# Talk:Quantum harmonic oscillator

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The momentum operator p, I think, should be (h_bar / i)*(d/dx), not (-h_bar * i)*(d/dx). Source: Modern Physics, Tipler, 5th Ed. page 244. Cheers, Justin. —Preceding unsigned comment added by 67.160.211.137 (talk) 10:05, 15 December 2008 (UTC)

They're the same thing. 1/i = -i. Reedbeta (talk) 20:42, 26 February 2011 (UTC)

->I think the article would improve by adding a definition of w (omega) — Preceding unsigned comment added by Brenogalvaowp (talkcontribs) 10:19, 2 February 2012 (UTC)

Can someone please include a section on the damped, driven quantum harmonic oscillator? This would add tremendously to the content. Thank you. —Preceding unsigned comment added by 128.36.183.217 (talk) 21:24, 8 December 2008 (UTC)

Answer to the previous comment: The damped quantum harmonic oscillator is a far (!) more advanced subject than the undamped oscillator. In fact, it has been a research subject for the past 30 years... The difference in the level of sophistication needed (undamped vs. damped quantum osc.) is far greater than e.g. in the classical case (where one can simply add a friction force). So it is probably better to leave that out in this article. — Preceding unsigned comment added by 131.188.166.21 (talk) 17:06, 14 November 2012 (UTC)

What is the "happy property of the r^2 potential", referred to in the section on the N-dimensional oscillator? It seems to me that we can also separate the potential energy of (uncoupled) anharmonic oscillators into terms depending on one coordinate each. -- Jitse Niesen 19:03, 14 Mar 2004 (UTC)

In classical mechanics there are exactly two central potentials whose orbits in N dimensions are ellipses, r^2 and 1/r. The r^2 potential is also the only potential (in one dimension) leading to oscillations whose period is intependent of the amplitude of the oscillations. The Born-Sommerfeld semiclassical quantization relates these two nice properties of the classical potential r^2 to the fact that the energy levels of the quantum harmonic orcillator are equally spaced. Hope that helps. Miguel 19:09, 2004 Mar 14 (UTC)

Wow, that's a quick answer. I get the point, thanks. -- Jitse Niesen 19:37, 14 Mar 2004 (UTC)

## 2 questions

I've deleted and transed it to copy at Wiki's reference desk.--HydrogenSu 12:10, 5 February 2006 (UTC)

"It is one of the most important model systems in quantum mechanics because, as in classical mechanics, a wide variety of physical situations can be reduced to it either exactly or approximately." Can be reduced to what? The word "it" is not specified as anything. It would be greatly appreciated if someone could clarify this sentence.

## Fanifol revert

On 13 June 2006, a user named User:Fanifol apparently reverted the article to the state of 10 March 2006. This was Fanifol's only contribution to the wikipedia, and his revision comment was "THE CERTAINTY PRINCIPLE WAR [1]". However, as in the last nine days no other wikipedia editor has bothered to revert this edit, I now hesitate to do so. Could somebody please provide some background on "THE CERTAINTY PRINCIPLE WAR"? — Tobias Bergemann 14:22, 22 June 2006 (UTC)

After reading the discussion on Uncertainty principle I now assume User:Fanifol to be a sock puppet of the banned user Hryun. I am going to revert his revert. — Tobias Bergemann 14:28, 22 June 2006 (UTC)

## error in diagram?

I think one of the diagrams (the one captioned as "Wavefunction representations for the first six bound eigenstates...") is incorrect. I was plotting Ψ for n=0..5 in maple for some work i'm doing and i noticed that my plots for n=2 and n=3 are reflected in the x axis relative to those shown in the article. This website [2]also agrees with my plots.

Have I missed something here, or are these actually wrong? If they are can some one fix them? Poobarb 17:15, 19 October 2006 (UTC)

The only difference is a minus sign in the wave function. Such things are irrelevant in QM. David 09:06, 21 March 2007 (UTC)

Well, the sign is not quite irrelevant, and it should be fixed: The minus sign becomes important when you use the annihilation/creation operator formalism to go down/up the ladder of states. Then every textbook has the convention that no minus sign appears in the matrix elements, i.e. a^dagger |n>=sqrt(n+1) |n+1>. This then fixes the relative signs of subsequent states (and only the overall sign of the ground state would be free to be chosen). And in this regard, the figure is unfortunately at least misleading, because readers will expect that the wave functions shown here are those that are stated further down in the article in terms of Hermite polynomials (which would have different signs).

## Anharmonic oscillator

The cubic potential is rather awkward...it will result in an unbounded system! I removed it for now. I think it's better to rewrite that section with a quartic potential instead, and probably in a different article. --HappyCamper 16:03, 14 April 2007 (UTC)

## 3-dimensional harmonic oscillator

This is why I edited the example of the 3-dimensional harmonic oscillator:

• A few small spelling mistakes.
• More importantly: a 3-dimensional harmonic oscillator is not necessarily isotropic. One can have different vibrational frequencies in different directions: ${\displaystyle \omega _{x}\neq \omega _{y}\neq \omega _{z}}$.
• I changed inline TeX letters to html italic, because for some reason, unknown to me, the inline TeX came out roman.

--P.wormer 08:53, 25 May 2007 (UTC)

I think that the normalization constant should be (re)written like this:

${\displaystyle N_{kl}={\sqrt {{\sqrt {\left({\frac {2\nu }{\pi }}\right)^{3}}}{\frac {2^{k+2l+3}\;k!\;\nu ^{l}}{(2k+2l+1)!!}}}}\,}$

For ${\displaystyle \nu =1}$:

Original formula integrated from r=0 to infty

New formula integrated from r=0 to infty — Preceding unsigned comment added by 192.135.11.195 (talk) 10:18, 30 May 2012 (UTC)

## "the zero of energy is not a physically meaningful quantity"

The article states "the lowest achievable energy is not zero...It is not obvious that this is significant, because normally the zero of energy is not a physically meaningful quantity, only differences in energies." If I'm not mistaken, the physically meaningful fact here is that the lowest energy is not equal to the minimum of the potential well. This is why there is zero-point "motion" in a QHO as opposed to a classical oscillator. We have set up the problem with the zero of energy at the bottom of the well, so when the ground state energy comes out as ${\displaystyle \hbar \omega /2}$ all it means is that the ground state energy is higher than the bottom of the well - there is really no issue with the zero of energy not being physically meaningful. Unless anyone objects, I'll edit the article to make this clear. Reedbeta (talk) 20:33, 26 February 2011 (UTC)

I've made the change. Reedbeta (talk) 19:35, 27 February 2011 (UTC)

## Proposition: adding "Analytical method"

Note: Apologies for the first edit that accidentally resulted in removing entire sections from the article. I realized it just as it was corrected by Cuzkatzimhut.

Beside the ladder operator method, another useful way of solving the one-dimensional quantum harmonic oscillator is to directly solve the Schrödinger equation for the system. We start by writing the equation:

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\psi =E\psi }$.

Now we introduce dimensionless variables ${\displaystyle \xi ={\sqrt {\frac {m\omega }{\hbar }}}x}$ and ${\displaystyle \epsilon ={\frac {E}{\hbar \omega }}}$, and rewrite the Schrödinger equation as:

${\displaystyle -{\frac {\hbar ^{2}}{2m}}({\frac {d\xi }{dx}}{\frac {d}{d\xi }})\psi +{\frac {1}{2}}m\omega ^{2}({\sqrt {\frac {\hbar }{m\omega }}}\xi )^{2}\psi =\epsilon \hbar \omega \psi }$,

which further simplifies to:

${\displaystyle -{\frac {1}{2}}{\frac {d^{2}}{d\xi ^{2}}}\psi +{\frac {1}{2}}\xi ^{2}\psi =\epsilon \psi }$,

and after moving terms becomes:

${\displaystyle {\frac {d^{2}\psi }{d\xi ^{2}}}=(\xi ^{2}-2\epsilon )\psi }$.

For very large values of ${\displaystyle \xi }$ (i.e., at very large ${\displaystyle x}$), ${\displaystyle \xi ^{2}}$ becomes much bigger than ${\displaystyle 2\epsilon }$, so we can approximated this equation as:

${\displaystyle {\frac {d^{2}}{d\xi ^{2}}}\psi \approx \xi ^{2}\psi }$,

Which has the general solution:

${\displaystyle \psi (\xi )=h(\xi )e^{-{\frac {\xi ^{2}}{2}}}}$.

Now our goal is to find ${\displaystyle h(\xi )}$. Let us start by finding the first and second derivatives:

${\displaystyle \psi '(\xi )=[h'(\xi )-\xi h(\xi )]e^{-\xi ^{2}/2}}$

and

${\displaystyle \psi ''(\xi )=[h''(\xi )-2\xi h'(\xi )+(\xi ^{2}-1)h(\xi )]e^{-\xi ^{2}/2}}$.

Note that according to the Schrödinger equation, ${\displaystyle {\frac {d^{2}\psi }{d\xi ^{2}}}=(\xi ^{2}-2\epsilon )\psi }$, so we substitute to achieve:

${\displaystyle h''(\xi )-2\xi h'(\xi )+(2\epsilon -1)h(\xi )=0}$.

Let us try to find a solution in the form of power series of ${\displaystyle h(\xi ):}$

${\displaystyle h(\xi )=\sum _{j=0}^{\infty }a_{j}\xi ^{j}}$

By substituting this in the equation and collecting similar powers we get:

${\displaystyle \sum _{j=0}^{\infty }\xi ^{j}[(j+1)(j+2)a_{j+2}-2ja_{j}+(2\epsilon -1)a_{j}]=0}$.

We can conclude that the coefficient of each power of ${\displaystyle \xi }$ must be the same on both sides, thus we have:

${\displaystyle a_{j+2}={\frac {2j+1-2\epsilon }{(j+1)(j+2)}}a_{j}}$, or:
${\displaystyle {\frac {a_{j+2}}{a_{j}}}={\frac {2j+1-2\epsilon }{(j+1)(j+2)}}}$.

This recursion formula must terminate for some ${\displaystyle j=n}$[1], so that ${\displaystyle a_{n+2}=0}$. This will truncate either the series ${\displaystyle h_{even}}$ or ${\displaystyle h_{odd}}$ and the other must be zero from the beginning. This gives:

${\displaystyle \epsilon =n+{\frac {1}{2}}={\frac {E}{\hbar \omega }}}$,

Which is favorable because it gives the allowed energy levels for the system:

${\displaystyle E=\hbar \omega (n+{\frac {1}{2}})}$

The properly normalized eigenfunctions are

${\displaystyle \psi _{n}(x)={\frac {1}{\sqrt {2^{n}n!}}}\left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}e^{-{m\omega x^{2}/2\hbar }}H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right)}$.

The first four hermite polynomials are:

${\displaystyle h_{0}(x)=1\,}$
${\displaystyle h_{1}(x)=2\xi \,}$
${\displaystyle h_{2}(x)=4\xi ^{2}-2\,}$
${\displaystyle h_{3}(x)=8\xi ^{3}-12\xi \,}$

And the rest can be calculated using the generating function:

${\displaystyle G(\xi ,s)=e^{-s^{2}+2s\xi }=\sum _{n}h_{n}(\xi ){\frac {s^{n}}{n!}}}$,

Or alternatively through the Rodrigues formula:

${\displaystyle h_{n}(\xi )=(-1)^{n}e^{\frac {+\xi ^{2}}{2}}{\frac {d^{n}}{d\xi ^{n}}}e^{-\xi ^{2}}}$.

Reference

1. ^ Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.

Unsigned talk comment by Mahrud, 15:40, 15 December 2014.

I personally don't see the point of this mini-tutorial, as the point and the results are already summarized in black and white in section 1.1! This looks like an attempt to introduce the novice to solving the Schr eqn, Hermite polynomials, the Rodrigues formula (not wikilinked??? WHY??) etc... at an introductory level. It is really a protracted explanatory footnote to section 1.1 to somebody shaky in QM. Such sections are often put in hide boxes, and this, suitably condensed and wikilinked, could find its way in a hidebox at the end of section 1.1. However, It really detracts from the continuity and logical cohesion of the treatment... If sufficiently many editors assist in paring off repetitive statements in it and endorsed its insertion, in a hidebox, at the end of section 1.1, I could help stick it in. Cuzkatzimhut (talk) 16:33, 15 December 2014 (UTC)

## Possibly overlooked solutions - regarding multidimensional harmonic oscillators

The section "Example: 3D isotropic harmonic oscillator" correlates with the notion that the ground state has energy of 3/2. (Permit me to omit the factor ħω.) But, the following wave function [a] satisfies the wave equation, [b] normalizes, and [c] correlates with an energy of 1/2.

Ψ(r, θ, φ) ∝ r−1 exp(−(r2)/(s2)) Y0,0(θ, φ)

• Here, s has dimensions of length.

I am reluctant to try to edit the page itself, because …

• I think other people should set the scope for the page.
• I may be the discoverer of the above and other solutions (for isotropic quantum harmonic oscillators) [a] that people might say are "non-traditional" and [b] for which I find possibly physics-relevant applications. If my findings are noteworthy, perhaps someone[s] else should report them on Wikipedia.
• I have little experience directly editing Wikipedia pages. Previously, I edited parts of only 1 page ["IDIQ"].

Some notes.

• Regarding normalization, a relevant integral is ∫ Ψ* Ψ r2 dr. The r2 factor allows for normalization.
• For each odd positive integer N with N≥3, there exist "non-traditional" solutions having energies of 1/2, 3/2, …, (N−2)/2, as well as the "traditional" solutions with energies N/2, (N+2)/2, … .
• In some attempted research (in math, particle physics, astrophysics, and cosmology), I find use for non-traditional solutions such as I note above.
• Should someone want [a] more information about the math and/or [b] references to cite, … please feel free to consult (on-line [zero-cost download] or in print) the book "Theory of Particles plus the Cosmos." The math I reprise above is in section 1.2. The specific solution I show above paraphrases equation (1.2.68). Here are links. On-line - https://www.academia.edu/11329222/Theory_of_Particles_plus_the_Cosmos_Small_Things_and_Vast_Effects_Extract_-_Front_pages_through_Part_3_ Print - http://www.amazon.com/Theory-Particles-plus-Cosmos-Effects/dp/1508610339/
• Permit me to note that my attempted research also shows possible uses of similar findings correlating with linear representations (as well as, per above, with radial/spherical representations). For example, for N=3, I find useful the non-traditional solution for which nx = −1, ny = 0, and nz = 0. (book table 1.2.3 and section 2.1) This solution correlates with an energy of 1/2.
• People might state concerns about using quantum numbers that people might say are negative numbers. But, the mathematics "works." And, the overall energies are all non-negative. (And, the math may have applications.)

Thomasjbuckholtz (talk) 21:29, 15 March 2015 (UTC)

You should be congratulated for your reluctance to edit, especially given the controlling WP:NOR rule. This is a threshold you must not cross. Normalizability is not the be–all and end–all, but regularity at the origin, which you are forfeiting, is. Martin has analyzed the issue of the neglected δ-function involved. The irregular solution, which Dirac knows about in his book and dismisses on physical grounds, does not solve the Schroedinger eqn for a regular potential like the oscillator, but matches a δ-function add-on at the origin---and violates the virial theorem. While technically informative, and definitely not new with you!, the strictly formal phenomenon does not belong here and has little bearing on oscillators or physics. Cuzkatzimhut (talk) 23:52, 15 March 2015 (UTC)

Thanks for your comments of 15 March 2016. Perhaps it would be appropriate to revisit this topic. Happenings during the last ~1.5 years include the following.

• Correlating a subset of solutions (of the type of solutions to which my 2015 comment alludes) with the set of known elementary particles (and some properties of those particles).
• Offering possibilities that other solutions (of the type of solutions to which my 2015 comment alludes) correlate with yet-to-be-found elementary particles (and some properties of those particles).
• Offering possibilities that applications of the expanded particle set may correlate with explanations of dark matter, dark energy, changes in the rate of expansion of the universe, and other phenomena.
• Showing such results in a book published by Springer. - http://www.springer.com/book/9789462391659 - {Some notes: (1) Springer makes available (as a no-cost download) the math-intensive chapter (Chapter 2) of the book - http://www.springer.com/cda/content/document/cda_downloaddocument/9789462391659-c2.pdf?SGWID=0-0-45-1565387-p177736899 . (2) Regarding lists of known and proposed particles, (2a) Chapters 3 and 7 of the book provide such lists and (2b) A YouTube-based video draft [I plan some touch-up] of a planned talk [for an American Physical Society Far West Section meeting in October 2016] summarizes the lists. - The page https://thomasjbuckholtz.wordpress.com/mathematical-physics/ currently provides a link to that draft.}

Perhaps, the above establishes the following.

• Some of the solutions have relevance to physics.
• Physics-relevance does not necessarily correlate with quantizing phenomena that correlate with traditional classical physics.
• Physics-relevance does not necessarily correlate with the Schrodinger equation.

Perhaps, you would like to consider establishing a new page (or a new section on the current page) to discuss these solutions and their physics-relevance. That relevance can include (at least) providing an analog - for elementary particles - to the periodic table for elements (during at least the era in which people had yet to formulate enough nuclear and atomic physics to understand atoms and isotopes). (Of course, paralleling history correlating with the periodic table, the extent to which solutions [that may correlate with new elementary particles] correlate with actual elementary particles is unresolved.)

Here are some additional notes.

• It is possible to consider that some of the applications of these solutions pertain to 'something other than space-time.' In this case, the book discusses using Poincare group symmetries to (in effect) map solutions into concepts correlating with space-time coordinates.
• On the other hand, at least for each non-zero-mass elementary particle, the work shows the following two concepts. (First) A solution that normalizes only in the limit of s squared approaching zero. (Here, s is the same as the s in my previous comment. The new book uses the symbol η, not s.) People might say that this solution correlates with a particle's participation in a point-like interaction vertex. (Second) A (different) solution that normalizes for any positive value of s squared. People might say that setting s to be proportional to the proper time that elapses after an interaction vertex correlates with propagation of a field associated with the particle. (The previous sentence assumes that the particle emerges from the interaction vertex.)

--Thomasjbuckholtz (talk) 22:41, 1 October 2016 (UTC)

## Commutators

"The following commutators can be easily obtained by substituting the canonical commutation relation," These follow immediately (mathematically) from the preceding lines. Why give the impression that they depend on something presented as a physics concept in another article. Keith McClary (talk) 20:40, 31 January 2016 (UTC)

Can you make clear what, exactly, you are proposing, and why? The reader who still needs reminding of where the following commutators came from is steered to use the canonical commutation relations linked to redo the elementary linear algebra to produce them. Cuzkatzimhut (talk) 21:07, 31 January 2016 (UTC)

## Question: Units in Wave Eqn for Quantum Harmonic Oscillator

Every formulation that I have seen for the quantum harmonic oscillator wave eqn has a term like
(mω/πћ)(1/4). Yet the units of this term before the root extraction are kg*sec-1/(m2*kg*sec-1), which cancels out to m-2. After the root is taken, the units of the term are m-1/2. Since the units cancel out in all the other terms of the wave eqn, the results of ψn(x), assuming x in meters or femtometers, have units 1/√m or 1/√fm. How is that possible? — Preceding unsigned comment added by CElliott316 (talkcontribs) 19:47, 15 March 2016 (UTC)

? What exactly is the problem? the probability density has then inverse units of length, so integrated over dx, it yields a dimensionless probability. You did notice, didn't you, the cube in the 3d formula in the suitable section? Taking a physics course should help. Cuzkatzimhut (talk) 20:08, 15 March 2016 (UTC)