Allow us to determine with good approximation in a multi-electron atom which electron configuration have more energy. It is a bit like the Aufbau principle, but at a finer resolution.

Note that this is not trivial since there is no explicit solution to the Schrödinger equation for multi-electron atoms like there is for hydrogen.

For example, consider carbon which has electron configuration 1s2 2s2 2p2.

If we were to populate the 3 p-orbitals with two electrons with spins either up or down, which has more energy? E.g. of the following two:

This example covered for example at Video 1. "Term Symbols Example 1 by TMP Chem (2015)".

Carbon has electronic structure 1s2 2s2 2p2.

For term symbols we only care about unfilled layers, because in every filled layer the total z angular momentum is 0, as one electron necessarily cancels out each other:

So in this case, we only care about the 2 electrons in 2p2. Let's list out all possible ways in which the 2p2 electrons can be.

There are 3 p orbitals, with three different magnetic quantum numbers, each representing a different possible z quantum angular momentum.

We are going to distribute 2 electrons with 2 different spins across them. All the possible distributions that don't violate the Pauli exclusion principle are:

m_l  +1  0 -1  m_L  m_S
     u_ u_ __    1    1
     u_ __ u_    0    1
     __ u_ u_   -1    1
     d_ d_ __    1   -1
     d_ __ d_    0   -1
     __ d_ d_   -1   -1
     u_ d_ __    1    0
     d_ u_ __    1    0
     u_ __ d_    0    0
     d_ __ u_    0    0
     __ u_ d_   -1    0
     __ d_ u_   -1    0
     ud __ __    2    0
     __ ud __    0    0
     __ __ ud   -2    0

where:

For example, on the first line:we have:and so the sum of them has angular momentum $0+1\hslash =1\hslash$. So the value of $m_L$ is 1, we just omit the $\hslash$.

TODO now I don't understand the logic behind the next steps... I understand how to mechanically do them, but what do they mean? Can you determine the term symbol for individual microstates at all? Or do you have to group them to get the answer? Since there are multiple choices in some steps, it appears that you can't assign a specific term symbol to an individual microstate. And it has something to do with the Slater determinant. The previous lecture mentions it: www.youtube.com/watch?v=7_8n1TS-8Y0 more precisely youtu.be/7_8n1TS-8Y0?t=2268 about carbon.

youtu.be/DAgEmLWpYjs?t=2675 mentions that ${}^{3}D$ is not allowed because it would imply $L=2,S=1$, which would be a state uu __ __ which violates the Pauli exclusion principle, and so was not listed on our list of 15 states.

He then goes for ${}^{1}D$ and mentions:and so that corresponds to states on our list:Note that for some we had a two choices, so we just pick any one of them and tick them off off from the table, which now looks like:

Then for ${}^{3}P$ the choices are:so we have 9 possibilities for both together. We again verify that 9 such states are left matching those criteria, and tick them off, and so on.

For the $m_S$, we have two electrons with spin up. The angular momentum of each electron is $1/2\hslash$, and so given that we have two, the total is $1\hslash$, so again we omit $\hslash$ and $m_S$ is 1.

Bibliography:

Can we make any ab initio predictions about it all?

A 2016 paper: aip.scitation.org/doi/abs/10.1063/1.4948309

Isomers were quite confusing for early chemists, before atomic theory was widely accepted, and people where thinking mostly in terms of proportions of equations, related: Section "Isomers suggest that atoms exist".

Molecules that are the same if you just look at "what atom is linked to what atom", they are only different if you consider the relative spacial positions of atoms.

Discrete quantum system model that can model both spin in the Stern-Gerlach experiment or photon polarization in polarizer.

Also known in quantum computing as a qubit :-)

One of the main reasons why physicists are obsessed by this topic is that position and momentum are mapped to the phase space coordinates of Hamiltonian mechanics, which appear in the matrix mechanics formulation of quantum mechanics, which offers insight into the theory, particularly when generalizing to relativistic quantum mechanics.

One way to think is: what is the definition of space space? It is a way to write the wave function ${\psi }_x(x)$ such that:And then, what is the definition of momentum space? It is of course a way to write the wave function ${\psi }_p(p)$ such that:

physics.stackexchange.com/questions/39442/intuitive-explanation-of-why-momentum-is-the-fourier-transform-variable-of-posit/39508#39508 gives the best idea intuitive idea: the Fourier transform writes a function as a (continuous) sum of plane waves, and each plane wave has a fixed momentum.

Bibliography:

This operator case is surprisingly not necessarily mathematically trivial to describe formally because you often end up getting into the Dirac delta functions/continuous spectrum: as mentioned at: mathematical formulation of quantum mechanics

One dimension in position representation:

In three dimensions In position representation, we define it by using the gradient, and so we see that

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