Quantum Numbers

The variables in Schrödinger’s equation are called quantum numbers and there are 4 of them. The names vary a little bit (depending on who you ask and their native language). The four are: 

• Principle Quantum Number 
• Angular Momentum Quantum Number 
• Magnetic Quantum Number 
• Spin Quantum Number 

On this page, we’ll look at each with a superficial lens which may be all you need.

Principal Quantum Number (n) 

The principal quantum number, n, tells the energy level of the electron. This is, in essence, the same idea as Bohr’s Energy levels. The numerical values are (not surprisingly 1, 2, 3, 4,...). 

A deeper understanding of the principle quantum number depends on your understanding of the presence of nodes in a wave and their relationship to energy. More on what energy level actually determines, can be found here.

Angular Momentum (ℓ) 

The Angular Momentum Quantum Number determines the shape of the orbital. These shapes are called s, p, d, and f. (There are orbital shapes beyond this, but they don’t appear in any ground state atoms and are, therefore, not of real importance for readers of this text.). 

If you need to understand the numerical values for l and how they determine the shape of the orbital, you can read about it here.

Magnetic (m) 

The Magnetic Quantum Number determines the spatial orientation the orbital has around the nucleus. For instance, the three different p orbitals are all basically the same but they are oriented at 90 degrees to each other. We describe them as oriented on the x, y, of z axis (although atoms don’t actually have axes). 

If you need to understand the possible numerical values of m and what they mean, you can read about them here.

Spin (m_s) 

The Spin Quantum Number allows electrons to share an orbital. Spin is a property of electrons (and other quantum particles) which is related to magnetic fields. In simple terms, electrons have “spin” which makes them act like little bar magnets. Just like bar magnets, if they are oriented in opposite directions (described as up or down), the magnets will attract.

It is important to remember that ALL electrons are negative and, therefore, ALWAYS repel each other. However, if two electrons have opposite spin, then the magnetic attraction “balances” the repulsion. This allows the two electrons to be in the same orbital. 

There can never be more than two electrons in an orbital, because the third electron will have the same magnetic orientation as one of the others and therefore will doubly repel (charge and magnetic field).

 

An annoying aside:

The image above is wrong. It is wrong because it pictures electrons as small objects that can spin. Schrödinger’s theory requires electrons to be considered as waves. So, the truth is that we don't really know what spin is. We know that electrons have it. We know that it creates magnetic fields. We know that there are two types. We know that electrons with opposite spin can share an orbital. But we really don't know what it is.