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 the following reasons: a. Set A is not allowed because ℓ=3, which is outside the allowed range (0 to n-1) for n=3. d. Set D is not allowed because m_s=-1, which is not a valid value (must be +1/2 or -1/2). e. Set E is not allowed because ℓ=-4, which is negative and not a valid value (must be within the range 0 to n-1). f. Set F is not allowed because m_ℓ=2 is outside the allowed range of -ℓ and +ℓ for a given value of ℓ=1.

Step-by-step solution

Set A: n=3, ℓ=3, m_ℓ=0, m_s=-1/2

Since the angular momentum quantum number (ℓ) must be in the range of 0 to (n-1), and here n = 3, the allowed values for ℓ would be 0, 1, and 2. As ℓ = 3, this set is not allowed.

Set B: n=4, ℓ=3, m_ℓ=2, m_s=-1/2

This set is allowed, as all the quantum numbers follow the restrictions: n = 4, ℓ ranges from 0 to 3, m_ℓ ranges from -3 to 3, and m_s can be either ±1/2.

Set C: n=4, ℓ=1, m_ℓ=1, m_s=+1/2

This set is allowed, as all the quantum numbers follow the restrictions: n = 4, ℓ ranges from 0 to 3, m_ℓ ranges from -1 to 1, and m_s can be either ±1/2.

Set D: n=2, ℓ=1, m_ℓ=-1, m_s=-1

In this set, the spin quantum number (m_s) is not allowed, as it must be either +1/2 or -1/2. Here it is given as -1, which is not a valid value.

Set E: n=5, ℓ=-4, m_ℓ=2, m_s=+1/2

In this set, the angular momentum quantum number (ℓ) is not allowed, as it must be in the range of 0 to (n-1) and cannot be negative. Here it is given as -4, which is not a valid value.

Set F: n=3, ℓ=1, m_ℓ=2, m_s=-1/2

In this set, the magnetic quantum number (m_ℓ) is not allowed, as it must be in the range of -ℓ to +ℓ for a given value of ℓ. Here, m_ℓ = 2 is not within the range of values for ℓ = 1 (which should be -1 to 1).