Quantum Numbers

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There are four quantum numbers; their symbols are n, ℓ, m and s. EVERY electron in an atom has a specific, unique set of these four quantum numbers. The story behind how these numbers came to be discovered is a complex one indeed and is one best left for another day.

A warning before I proceed: this is a 100% non-mathematical discussion. The equations governing electron behavior in atoms are complex. This area of study generated LOTS of Nobel Prizes and the reasoning leading to the above mentioned equations is sophisticated and sometimes quite subtle.

Just keep this in mind: EVERY electron's behavior in an atom is governed by a set of equations and that n, ℓ, m, and s are values in those equations. EVERY electron in an atom has a unique set of quantum numbers.

Lastly, I'm going to reserve to another discussion what these numbers mean. I will just describe their existence and the rules for how to determine them in this tutorial. The next tutorial will start with hydrogen and assign quantum numbers to its electron, then proceed to helium and do the same, then lithium, beryllium, and so on.

I. The Principal Quantum Number (signified by the letter 'n'): This quantum number was the first one discovered and it was done so by Niels Bohr in 1913. Bohr thought that each electron was in its own unique energy level, which he called a "stationary state," and that each electron would have a unique value of 'n.'

In this idea, Bohr was wrong. It very quickly was discovered that more than one electron could have a given 'n' value. For example, it was eventually discovered that when n = 3, eighteen different electrons could have that value.

Keep in mind that it is the set of four quantum numbers that is important. As you will see, each of the 18 electrons just mention will have its own unique set of n, ℓ, m, and s.

Finally, there is a rule for what values 'n' can assume. It is:

n = 1, 2, 3, and so on.

n will always be a whole number and NEVER less than one.

One point: n does not refer to any particular location in space or any particular shape. It is one component (of four) that will uniquely identify each electron in an atom.

II. The Azimuthal Quantum Number (signified by the letter 'ℓ'): about 1914-1915, Arnold Sommerfeld realized that Bohr's 'n' was insufficient. In other words, more equations were needed to properly describe how electrons behaved. In fact, Sommerfeld realized that TWO more quantum numbers were needed.

The first of these is the quantum number signified by 'ℓ.' When Sommerfeld started this work, he used n' (n prime), but he shifted it to 'ℓ' after some years. I'm not sure why, but it seems easier to print ℓ than n prime and what if the printer (of a textbook) accidently dumps a few prime symbols, leaving just the letter 'n?' Ooops!

The rule for selecting the proper values of 'ℓ' is as follows:

ℓ = 0, 1, 2, . . . , n-1

ℓ will always be a whole number and will NEVER be as large as the 'n' value it is associated with.

III. The Magnetic Quantum Number (signified by the letter 'm' or m): this quantum number was also discovered by Sommerfeld in the same 1914-1915 time frame. I don't think he discovered one and then the other, I think that him realizing the need for two runs together somewhat. I could be wrong in this, so don't take my word for it!

The rule for selecting m is as follows:

m starts at negative 'ℓ,' runs by whole numbers to zero and then goes to positive 'ℓ.'

For example, when ℓ = 2, the m values used are -2, -1, 0, +1, +2, for a total of five values.

IV. The Spin Quantum Number (signified by the letter 's' or ms): spin is a property of electrons that is not related to a sphere spinning. It was first thought to be this way, hence the name spin, but it was soon realized that electrons cannot spin on their axis like the Earth does on its axis. If the electron did this, its surface would be moving at about ten times the speed of light (if memory serves correctly!). In any event, the electron's surface would have to move faster than the speed of light and this isn't possible.

In 1925, Wolfgang Pauli demonstrated the need for a fourth quantum number. He closed the abstract to his paper this way:

"On the basis of these results one is also led to a general classification of every electron in the atom by the principal quantum number n and two auxiliary quantum numbers k1 and k2 to which is added a further quantum number m in the presence of an external field. In conjunction with a recent paper by E. C. Stoner this classification leads to a general quantum theoretical formulation of the completion of electron groups in atoms."

In late 1925, two young researchers named George Uhlenbeck and Samuel Goudsmit discovered the property of the electron responsible for the fourth quantum number being needed and named this property spin.

The rule for selecting s is as follows:

after the n, ℓ and m to be used have been determined, assign the value +1/2 to one electron, then assign -1/2 to the next electron, while using the same n, ℓ and m values.

For example, when n, ℓ, m = 1, 0, 0; the first s value used is +1/2, however a second electron can also have n, ℓ, m = 1, 0, 0; so assign -1/2 to it.

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