Intro to the Quantum Harmonic Oscillator in 9 Minutes #PaCE1

A brief conceptual overview of the quantum harmonic oscillator. Hopefully this video provides some visual intuition for what the quantum harmonic oscillator is all about.

Wikipedia page on the Quantum Harmonic Oscillator:

Derivation of the stationary states for the Quantum Harmonic Oscillator:

Wikipedia page on the Hermite polynomials:

Please be careful to use the *physicist’s* Hermite polynomials, which are the cousins of the mathematician’s Hermite polynomials. The Wikipedia article has a list for both.

Thanks to Seth Cottengim for letting me borrow your microphone and helping with the audio, thanks Patrick and Uncle Art for reviewing the rough draft and providing feedback, and thank you for watching! :)

[nicholasspicer]

10/10, im just a tourist when it comes to physics but this was a nice dive into something most youtubers just dip their toes in. the technical details and showing the big scary equations give me something to google! i had never heard of stationary states and many other things until your video. thanks for sharing : )

[KevinCarretoMendoza]

Loved the video, i start my Quantum Mechanics class next week and i'm already looking forward to watch more of your content about this topic:)

[temp]

Excellent. Haven't seen that summation before.

[douglasstrother]

"Quantum Mechanics: Foundations and Applications" by Arno Bohm This is a graduate-level text (or advanced undergrad text), so don't start here; however, the writing style is very clear and crisp.
The author approaches Quantum Theory as an algebraic structure built to explain measured phenomena, like Werner Heisenberg. It treats states as vectors and observables as operators on those vectors. He ties in measurements early on as well.

[MichaelOrr]

Hope you release more vids, they're brilliant.

[TheoriesofEverything]

Thank you so much for your submission! (final video announcement with winners / runner ups out now, by the way)

[vwcanter]

Trying to understand how the superposition of these many modes can result in a wave "packet". A piano string also has a superposition of many modes. But they do not add up to anything that resembles a "packet" moving up and down the string. So there must be some other factor, right?

The other part I don't see yet is how the expression for the Hamiltonian is going to work in general case. It appears to assume the particle is in a one dimensional valley of potential, or tied to a one dimensional spring for a restoring force. But that can't be correct in general, only in certain conditions. Right?

[EmergentUniverse]

Why does the oscillator have fixed endpoints? I would have thought an integer frequency would be more natural.

[sensorer]

Is the state you chose a truncated sum of the coherent state decomposition? Or did you choose some arbitrary coefficients that looked good?

[DavyOneness]

Why is it assumed "quantum waves" are transverse waves and not longitudinal waves?

[xelth]

Do I understand correctly? Superposition as in the video is only possible through energy blurring?

[SerGio-xsss]

Is it possible to get this site to play with harmonics ? Thank you.

[andrewferris]

So, is that why Electric and Magnetic fields can bend a particle's path? Because they are coupled to the phase of the wave function, and can therefore change the interference patterns and probable paths of the particle?

[Dyslexic-Artist-Theory-on-Time]

Very good video! Could the mathematics of quantum mechanics represent geometry as in geometrical process? The Planck Constant ħ=h/2π is linked to 2π circular geometry representing a two-dimensional aspect of 4π spherical three-dimensional geometry. We have to square the wave function Ψ² representing the radius being squared r² because the process is relative to the two-dimensional spherical 4π surface. We then see 4π in Heisenberg’s Uncertainty Principle ∆×∆pᵪ≥h/4π representing our probabilistic temporal three dimensions life. The charge of the electron e² and the speed of light c² are both squared for the same geometrical reason. The spontaneous absorption and emission of light photon ∆E=hf energy is forming potential photon energy into the kinetic energy of electrons. Kinetic Eₖ=½mv² energy is the energy of what is actually happening. Could we have an uncertain probabilistic future continuously coming into existence with the exchange of photon energy?

[dantucho]

Weird question, but are you that tungsten cube guy?

[haniamritdas]

"you can see how the probability 3:38 densities of the stationary states do not change over time. Their phase changes but the amplitude stays constant so the stationary states are like standing probability waves"

They are nothing more or less than standing probability waves predicting a parabola on a number line.

The probability amplitudes are stable because _the wave equation is nothing but a mathematical machine for using linear algebra to create discrete transforms for geometrical curves_

The curves that can be linearized by algebra, which is to say quadratic equations, or partial higher order versions of them. There's no physical theory in these maths. It is a general purpose device which was supposed to provide a theory, but no one outside the room could be convinced of the difference between a mathematical tautology and a testable, falsifiable physical hypothesis.

The "theory" of the Standard Model is the proposition that you can use combinations of linear functions to generate functions with characteristic "probability densities that the solution lies in a range of value from zero to Infinity". That's just another way of saying "quadratic equations".

There is nothing predictive about the wave equation or the standard model that isn't a statistical statement about algebraic geometry. The weirdness of it is a confidence game, played by physicists for a century: yeah it's so hard to understand how nature can be probabilistic! What a mystery! Well you lay people will never understand our wall of magical numbers! Just give us billions of dollars to come up with more magical probability combinations, and we will keep on keeping you mystified about the simplicity of the world.

The show given here there and everywhere is obfuscational of this crucial fact: the Standard Model is no more falsifiable than linear algebra. Because it is only a framework for calculating Hermitian equations. It is not a hypothesis, but a tool for generating geometric probabilistic speculations with no analytic expressions related to the coordinate system, only rules of arithmetic; so it cannot give you the orbit of a planet, just the probability density that the orbit is within a range of values that can only be measured singly. Just apply orbital mechanics to the wave function, and a solar system looks like an energy density probability plot. Because that's what it is from a logical mathematical perspective: that is what "everything" is from that point of view! So then the "function collapses into a point when measured" because it is impossible to generate a meaningful equation of motion from a probabilistic measurement of a single datum. The purpose was to string these measurements together, with physical explanatory power, which has not yet happened. And in order to have explanatory power, it has to be falsifiable, not a statistical inevitability. That is true of a hypothesis, not to mention "theory".

A century of cons and shysters worthy of the economy their children built. We so smart!🤓 Trust Science!

[RPG_Guy-fxns]

This is circular logic.

Making waves out of waves doesn't really get you anywhere.
There needs to be a reason things move towards an equilibrium in order to have the spring forces required to oscillate the sine waves.
Adding up sine waves cannot be the reason for oscillation, because sine waves require oscillation.
At the core of spring forces, underneath the mechanics of attraction, repulsion, and equilibrium, they are made of transactions between neighbors.
The core of physics is some kind of local pressure balancing mechanic. Some kind of matrix of numbers that for some unknowable reason, have rules for balancing with their neighbors.
Like Conway's game of life, there needs to be some kind of neighbor checking rule system that animates at the core of physics.

You can make animated things out of other animated things. you can make springs out of things that already have springs in them. but you can't get from inanimate to animate. There is no reason any exchange of any information exists. We will never know why any motion exists, rather than an entirely static universe that was completely uncorrelated with its neighbors.

So fundamentally, you need counting numbers, that update their value, connected with the update of nearby numbers in local transactions, to make sine waves and spring forces and scalar fields and conservation of momentum, etc...

we don't know where existence got counting numbers from or how it formed the rules that entangled these numbers into forces that try to find equilibrium. Values are the most fundamental thing. In computer science, we make values out of binary bits. We have no idea what kinds of numbers the universe uses for its spring forces. if you can arbitrarily entangle numbers, it doesn't matter what size they are, you can always add more precision by entangling more numbers. in programming, we have functions that operate on values, entangling them in a transaction. The universe has forces that serve as functions, and we have no idea how they were defined.

All the universe really needs is 2 things:
binary bits: some kind of contrast, in 1 dimension.
and the ability to correlate those numbers.

[daltanionwaves]

If you removed everything that is a wave from the Would there be anything left? 🤔 😸

[dadsonworldwide]

Be honest something in you already imagine this or if you built it this is how you would do it.
philosophical categorically weird to think that like the love for your parents can't be observed but we can build lines of evidence about the secondary effects your love has on the environment.
It's not any different than some idealism found in the quantum world.

[buddysnackit]

Except....how does a pull work? I ask this of everyone involved in physics. It points out you are only skimming over the realities of physics. Another way to ask this is to ask you to "Describe the mechanics behind a pull.". Ultimately you won't be able to and this will reveal your surface-level understanding of physics. So everything you said is correct except...how does that pull work?

It is a wavy wave is about all I expect from you. Yes Richard it is a wavy wave. :)

I would argue that mathematics limits imagination.

Now explain the physical connection and reason why I should believe your math....ah...sigh...yeah there is that pull question again...

If you want an example I can describe why gravity works using math and a model which you should agree with.

Ugggg, not a quote from a priest of physics. You should base your physics on the truth and not one of these priests. They didn't figure out the truth of physics and all your doing is accepting their flawed ideas. Assume Bohr, Dirac, Einstein, Michelson, Morley, Hertz, Maxwell, and many others were wrong (especially Mr. bongos). Because they were and are. How do I know? See if they ever assume gravity is a magical pull...and warped space is still a magical pull (hoping you can see that).