## The quantum harmonic oscillator

...and how to do so...
RiverLyle
KVRer
10 posts since 25 Jul, 2016
The quantum harmonic oscillator

This name sounded great even for a DIY project of making a synth or a song even.
Source here (scroll down): https://en.wikipedia.org/wiki/Correspondence_principle

If anyone already got it share it! (or let us know where you bought it!)

RiverLyle
KVRer
10 posts since 25 Jul, 2016
The quantum harmonic oscillator
Here is a demonstration[8] of how large quantum numbers can give rise to classical (continuous) behavior.

Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic and potential) energy of the oscillator, E, has a set of discrete values,

{\displaystyle E=(n+1/2)\hbar \omega ,\ n=0,1,2,3,\dots ~,}E=(n+1/2)\hbar \omega ,\ n=0,1,2,3,\dots ~,
where ω is the angular frequency of the oscillator.

However, in a classical harmonic oscillator such as a lead ball attached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopic system appears to vary over a continuum of values. We can verify that our idea of macroscopic systems fall within the correspondence limit. The energy of the classical harmonic oscillator with amplitude A, is

{\displaystyle E={\frac {m\omega ^{2}A^{2}}{2}}.}E={\frac {m\omega ^{2}A^{2}}{2}}.
Thus, the quantum number has the value

{\displaystyle n={\frac {E}{\hbar \cdot \omega }}-{\frac {1}{2}}={\frac {m\omega A^{2}}{2\hbar }}-{\frac {1}{2}}}n={\frac {E}{\hbar \cdot \omega }}-{\frac {1}{2}}={\frac {m\omega A^{2}}{2\hbar }}-{\frac {1}{2}}
If we apply typical "human-scale" values m = 1kg, ω = 1 rad/s, and A = 1 m, then n ≈ 4.74×1033. This is a very large number, so the system is indeed in the correspondence limit.

It is simple to see why we perceive a continuum of energy in this limit. With ω = 1 rad/s, the difference between each energy level is ħω ≈ 1.05 × 10−34J, well below what we normally resolve for macroscopic systems. One then describes this system through an emergent classical limit.

resynthesis
KVRist
457 posts since 17 Sep, 2007 from Planet Thanet
LaTeX may not be the best choice of markup here

Meffy
Skunk Mod
20866 posts since 10 Jun, 2004 from Pony Pasture
resynthesis wrote:
Thu Jan 23, 2020 5:23 pm
LaTeX may not be the best choice of markup here
Maybe https://www.quicklatex.com/ ? Paste in your LaTeX code, click button, copy and paste the link they provide. You get a rendered image, they get a backlink.

whyterabbyt
Beware the Quoth
27856 posts since 4 Sep, 2001 from R'lyeh Oceanic Amusement Park and Funfair
Meffy wrote:
Thu Jan 23, 2020 7:23 pm
resynthesis wrote:
Thu Jan 23, 2020 5:23 pm
LaTeX may not be the best choice of markup here
Maybe https://www.quicklatex.com/ ? Paste in your LaTeX code, click button, copy and paste the link they provide. You get a rendered image, they get a backlink.
Im not sure if its his LaTeX code, the entirety of his second post looks like its cut and paste from the Wiki article.
Im not sure how informed the user is about the subject, its a bit 'hey this is some thing that mentions oscillators, so SMOP, where do I get?'
"The bearer of this signature is a genuine and authorised pope."

BertKoor
KVRAF
11480 posts since 8 Mar, 2005 from Utrecht, Holland
How practical is it? It's probably limited to sine waves anyway.
Surely it oscillates, but can you change its frequency? How to get out a voltage of around 1V?
Does it need a cubic meter of equipment, consuming 15kW power?

So many questions, and no answer on wikipedia
We are the KVR collective. Resistance is futile. You will be assimilated.
My MusicCalc is back online!!

vurt
58951 posts since 26 Jan, 2003 from through the looking glass
does it work like quantum computing?
im not sure drilling holes to alternate dimensions and bringing back extra resources, is advisable for music related projects.
the risks outweigh the benefits, better tuning/turning the universe inside out

vurt
58951 posts since 26 Jan, 2003 from through the looking glass
but, send the wiki page to game changer fx in latvia, bet theyd have fun with it

Meffy
Skunk Mod
20866 posts since 10 Jun, 2004 from Pony Pasture
whyterabbyt wrote:
Fri Jan 24, 2020 2:41 am
Im not sure if its his LaTeX code, the entirety of his second post looks like its cut and paste from the Wiki article.
Yes, the [8] was a bit of a give-away. By "his" I meant whatever code he wants to render.

My standard reply to explicit declarations of SMOP is "Best hit the books. It'll be simple."

vurt