User:Jmkinder1/sandbox

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WKB Approximation

There are some issues with the mathematical typesetting on WKB approximation. Two formulae do not render at all (at least with my browser settings) and others look strange.

The problem: Just typing primes and superscripts.

$S_{1}''+S_{0}'^{2}$

On my browser, it appears the rendering program tries to attach the superscript "2" to the apostrophe in the second term. When I generate my own MathJax HTML and view it on my browser, this renders properly. On Wikipedia, it appears not to.

Solution 1: Use \prime as superscript.

$S_{1}^{\prime \prime }+S_{0}^{\prime }{}^{2}$

Solution 2: Use {}^2 for superscript.

$S_{1}''+S_{0}'{}^{2}$

Solution 3: Place function and derivatives in a group.

${S_{1}''}+{S_{0}'}^{2}$

Solution 1 seems best.

Attempted Solution to Page Text

This example comes from the text of Carl M. Bender and Steven Orszag. Consider the second-order homogeneous linear differential equation $$\epsilon ^{2}{\frac {d^{2}y}{dx^{2}}}=Q(x)y,$$ where $Q(x)\neq 0$. Substituting $$y(x)=\exp \left[{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}(x)\right]$$ results in the equation $$\epsilon ^{2}\left[{\frac {1}{\delta ^{2}}}\left(\sum _{n=0}^{\infty }\delta ^{n}S_{n}^{\prime }\right)^{2}+{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}^{\prime \prime }\right]=Q(x).$$

To leading order in ϵ (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as Failed to parse (Conversion error. Server ("/media/api/rest_") reported: "Cannot get mml. TeX parse error: Double exponent: use braces to clarify"): {\displaystyle {\frac {\epsilon ^{2}}{\delta ^{2}}}S_{0}^{\prime }^{2}+{\frac {2\epsilon ^{2}}{\delta }}S_{0}^{\prime }S_{1}^{\prime }+{\frac {\epsilon ^{2}}{\delta }}S_{0}^{\prime \prime }=Q(x).}

In the limit δ → 0, the dominant balance is given by Failed to parse (Conversion error. Server ("/media/api/rest_") reported: "Cannot get mml. TeX parse error: Double exponent: use braces to clarify"): {\displaystyle {\frac {\epsilon ^{2}}{\delta ^{2}}}S_{0}^{\prime }^{2}\sim Q(x).}

So δ is proportional to ϵ. Setting them equal and comparing powers yields Failed to parse (Conversion error. Server ("/media/api/rest_") reported: "Cannot get mml. TeX parse error: Double exponent: use braces to clarify"): {\displaystyle \epsilon ^{0}:\quad S_{0}^{\prime }^{2}=Q(x),} which can be recognized as the eikonal equation, with solution $$S_{0}(x)=\pm \int _{x_{0}}^{x}{\sqrt {Q(x')}}\,dx'.$$

Considering first-order powers of ϵ fixes $$\epsilon ^{1}:\quad 2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }=0.$$ This has the solution $$S_{1}(x)=-{\frac {1}{4}}\ln Q(x)+k_{1},$$ where k₁ is an arbitrary constant.

We now have a pair of approximations to the system (a pair, because S₀ can take two signs); the first-order WKB-approximation will be a linear combination of the two: $$y(x)\approx c_{1}Q^{-{\frac {1}{4}}}(x)\exp \left[{\frac {1}{\epsilon }}\int _{x_{0}}^{x}{\sqrt {Q(t)}}\,dt\right]+c_{2}Q^{-{\frac {1}{4}}}(x)\exp \left[-{\frac {1}{\epsilon }}\int _{x_{0}}^{x}{\sqrt {Q(t)}}\,dt\right].$$

Higher-order terms can be obtained by looking at equations for higher powers of δ. Explicitly, $$2S_{0}^{\prime }S_{n}^{\prime }+S_{n-1}^{\prime \prime }+\sum _{j=1}^{n-1}S_{j}^{\prime }S_{n-j}^{\prime }=0$$ for n ≥ 2.

Arctic Circle

In Arctic circle there are conflicting definitions of sunrise and sunset, and hence, of the location of the Arctic Circle. One part of the article says polar night is when the sun sets below the horizon and does not rise again for 24 hours or more, while another says it is when the center of the sun does not dip below the horizon. One of the references on "polar night" seems to give proper astronomical definitions, but the article could use more primary sources.

It seems this is the offending statement:

The Arctic Circle marks the southernmost latitude at which, on the December solstice, the shortest day of the year in the northern hemisphere, the sun will not rise all day, and on the June solstice, the longest day of the year in the northern hemisphere, the sun will not set.

According to Ref. 2, the Arctic Circle is where the center of the sun meets the horizon on these days — not where is "does not rise" or "does not set". Ref. 2 is a secondary source. It would be nice to have a definition from a primary source, like the International Astronomers Union or the National Geographic Society instead.

The next statement is false, according to Ref. 2:

These phenomena are referred to as polar night and midnight sun respectively,

The definition of polar night refers to civil twilight, not sunset, and polar night is not experienced at the Arctic circle.

Moreover, in Polar night, there is a statement that requires clarification:

"Night" is understood as the center of the Sun being below a free horizon. Since the atmosphere refracts sunlight, the polar day is longer than the polar night, and the area that is affected by polar night is somewhat smaller than the area of midnight sun. The polar circle is located at a latitude between these two areas, at approximately 66.5°. While it is day in the Arctic Circle, it is night in the Antarctic Circle, and vice versa.

The author has subjectively defined "night" in contradiction to common sense and other sources.

Chain physics

I just read Neal Stephenson's _Seveneves._ I was interested in the physics of chains described in the book, but had little luck tracking down much information. The Wikipedia article on "Whip cracking" had a couple of outdated links. There was a little more on the "Chain fountain", but this seems outdated and incomplete, too.

Atom chip

The Wikipedia article on "atom chip" refers to an Intel processor. However, an "atom chip" is also the name of a MEMS device used to manipulate Bose-Einstein condensates in cold atom physics, as described here: https://www.imperial.ac.uk/centre-for-cold-matter/research/cold-atoms/interferometer/.

This technology was part of a recent experiment that achieved an incredibly low temperature of 38 pK, as described here (https://physics.aps.org/articles/v14/119?utm_campaign=weekly&utm_medium=email&utm_source=emailalert) and in the accompanying Physical Review Letters.

It seems like a new page should be created for the "atom chip" of cold atom physics and a disambiguation page should be created for "atom chip".

Many-body problem

This article on the quantum many body problem needs a more descriptive lead. It has a lot of references and resources, but it does not orient the reader to the problem or explain much about it. Jmkinder1 (talk) 19:52, 9 February 2021 (UTC)

Matrix Product States

Another article that could use improvement is the one on matrix product states. It is related to tensor networks, and it needs a lot of work. I have extensive sources for this article already.

Jmkinder1 (talk) 07:43, 25 January 2021 (UTC)

Hubbard Model

One article I will edit is The Hubbard Model. I like the topic, know a bit about it, and the article seems only partially formed. A connection to quantum science might be recent atomic lattice simulations of the model, and the use of quantum annealing to solve it.

Jmkinder1 (talk) 06:36, 17 January 2021 (UTC)

Tensor Network

Another article I will edit is on tensor network states. The page does not exist! With all of their use in quantum information and quantum computing, I find this surprising. Jmkinder1 (talk) 10:08, 19 January 2021 (UTC)

I created a draft page in my sandbox.

It has sense been moved to the main space: Tensor network.

Jmkinder1 (talk) 19:55, 9 February 2021 (UTC)

From "Quantum Tensor Networks in a Nutshell" ...

1. Tensor network state is a multilinear ansatz --- linear in each variational parameter.
2. Deutsch and Feynman used tensor network notation in describing quantum circuits and quantum computers.
3. Varieties of tensor networks
   • matrix product states
   • tensor trains (different than MPS?)
   • snake states
   • tree tensor networks
   • multiscale entanglement renormalization ansatz
   • projected entangled pair states
   • correlator product states (equivalent to neural network quantum states)

Degrees of freedom (physics and chemistry)

There is a contradiction between the text and a table in Degrees of freedom (physics and chemistry). It seems the confusion comes from kinetic degrees of freedom versus total degrees of freedom. There is a lot of material in the talk page, and it looks like a heated (but uninformed) dispute. I think it would be helpful to clean up the discussion — or at least eliminate the contradiction.

The article on the Equipartition theorem seems better written and less contentious. Perhaps it would be wise to draw from this article in improving the other.

Piezoresistive effect

The page on the Piezoresistive effect is seriously lacking in citations, and it seems the mathematical notation is not consistent throughout.

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