s\) is not within the allowed range. Set e is incorrect because \(\ell\) is not within the allowed range. Set f is incorrect because \(m_\ell\) is not within the allowed range.

Step-by-step solution

Check the set a. \(n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2}\)

First, let's evaluate the given values of the quantum numbers. Since \(n=3\) is a positive integer, it is valid. However, since \(\ell=n\), it is not within the allowed range of \(0 \leq \ell \leq n-1\), so this set is not allowed. The reason is that the value of \(\ell\) is not within the allowed range.

Check the set b. \(n=4, \ell=3, m_{\ell}=2, m_{s}=-\frac{1}{2}\)

The given values of the quantum numbers are all within their respective allowed ranges. \(n=4\) is a positive integer, \(\ell=3\) is within the range 0 to \(n-1=3\), \(m_\ell=2\) is within the range of \(-\ell\) to \(+\ell\), and \(m_s=-\frac{1}{2}\) can either be \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Hence, this set is allowed.

Check the set c. \(n=4, \ell=1, m_{\ell}=1, m_{s}=+\frac{1}{2}\)

The given values of the quantum numbers are all within their respective allowed ranges. Therefore, this set is allowed.

Check the set d. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-1\)

The values of \(n\), \(\ell\), and \(m_\ell\) are within their allowed ranges. However, \(m_s=-1\) does not meet the criteria of either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Hence, this set is not allowed because the value of \(m_s\) is not within the allowed range.

Check the set e. \(n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2}\)

In this set, \(n=5\) is a positive integer, but \(\ell=-4\) is not within the allowed range of 0 to \(n-1=4\). Hence, this set is not allowed because the value of \(\ell\) is not within the allowed range.

Check the set f. \(n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2}\)

In this set, \(n\), \(\ell\), and \(m_s\) are within their respective allowed ranges. However, \(m_\ell =2\) is not within the allowed range, which should be from \(-\ell\) to \(+\ell\) or -1 to +1. Hence, this set is not allowed because the value of \(m_\ell\) is not within the allowed range. In conclusion, sets a, d, e, and f are not allowed due to their quantum numbers being outside the allowed range.