# Solutions of the Schrodinger equation

As always, the best way to get some intuition about an equation is to solve it for some simple cases, so let's give that a try with different fixed potentials.

## Schrödinger equation for a one dimensional particle

We select for the general Equation "Schrodinger equation":
giving the full explicit partial differential equation:
Equation 1. Schrödinger equation for a one dimensional particle.
The corresponding time-independent Schrödinger equation for this equation is:
Equation 2. time-independent Schrödinger equation for a one dimensional particle.

## Schrödinger equation for a free one dimensional particle

Then, for each energy , from the discussion at Section "Solving the Schrodinger equation with the time-independent Schrödinger equation", the solution is:
Therefore, we see that the solution is made up of infinitely many plane wave functions.

## Plane wave function

In this solution of the Schrödinger equation, by the uncertainty principle, position is completely unknown (the particle could be anywhere in space), and momentum (and therefore, energy) is perfectly known.
The plane wave function appears for example in the solution of the Schrödinger equation for a free one dimensional particle. This makes sense, because when solving with the time-independent Schrödinger equation, we do separation of variable on fixed energy levels explicitly, and the plane wave solutions are exactly fixed energy level ones.

## Time-independent Schrödinger equation for a free one dimensional particle

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

## Quantum harmonic oscillator

The first is the stupid "here's a guess" + "hey this family of solutions forms a complete bases"! This is exactly how we solved the problem at Section "Solving partial differential equations with the Fourier series", except that now the complete basis are the Hermite functions.
The second is the much celebrated ladder operator method.

## Quantum LC circuit

A quantum version of the LC circuit!
TODO are there experiments, or just theoretical?

## Hermite polynomials

I.e.: they are both:

## Hermite functions

Not the same as Hermite polynomials.

The operators are a natural guess on the lines of "if p and x were integers".
And then we can prove the ladder properties easily.
The commutator appear in the middle of this analysis.

## Quantum tunnelling

Examples:
• flash memory uses quantum tunneling as the basis for setting and resetting bits
• alpha decay is understood as a quantum tunneling effect in the nucleus

## Schrödinger equation solution for the hydrogen atom

Is the only atom that has a closed form solution, which allows for very good predictions, and gives awesome intuition about the orbitals in general.
It is arguably the most important solution of the Schrodinger equation.
In particular, it predicts:
The explicit solution can be written in terms of spherical harmonics.

## Atomic orbital

In the case of the Schrödinger equation solution for the hydrogen atom, each orbital is one eigenvector of the solution.
Remember from time-independent Schrödinger equation that the final solution is just the weighted sum of the eigenvector decomposition of the initial state, analogously to solving partial differential equations with the Fourier series.
This is the table that you should have in mind to visualize them: en.wikipedia.org/w/index.php?title=Atomic_orbital&oldid=1022865014#Orbitals_table

## Quantum number

Quantum numbers appear directly in the Schrödinger equation solution for the hydrogen atom.
However, it very cool that they are actually discovered before the Schrödinger equation, and are present in the Bohr model (principal quantum number) and the Bohr-Sommerfeld model (azimuthal quantum number and magnetic quantum number) of the atom. This must be because they observed direct effects of those numbers in some experiments. TODO which experiments.
E.g. The Quantum Story by Jim Baggott (2011) page 34 mentions:
As the various lines in the spectrum were identified with different quantum jumps between different orbits, it was soon discovered that not all the possible jumps were appearing. Some lines were missing. For some reason certain jumps were forbidden. An elaborate scheme of ‘selection rules’ was established by Bohr and Sommerfeld to account for those jumps that were allowed and those that were forbidden.
This refers to forbidden mechanism. TODO concrete example, ideally the first one to be noticed. How can you notice this if the energy depends only on the principal quantum number?

## Principal quantum number (n)

Determines energy. This comes out directly from the resolution of the Schrödinger equation solution for the hydrogen atom where we have to set some arbitrary values of energy by separation of variables just like we have to set some arbitrary numbers when solving partial differential equations with the Fourier series. We then just happen to see that only certain integer values are possible to satisfy the equations.

## Azimuthal quantum number (l)

The direction however is not specified by this number.
To determine the quantum angular momentum, we need the magnetic quantum number, which then selects which orbital exactly we are talking about.

## Magnetic quantum number ()

Fixed quantum angular momentum in a given direction.
Can range between .
E.g. consider gallium which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1:
• the electrons in s-orbitals such as 1s, 2d, and 3d are , and so the only value for is 0
• the electrons in p-orbitals such as 2p, 3p and 4p are , and so the possible values for are -1, 0 and 1
• the electrons in d-orbitals such as 2d are , and so the possible values for are -2, -1, 0 and 1 and 2
The z component of the quantum angular momentum is simply:
so e.g. again for gallium:
• s-orbitals: necessarily have 0 z angular momentum
• p-orbitals: have either 0, or z angular momentum
Note that this direction is arbitrary, since for a fixed azimuthal quantum number (and therefore fixed total angular momentum), we can only know one direction for sure. is normally used by convention.

## Spectroscopic notation

This notation is cool as it gives the spin quantum number, which is important e.g. when talking about hyperfine structure.
But it is a bit crap that the spin is not given simply as but rather mixes up both the azimuthal quantum number and spin. What is the reason?

Bibliography:

## Solutions of the Schrodinger equation for two electrons

TODO. Can't find it easily. Anyone?
This is closely linked to the Pauli exclusion principle.
What does a particle even mean, right? Especially in quantum field theory, where two electrons are just vibrations of a single electron field.
Another issue is that if we consider magnetism, things only make sense if we add special relativity, since Maxwell's equations require special relativity, so a non approximate solution for this will necessarily require full quantum electrodynamics.
As mentioned at lecture 1 youtube.com/watch?video=H3AFzbrqH68&t=555, relativistic quantum mechanical theories like the Dirac equation and Klein-Gordon equation make no sense for a "single particle": they must imply that particles can pop in out of existence.
Bibliography:

## Orbital approximation

Just ignore the electron electron interactions.

## Schrödinger equation solution for the helium atom

No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.

## Why do multiple electrons occupy the same orbital if electrons repel each other?

That is, two electrons per atomic orbital, each with a different spin.
As shown at Schrödinger equation solution for the helium atom, they do repel each other, and that affects their measurable energy.
However, this energy is still lower than going up to the next orbital. TODO numbers.
This changes however at higher orbitals, notably as approximately described by the aufbau principle.

## Aufbau principle

Boring rule that says that less energetic atomic orbitals are filled first.
Much more interesting is actually determining that order, which the Madelung energy ordering rule is a reasonable approximation to.

## Electron configuration notation (, 1s2 2s2 2p6)

We will sometimes just write them without superscript, as it saves typing and is useless.

## Why does 2s have less energy than 1s if they have the same principal quantum number?

The principal quantum number thing fully determining energy is only true for the hydrogen emission spectrum for which we can solve the Schrödinger equation explicitly.
For other atoms with more than one electron, the orbital names are just a very good approximation/perturbation, as we don't have an explicit solution. And the internal electrons do change energy levels.
Note however that due to the more complex effect of the Lamb shift from QED, there is actually a very small 2p/2s shift even in hydrogen.

Looking at the energy level of the Schrödinger equation solution for the hydrogen atom, you would guess that for multi-electron atoms that only the principal quantum number would matter, azimuthal quantum number getting filled randomly.
However, orbitals energies for large atoms don't increase in energy like those of hydrogen due to electron-electron interactions.
In particular, the following would not be naively expected:
• 2s fills up before 2p. From the hydrogen solution, you might guess that they would randomly go into either one as they'd have the same energy
• in potassium fills up before 3d, even though it has a higher principal quantum number!
This rule is only an approximation, there exist exceptions to the Madelung energy ordering rule.

## Exception to the Madelung energy ordering rule

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?

## Term symbol (, , )

This notation is so confusing! People often don't manage to explain the intuition behind it, why this is an useful notation. When you see Indian university entry exam level memorization classes about this, it makes you want to cry.
The key reason why term symbols matter are Hund's rules, which allow us to predict with some accuracy which electron configurations of those states has more energy than the other.
web.chem.ucsb.edu/~devries/chem218/Term%20symbols.pdf puts it well: electron configuration notation is not specific enough, as each such notation e.g. 1s2 2s2 2p2 contains several options of spins and z angular momentum. And those affect energy.
This is why those symbols are often used when talking about energy differences: they specify more precisely which levels you are talking about.
Basically, each term symbol appears to represent a group of possible electron configurations with a given quantum angular momentum.
We first fix the energy level by saying at which orbital each electron can be (hyperfine structure is ignored). It doesn't even have to be the ground state: we can make some electrons excited at will.
The best thing to learn this is likely to draw out all the possible configurations explicitly, and then understand what is the term symbol for each possible configuration, see e.g. term symbols for carbon ground state.
It also confusing how uppercase letters S, P and D are used, when they do not refer to orbitals s, p and d, but rather to states which have the same angular momentum as individual electrons in those states.
It is also very confusing how extremelly close it looks to spectroscopic notation!
The form of the term symbol is:
The can be understood directly as the degeneracy, how many configurations we have in that state.

## Hund's rules (1927)

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:
m_L -1  0  1
u_ u_ __
u_ __ u_
__ ud __

## Hund's first rule

Higher spin multiplicity means lower energy. I.e.: you want to keep all spins pointin in the same direction.

## Term symbols for carbon ground state

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:
• magnetic quantum number varies from -l to +l, each with z angular momentum to and so each cancels the other out
• spin quantum number is either + or minus half, and so each pair of electron cancels the other out
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:
• m_l is , the magnetic quantum number of each electron. Remember that this gives a orbital (non-spin) quantum angular momentum of to each such electron
• m_L with a capital L is the sum of the of each electron
• m_S with a capital S is the sum of the spin angular momentum of each electron
For example, on the first line:
m_l  +1  0 -1  m_L  m_S
u_ u_ __    1    1
we have:
• one electron with , and so angular momentum
• one electron with , and so angular momentum 0
and so the sum of them has angular momentum . So the value of is 1, we just omit the .
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 is not allowed because it would imply , 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 and mentions:
• S = 1 so can only be 0
• L = 2 (D) so ranges in -2, -1, 0, 1, 2
and so that corresponds to states on our list:
ud __ __    2    0
u_ d_ __    1    0
u_ __ d_    0    0
__ u_ d_   -1    0
__ __ ud   -2    0
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:
 +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
d_ u_ __    1    0
d_ __ u_    0    0
__ d_ u_   -1    0
__ ud __    0    0
Then for the choices are:
• S = 2 so is either -1, 0 or 1
• L = 1 (P) so ranges in -1, 0, 1
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 , we have two electrons with spin up. The angular momentum of each electron is , and so given that we have two, the total is , so again we omit and is 1.

## Schrödinger equation solution for the hydrogen molecule

Can we make any ab initio predictions about it all?

## Isomer

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 (1874)".

## Cis-trans isomerism

Exist because double bonds don't rotate freely. Have different properties of course, unlike enantiomer.

## Enantiomer

Mirror images.
Key exmaple: d and L amino acids. Enantiomers have identical physico-chemical properties. But their biological roles can be very different, because an enzyme might only be able to act on one of them.

## Polymorphism (materials science)

TODO definition. Appears to be isomers
Example:

## Stereochemistry

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.

## Two-state quantum system

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 :-)

## Bloch sphere

A more concrete and easier to understand version of it is the more photon-specific Poincaré sphere, have a look at that one first.