Laidler, Meiser, Sanctuary textbook – Support Portal

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 “*aha*” moment passed.

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_{ℓ}, m_{s}.

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_{c} and 1Aa_{d}.

The second floor is split into two departments. A is similar to the one below it: 2Aa_{c} and 2Aa_{d }; plus one new department, B, that has three offices (a,b,c) for a total of six pieces of furniture: 2Ba_{c}, 2Ba_{d}, 2Bb_{c}, 2Bb_{d}, 2Bc_{c}, 2Ba_{d}.

The third floor gets the same as the second: 3Aa_{c}, 3Aa_{d} and 3Ba_{c}, 3Ba_{d}, 3Bb_{c}, 3Bb_{d}, 3Bc_{c}, 3Ba_{d}, plus one more department, C, that has five offices (a,b,c,d,e) with ten pieces of furniture! 3Ca_{c}, 3Ca_{d}, 3Cb_{c}, 3Cb_{d}, 3Cc_{c}, 3Cc_{d}, 3Cd_{c}, 3Cd_{d}, 3Ce_{c}, 3Ce_{d}

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_{d} meant, what is the answer? It is a desk from the third floor and the (c) office in the B department. How about 3Bc_{c}? Notice that the first three symbols are identical which indicates again the third floor and the (c) office in the B department but the fourth symbol represents a chair.

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_{d}? The location is supposed to be the second floor, the (a) office in the C department that contains a desk. The problem is that there is no C department on the second floor! The quantum numbers are exactly the same!

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 *l*) divides the shells into smaller groups called sub-shells (departments) and provides the shape required to use up the entire three dimensional area. This can be designated by both *a letter* and *a number*, depending on what the information is going to be used for.

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_{ℓ}) are called the orbitals or offices. This number gives the orientation in 3D space. For example the s sub-shell is a sphere, which takes up all of 3D space so its magnetic quantum number is zero (m_{ℓ} = 0). The p sub-shell (ℓ = 1) needs three orbitals to occupy 3D space and the magnetic quantum number tells us that one needs to be aligned in the x, y, and z axes designated by -1, 0, 1 numerically (m_{ℓ} = -ℓ…ℓ).

The last is the spin quantum number (m_{s}). This number indicates the actual spin on the electron, which can be visualized much like a planet either spinning clockwise or counterclockwise.

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_{s} = +½ or -½.

** ****Summary of quantum numbers**

n ℓ m_{ ℓ} Subshell(ℓ) # orbitals(m_{ ℓ}) # electrons

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_{ ℓ},m_{s}) meant, what is the answer? It is an electron with a “positive” spin in the third shell, the p or dumbbell sub-shell, in the x axes. How about 2,1,1, -½? It is an electron with a “negative” spin in the second shell, the p or dumbbell sub-shell, in the z axes.

What about 1,1,1, +½? Right! It isn’t *possible* because the first shell does not contain p sub-shells.

Aha!

Nice metaphor to use a shop with many floors to understand the spdf orbitals. It is hard to tell students that the quantum numbers arise due to boundary conditions of the Schrodinger equation. Thank you for that contribution.