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When learning about quantum numbers for the first time, it can be overwhelming and confusing. However, in all my years of teaching, no other topic so feared at first; has been so rewarding and enjoyed after the “

Sections11.8 and 11.9 of the Physical Chemistry textbook, gives a very comprehensive explanation about the origins of each of the quantum numbers, but it is sometimes nice to have an overview of what is being discussed beforehand.

While there is a plethora of knowledge pertaining to these innocent looking symbols/numbers, I believe a very basic visual approach for all of them simultaneously works best. Once the overall picture is clear, details of each piece can be explored.

In a nutshell, the quantum numbers describe the identity and location of the electrons. Since no two electrons can be at exactly the same location at exactly the same time, no two electrons will ever have EXACTLY the same quantum numbers. This identity/location, is described by four numbers: n, ℓ, m

Let’s use the visual example of a building with each floor containing different numbers of departments and offices. The ground floor is one huge department with its office taking up the whole space. Every new floor is split so that it has one more department than the floor below it. Each new department is split so that it always has two more offices than the one below it and every office may contain only two pieces of furniture, a desk and a chair.

Each floor is identified by its number: 1, 2, 3…; each department is assigned a capital letter: A, B, C…; each office in that department is given a lower case letter: a, b, c…; and the chair and desk are identified using their first letters as a subscript.

The first floor has one department, A, one office (a) and can hold two pieces of furniture. They are identified with: 1Aa

The second floor is split into two departments. A is similar to the one below it: 2Aa

The third floor gets the same as the second: 3Aa

The fourth floor would get one more department, D, that is split into seven offices (a,b,c,d,e,f) and so on.

Now, if I was to ask you what 3Bc

Will two designations ever be exactly the same? NO! The location is given by the first three symbols, which may be the same; but the fourth will always be different designating what goes into the assigned office.

What is wrong with 2Ca

The principle quantum number (n) determines which “shell” or floor. It relates both energy and size with the first shell or ground floor being the lowest in energy and smallest in size.

The secondary (azimuthal) quantum number (ℓ or

The lowest sub-shell ℓ = 0 or in the building example 1Aa, takes up the entire space. What is the best shape to occupy all of 3D space? A sphere!

The next sub-shell ℓ = 1 or in the building example 1B has three orbitals or offices. What is the best way for three items to occupy all 3D space? Picture a dumbbell shape (∞) with one occupying the x axis, y axis and z axis simultaneously. The dumbbell shape is required so that the centers can be aligned with the large bulb extending out in each direction. Combined together, it would cover the entire space. So the shape of the second sub-shell is a dumbbell and there are three of them. This of course, is explained in detail in the physical chemistry textbook, along with the mathematical proof.

The sub-shells can be designated as either a number, ℓ = 0 … (n –1), or as a letter that represents each shape. The lowest energy sub-shell that is a sphere is s, the next (the dumbbells) are p and so on with the energy of each sub-shell increasing s<p<d<f.

ℓ = 0, 1, 2, 3, 4, 5….

s, p, d, f, g, h….

The magnetic quantum number (m

The last is the spin quantum number (m

Each orbital may contain a maximum of two electrons with opposite or complementary spins or in the office example, each office must contain two complementary pieces of furniture, a desk and a chair. To avoid confusion with the other quantum numbers, it is usually designated as, m

n ℓ m

1 0 0 1s 1 2

2 0 0 2s 1 2

. 1 -1,0,1 2p 3 6

3 0 0 3s 1 2

1 -1,0,1 3p 3 6

. 2 -2,-1,0,1,2 3d 5 10

Now, if I was to ask you what 3,1,-1, +½ (n,ℓ,m

What about 1,1,1, +½? Right! It isn’t

Aha!

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