TODO definition. Appears to be isomers

Example:

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:

The CNOT gate is a controlled quantum gate that operates on two qubits, flipping the second (operand) qubit if the first (control) qubit is set.

This gate is the first example of a controlled quantum gate that you should study.

To understand why the gate is called a CNOT gate, you should think as follows.

First let's produce a generic quantum state vector where the control qubit is certain to be 0.

On the standard basis:we see that this means that only $|00⟩$ and $|01⟩$ should be possible. Therefore, the state must be of the form:where $x$ and $y$ are two complex numbers such that $|x|+|y|=1.0$

If we operate the CNOT gate on that state, we obtain:and so the input is unchanged as desired, because the control qubit is 0.

If however we take only states where the control qubit is for sure 1:

Therefore, in that case, what happened is that the probabilities of $|10⟩$ and $|11⟩$ were swapped from $x$ and $y$ to $y$ and $x$ respectively, which is exactly what the quantum NOT gate does.

So from this we understand more concretely what "the gate only operates if the first qubit is set to one" means.

Now go and study the Bell state and understand intuitively how this gate is used to produce it.

The most notable exception is the borrowing of 3d-orbital electrons to 4s as in chromium, leading to a 3d5 4s1 configuration instead of the 3d4 4s2 we would have with the rule. TODO how is that observed observed experimentally?

Just like as for classic gates, we would like to be able to select quantum computer physical implementations that can represent one or a few gates that can be used to create any quantum circuit.

Unfortunately, in the case of quantum circuits this is obviously impossible, since the space of N x N unitary matrices is infinite and continuous.

Therefore, when we say that certain gates form a "set of universal quantum gates", we actually mean that "any unitary matrix can be approximated to arbitrary precision with enough of these gates".

Or if you like fancy Mathy words, you can say that the subgroup of the unitary group generated by our basic gate set is a dense subset of the unitary group.

Not the same as Hermite polynomials.

Show up in the solution of the quantum harmonic oscillator after separation of variables leading into the time-independent Schrödinger equation, much like solving partial differential equations with the Fourier series.

I.e.: they are both:

so the solution is:We notice that the solution has continuous spectrum, since any value of $E$ can provide a solution.

The time-independent Schrödinger equation is a variant of the Schrödinger equation defined as:

So we see that for any Schrödinger equation, which is fully defined by the Hamiltonian $\hat{H}$, there is a corresponding time-independent Schrödinger equation, which is also uniquely defined by the same Hamiltonian.

The cool thing about the Time-independent Schrödinger equation is that we can always reduce solving the full Schrödinger equation to solving this slightly simpler time-independent version, as described at: Section "Solving the Schrodinger equation with the time-independent Schrödinger equation".

Because this method is fully general, and it simplifies the initial time-dependent problem to a time independent one, it is the approach that we will always take when solving the Schrodinger equation, see e.g. quantum harmonic oscillator.

As of 2022, this tends to be the more "default" when you talk about a quantum computer.

But there are some serious analog quantum computer contestants in the field as well.

One single universal wavefunction, and every possible outcomes happens in some alternate universe. Does feel a bit sad and superfluous, but also does give some sense to perceived "wave function collapse".