Question

Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom. The four quantum numbers, the principal quantum number (n), the angular momentum quantum number (ℓ), the magnetic quantum number (mℓ), and the spin quantum number (ms) have strict rules which govern the possible values. Identify allowable combinations of quantum numbers for an electron. Select all that apply. n = 6, ℓ= 6, mℓ= 0, ms= 1/2 n = 2, ℓ= 1, mℓ= –1, ms= 0 n = 3, ℓ= –1, mℓ= 0, ms= –1/2 n = 3, ℓ= 0, mℓ= 0, ms= –1/2 n = 4, ℓ= 2, mℓ= 3, ms= –1/2 n = 5, ℓ= 2, mℓ= 0, ms= –1/2

Answer 1

Quantum numbers describe values of conserved quantities in the dynamics of a quantum system. In the case of electrons, the quantum numbers can be defined as "the sets of numerical values which give acceptable solutions to the Schrödinger wave equation for the hydrogen atom". An important aspect of quantum mechanics is the quantization of observable quantities, since quantum numbers are discrete sets of integers or half-integers, although they could approach infinity in some cases. This distinguishes quantum mechanics from classical mechanics where the values that characterize the system such as mass, charge, or momentum, range continuously. Quantum numbers often describe specifically the energy levels of electrons in atoms, but other possibilities include angular momentum, spin, etc. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.

Name	Symbol	Orbital meaning	Range of values	Value examples
principal quantum number	n	shell	1 ≤ n	n = 1, 2, 3, …
azimuthal quantum number (angular momentum)	ℓ	subshell (s orbital is listed as 0, p orbital as 1 etc.)	0 ≤ ℓ ≤ n − 1	for n = 3: ℓ = 0, 1, 2 (s, p, d)
magnetic quantum number, (projection of angular momentum)	mℓ	energy shift (orientation of the subshell's shape)	−ℓ ≤ mℓ ≤ ℓ	for ℓ = 2: mℓ = −2, −1, 0, 1, 2
spin projection quantum number	ms	spin of the electron (−½ = "spin down", ½ = "spin up")	−s ≤ ms ≤ s	for an electron s = ½, so ms = −½, ½

allowable combinations of quantum number for the eletrocon is

a. n = 6, ℓ= 6, mℓ= 0, ms= 1/2

this is not woks because the ℓ is not in between n and n-1.

b..n = 2, ℓ= 1, mℓ= –1, ms= 0

It does not work because ms is not in the range of -1/2 and 1/2

c..n = 3, ℓ= –1, mℓ= 0, ms= –1/2

It doesnot work beacuse ℓ not in the range of 0 to n-1.

d...n = 3, ℓ= 0, mℓ= 0, ms= –1/2

It woks beacuse ℓ in the range of n=0 to n-1 and the mℓ is also in the −ℓ ≤ m_ℓ ≤ ℓ range.

e.. n = 4, ℓ= 2, mℓ= 3, ms= –1/2

It doesnt work because ml not in range of −ℓ ≤ m_ℓ ≤ ℓ

f..n = 5, ℓ= 2, mℓ= 0, ms= –1/2

It woks..