- via the Schrödinger equation solution for the hydrogen atom it predicts:
- Schrödinger equation solution for the helium atom: perturbative solutions give good approximations to the energy levels
- double-slit experiment: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments
The easiest to understand case of the equation which you must have in mind initially that of the Schrödinger equation for a free one dimensional particle.
Then, with that in mind, the general form of the Schrödinger equation is:
- is the reduced Planck constant
- is the wave function
- is the time
- is a linear operator called the Hamiltonian. It takes as input a function , and returns another function. This plays a role analogous to the Hamiltonian in classical mechanics: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.
The argument of could be anything, e.g.:
Note however that there is always a single magical time variable. This is needed in particular because there is a time partial derivative in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly non-relativistic.
- we could have preferred polar coordinates instead of linear ones if the potential were symmetric around a point
- we could have more than one particle, e.g. solutions of the Schrodinger equation for two electrons, which would have e.g. and for different particles. No matter how many particles there are, we have just a single , we just add more arguments to it.
- we could have even more generalized coordinates. This is much in the spirit of Hamiltonian mechanics or generalized coordinates
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".
Once that example is clear, we see that the exact same separation of variables can be done to the Schrödinger equation. If we name the constant of the separation of variables for energy, we get:
- a time-only part that does not depend on space and does not depend on the Hamiltonian at all. The solution for this part is therefore always the same exponentials for any problem, and this part is therefore "boring":
- a space-only part that does not depend on time, bud does depend on the Hamiltonian:Note that the here is not the same as the in the time-dependent Schrodinger equation of course, as that psi is the result of the multiplication of the time and space parts. This is a bit of imprecise terminology, but hey, physics.
Because the time part of the equation is always the same and always trivial to solve, all we have to do to actually solve the Schrodinger equation is to solve the time independent one, and then we can construct the full solution trivially.
Once we've solved the time-independent part for each possible , we can construct a solution exactly as we did in heat equation solution with Fourier series: we make a weighted sum over all possible to match the initial condition, which is analogous to the Fourier series in the case of the heat equation to reach a final full solution:
- if there are only discretely many possible values of , each possible energy . we proceed such that at time we match the initial condition:
- if there are infinitely many values of E, we do something analogous but with an integral instead of a sum. This is called the continuous spectrum. One notable
The fact that this approximation of the initial condition is always possible from is mathematically proven by some version of the spectral theorem based on the fact that The Schrodinger equation Hamiltonian has to be Hermitian and therefore behaves nicely.
It is interesting to note that solving the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation where:matrix eigenvectors is that we are now dealing with an infinite dimensional vector space.
- we immediately see from the equation that the time-independent solutions are states of deterministic energy because the energy is an eigenvalue of the Hamiltonian operator
- by looking at Equation 3. "Solution of the Schrodinger equation in terms of the time-independent and time dependent parts", it is obvious that if we take an energy measurement, the probability of each result never changes with time, because it is only multiplied by a constant
Where derivation == "intuitive routes", since a "law of physics" cannot be derived, only observed right or wrong.
TODO also comment on why are complex numbers used in the Schrodinger equation?.
- en.wikipedia.org/w/index.php?title=Schr%C3%B6dinger_equation&oldid=964460597#Derivation: holy crap, this just goes all in into a Lie group approach, nice
- Richard Feynman's derivation of the Schrodinger equation:
- www.youtube.com/watch?v=xQ1d0M19LsM "Class Y. Feynman's Derivation of the Schrödinger Equation" by doctorphys (2020)
- www.youtube.com/watch?v=zC_gYfAqjZY&list=PL54DF0652B30D99A4&index=53 "I5. Derivation of the Schrödinger Equation" by doctorphys
Why is there a complex number in the equation? Intuitively and mathematically:
The Schrödinger equation Hamiltonian has to be a Hermitian so we will have only positive energies I think: quantumcomputing.stackexchange.com/questions/12113/why-does-a-hamiltonian-have-to-be-hermitian
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.
- www.youtube.com/watch?v=1Z9wo2CzJO8 "Schrodinger equation solved numerically in 3D" by Tetsuya Matsuno. 3D hydrogen atom, code may be hidden in some paper, maybe
- www.youtube.com/playlist?list=PLdCdV2GBGyXM0j66zrpDy2aMXr6cgrBJA "Computational Quantum Mechanics" by Let's Code Physics. Uses a 1D trinket.io.
- www.youtube.com/watch?v=BBt8EugN03Q Simulating Quantum Systems [Split Operator Method] by LeiosOS (2018)
- www.youtube.com/watch?v=86x0_-JGlGQ Simulating the Quantum World on a Classical Computer by Garnet Chan (2016) discusses how modeling only local entanglement can make certain simulations feasible
This is basically how quantum computing was first theorized by Richard Feynman: quantum computers as experiments that are hard to predict outcomes.
TODO answer that: quantumcomputing.stackexchange.com/questions/5005/why-it-is-hard-to-simulate-a-quantum-device-by-a-classical-devices. A good answer would be with a more physical example of quantum entanglement, e.g. on a photonic quantum computer.
Schrödinger equation for a one dimensional particle with . The first step is to calculate the time-independent 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: plane wave functions.
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.
This equation is a subcase of Equation "Schrödinger equation for a one dimensional particle" with .
We get the time-independent Schrödinger equation by substituting this into Equation "time-independent Schrödinger equation for a one dimensional particle":
Now, there are two ways to go about this.
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.
A quantum version of the LC circuit!
TODO are there experiments, or just theoretical?
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.
Not the same as Hermite polynomials.
www.physics.udel.edu/~jim/PHYS424_17F/Class%20Notes/Class_5.pdf by James MacDonald shows it well.
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.
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.
The explicit solution can be written in terms of spherical harmonics.
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 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?
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.
Fixed total angular momentum.
The direction however is not specified by this 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:
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.
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?
- Quantum Mechanics for Engineers by Leon van Dommelen (2011) "5. Multiple-Particle Systems"
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.
- www.youtube.com/watch?v=Og13-bSF9kA&list=PLDfPUNusx1Eo60qx3Od2KLUL4b7VDPo9F "Advanced quantum theory" by Tobias J. Osborne says that the course will essentially cover multi-particle quantum mechanics!
- physics.stackexchange.com/questions/54854/equivalence-between-qft-and-many-particle-qm "Equivalence between QFT and many-particle QM"
- Course: Quantum Many-Body Physics in Condensed Matter by Luis Gregorio Dias (2020) from course: Quantum Many-Body Physics in Condensed Matter by Luis Gregorio Dias (2020) give a good introduction to non-interacting particles
Just ignore the electron electron interactions.
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.
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.
We will sometimes just write them without superscript, as it saves typing and is useless.
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.
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.
This rule is only an approximation, there exist exceptions to the Madelung energy ordering rule.
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.
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.
- chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/08%3A_Multielectron_Atoms/8.08%3A_Term_Symbols_Gives_a_Detailed_Description_of_an_Electron_Configuration The PDF origin: web.chem.ucsb.edu/~devries/chem218/Term%20symbols.pdf
- physics.stackexchange.com/questions/8567/how-do-electron-configuration-microstates-map-to-term-symbols How do electron configuration microstates map to term symbols?
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 __
Higher spin multiplicity means lower energy. I.e.: you want to keep all spins pointin in the same direction.
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.
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
For example, on the first line:
m_l +1 0 -1 m_L m_S u_ u_ __ 1 1
and so the sum of them has angular momentum . So the value of is 1, we just omit the .
- one electron with , and so angular momentum
- one electron with , and so angular momentum 0
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.
He then goes for and mentions:
and so that corresponds to states on our list:
- S = 1 so can only be 0
- L = 2 (D) so ranges in -2, -1, 0, 1, 2
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:
ud __ __ 2 0 u_ d_ __ 1 0 u_ __ d_ 0 0 __ u_ d_ -1 0 __ __ ud -2 0
+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:
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.
- S = 2 so is either -1, 0 or 1
- L = 1 (P) so ranges in -1, 0, 1
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.
Can we make any ab initio predictions about it all?
A 2016 paper: aip.scitation.org/doi/abs/10.1063/1.4948309
TODO definition. Appears to be isomers
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.
Also known in quantum computing as a qubit :-)
The wave equation contains the entire state of a particle.
And a single vector can be represented in many different ways in different basis, and two of those ways happen to be the position and the momentum representations.
When you represent a wave equation as a function, you have to say what the variable of the function means. And depending on weather you say "it means position" or "it means momentum", the position and momentum operators will be written differently.
This is well shown at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)".
Then the uncertainty principle follows immediately from a general property of the Fourier transform: en.wikipedia.org/w/index.php?title=Fourier_transform&oldid=961707157#Uncertainty_principle
We can visualize the uncertainty principle more intuitively by thinking of a wave function that is a real flat top bump function with a flat top in 1D. We can then change the width of the support, but when we do that, the top goes higher to keep probability equal to 1. The momentum is 0 everywhere, except in the edges of the support. Then:
- to localize the wave in space at position 0 to reduce the space uncertainty, we have to reduce the support. However, doing so makes the momentum variation on the edges more and more important, as the slope will go up and down faster (higher top, and less x space for descent), leading to a larger variance (note that average momentum is still 0, due to to symmetry of the bump function)
- to localize the momentum as much as possible at 0, we can make the support wider and wider. This makes the bumps at the edges smaller and smaller. However, this also obviously delocalises the wave function more and more, increasing the variance of x
- www.youtube.com/watch?v=bIIjIZBKgtI&list=PL54DF0652B30D99A4&index=59 "K2. Heisenberg Uncertainty Relation" by doctorphys (2011)
- physics.stackexchange.com/questions/132111/uncertainty-principle-intuition Uncertainty Principle Intuition on Physics Stack Exchange
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 such that:
And then, what is the definition of momentum space? It is of course a way to write the wave function such that:
- the position operator is the multiplication by
- the momentum operator is the derivative by
- the momentum operator is the multiplication by
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.
There is an uncertainty principle between the x, y and z angular momentums, we can only measure one of them with certainty at a time. Video 1. "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)" justifies this intuitively by mentioning that this is analogous to precession: if you try to measure electrons e.g. with the Zeeman effect the precess on the other directions which you end up modifing.
TODO experiment. Likely Zeeman effect.
TODO is there any good intuitive argument or proof of conservation of energy, momentum, angular momentum?
Proof that the probability 1 is conserved by the time evolution:
It can be derived directly from the Schrödinger equation.
- That proof also mentions that if the potential
Vis not real, then there is no conservation of probability! Therefore the potential must be real valued!
Contains the full state of the quantum system.
This is in contrast to classical mechanics where e.g. the state of mechanical system is given by two real functions: position and speed.
Relates particle momentum and its wavelength, or equivalently, energy and frequency.
Particle wavelength can be for example measured very directly on a double-slit experiment.
So if we take for example electrons of different speeds, we should be able to see the diffraction pattern change accordingly.