Quantum numbers

Definitions & Key takeaways

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". In more general cases, quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be known with precision at the same time as the system's energy—and their corresponding eigenspaces. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together.
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Quantum numbers can be used to understand the electronic structure of atoms. Quantum numbers come from Quantum Mechanics which was the successor to the Bohr Model.
The Bohr model traces to the early 20th century physicist Niels Bohr, who realized that he could use ideas from classical mechanics to formulate a simple picture of an atom.
His main idea was that an electron orbits around the nucleus at a specific radius in the same way that planets orbit around the Sun.
Unfortunately, the Bohr model does not remain accurate when dealing with atoms with more electrons than hydrogen. For many electron atoms, more sophisticated theories based on quantum mechanics have been developed.
A main point of these theories is that, rather than imagining an atom as a collection of electrons orbiting the nucleus, it is better to think of the electrons as a diffuse cloud around the nucleus -- corresponding to a probability of electrons existing in various locations.
Therefore, rather than stating that an electron lies at a given, fixed distance from the nucleus like the Bohr model, these more sophisticated models provide the probability of finding electrons in various locations.These clouds of likely electron locations have a name -- they are called orbitals.
Looking at a picture of an electron density model, we can see that all these dots indicate the probability of finding an electron.
We don't know exactly where the electron is, but it is somewhere in this cloud around the nucleus. If we think about this orbital as being a three-dimensional sphere, we would say there is a 90% probability that the electron is somewhere in that sphere around the nucleus.Electrons in orbitals are described by four quantum numbers.
The first quantum number is called the principal quantum number, and it is symbolized by N. N is a whole number, like 1, 2, 3, or higher.
In a diagram of an atom, electrons with higher values of N -- and thus higher principal quantum numbers -- are found further away from the nucleus.
We can think of electrons that are further away from the nucleus as having higher energy, because they have a higher potential energy relative to the nucleus.
And so the principal quantum number N tells us the energy level of an electron. Also, an electron’s principal quantum number is sometimes referred to as a shell.The second quantum number is called the angular momentum quantum number.
It is symbolized by “L”, and it refers to the shape of the orbital. “L” can be any whole number between 0 and N -1 -- so the energy level, or first quantum number, determines the possible values of the angular momentum quantum number.So for the case of the first energy level, N = 1, we know that L can be at most equal to L-1.
So that means that when N = 1, L always equals 0. An orbital with L = 0 corresponds to a spherically shaped orbital called an S orbital.
So the first energy level has only one orbital, an s orbital shaped like a sphere. In the case of the second energy level, N is equal to 2.
When N is equal to 2, our values for L can be both 0 or N minus one which equals one. When L = zero, we are dealing with an S orbital which is shaped like a sphere.
When L = one, we are dealing with a P orbital. P orbitals are shaped like dumbbells.
Since we can refer to energy levels as shells, we can also refer to our S and P orbitals as subshells. So in the second shell, there are two subshells.
The third quantum number is called the magnetic quantum number. It is symbolized by “m sub l,” and it tells us the orientation of an orbital.
“m sub l” can be any whole number between negative L and positive L. So if L equals 0, the only possible value for “m sub L” is zero.
If L equals 1, then “m sub L” can equal -1, 0, or 1. If L equals zero, we know that we are dealing with an S orbital which is shaped like a sphere.
And since we have only one value for “m sub L,” then we have only one orientation. There is no way to change the orientation of a sphere -- it’s still a sphere if you turn it upside down -- and so it makes sense that m sub L would only have one possible value.If L equals one, we know that we are dealing with a P orbital which is shaped like a dumbbell.
If we have three values for “m sub L,” then we have three possible orientations. In a 3D coordinate system, we would refer to these three differently-oriented p orbitals as px, py, and pz.
This is consistent with the rule that m sub L has three allowed values: -1, 0, 1. The fourth and final quantum number is the spin quantum number.
It is symbolized by “m sub s”, and it tells is the spin of an electron. For electrons, m sub s can be either positive ½, or negative ½.To get some intuition for what this number means, we can visualize electrons as constantly spinning.
A spinning charge creates a magnetic field, which points upwards or downwards depending on whether the electron is spinning clockwise or counterclockwise.
These two orientations correspond to the two possible values for m sub s of positive ½ or negative ½Let’s use the quantum numbers to determine the total number of electrons in the first four energy levels.
Let’s start with the first energy level. N equals one, so we have only one possible value for L, which is L is equal to zero.
L is equal to zero refers to an S orbital. We also have only one possible value for m sub L, which is also zero.
Since we have only one value for m sub L, there is only one orientation, or one S orbital. Each orbital can hold a maximum of two electrons, one with spin positive one-half and one with spin negative one-half.
Therefore, we can fit a maximum of two electrons into this S orbital in the first energy level. So the total number of electrons the first energy level can hold is two electrons.
Alternatively, we can just use a formula 2 n^2: the number of electrons in the energy level equals two times n squared, where n is the principal quantum number.
Plugging in n = 1, this formula again tells us that there are two electrons in the first energy level.Now, let’s consider the second energy level where n is equal to 2.
In this case, L can have values of 0 or 1. When L is equal to zero, we are dealing with an S orbital.
So there is one S orbital in the second energy level that can hold a maximum of two electrons. When L is equal to 1, we are dealing with a P orbital and there are three possible values of m sub L: negative one, zero, and positive one.
Three possible values means three orientations and three orbitals. Since each orbital can hold two electrons and there are three orbitals, three times two is six.
So the P orbitals in the second energy level can hold a maximum of six electrons. In order to get the total number of electrons in this shell, we need to add these six electrons to the two electrons that were in the S orbital, giving us a total of “six plus two equals eight” electrons in the second shell.
We compare this to our formula, which tells us that there are two times two squared electrons, which once again equals eight electrons overall.When we are in the third energy level, N is equal to three and there are three possible values for L.
L can equal zero, one, or two. When L is equal to zero, it’s an s orbital that can hold two electrons.
When L is equal to one, we are dealing with a P orbital and there are three possible values of m sub L: negative one, zero, and positive one.
Just like before, the three values tell us there are three orbitals and each orbital can hold two electrons. Three times two is six electrons in the three p orbitals in the third energy level.
The case where L equals two corresponds to a new type of orbital, the “d” orbital. A “d” orbital can have five possible orientations, corresponding to the five possible values of m sub L.
When L = 2, the possible values for m sub L are -2, -1, 0, 1, 2.Each of these orbitals can hold two electrons resulting in a total of ten electrons in the d orbitals.
Adding these 10 electrons from the “d” orbitals, to the 6 electrons from the “p” orbitals, to the two electrons from the “s” orbital, gives us a total of eighteen electrons in the third energy level.
This once again agrees with our formula, two times three squared equals eighteen.When N = 4, there are four possible values of L, 0, 1, 2, and 3.When L is equal to zero, it’s an S orbital with maximum of two electrons.
When L is equal to one, there are three p orbitals and each orbital can hold a maximum of two electrons for a total of six electrons.
When L is equal to two, we are talking about a d orbital. There are five d orbitals and each orbital can hold two electrons for a total of ten electrons.
When L is equal to three, we are now talking about an “f” orbital. Since L is equal to three, the possible values for “m sub L” are negative three, negative two, negative one, zero, one, two, and three.
Since there are seven possible values for “m sub L,” there are seven orientations and seven f orbitals. Since each orbital can hold two electrons, there are fourteen electrons that can fit into the f orbitals.
Two plus six plus ten plus fourteen is thirty-two. Therefore, there are thirty-two electrons in the fourth energy level.
##SummaryTo recap, quantum numbers can be used to understand the electronic structure of atoms. The first quantum number, n, tells us the energy level, or shell, of the electrons.
The second quantum number, L, tells us the shape of the orbital. The third quantum number, “m sub L”, tells us how the orbital is oriented in space.
And the fourth quantum number, “m sub s”,