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Cantore Arithmetic is able to state:  The LibreTexts libraries are Powered by NICE CXone Expert and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Legal. Accessibility Statement For more information contact us at info@libretexts.org.:  2.1: Examples of PDE - Mathematics LibreTexts

• Klein-Gordon equation
  ∇2𝑢−1𝑐2∂2𝑢∂𝑡2+𝜆2𝑢=0$${\mathrm{\nabla }}^{2}u-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}}+{\lambda }^{2}u=0$$
  Relativistic quantum particles,|𝑢|2$|u{|}^2$ has a probability interpretation.

1.  LA X

a.  Before the 1930s, US airports used a two-letter abbreviation and "LA" served as the designation for Los Angeles Airport. With rapid growth in the aviation industry, in 1947, the identifiers were expanded to three letters, and "LA" received an extra letter to become "LAX". The "X" does not have any specific meaning.

2.  SF O

b.  The 'S' and 'F' come from the first letters of San Francisco. As the museum shares, “Some of the two-letter codes that existed before that simply had an 'X' added for international airports, such as LAX and PHX.

3.  Equation word Rolex[Watch[watch[two[2[Hands[four[4]]]]]]

4.  The Ocean equated word Pond

What's the meaning of the phrase 'Across the pond’?

anchor

The pond refers to the North Atlantic Ocean. Going across the pond refers to travel between Europe (especially the UK) and the Americas (especially the USA).

What's the origin of the phrase 'Across the pond’?

The OED defines a pond as ‘a small body of still water of artificial formation’.

Given that the North Atlantic Ocean is a large stormy body of water of natural formation, there was clearly some irony at work in the naming of ‘The Pond’. We might guess it to be a fairly recent coinage.

In fact, that naming of the ocean came about not all that long after it was first sailed across. Whoever coined the Pond as the name for the Atlantic must have had a more than usually acute sense of irony, the journey across it then being measured in months.

Initially, ‘The Pond’ was the name given to any sea. In 1612 Bishop Joseph Hall called the collective oceans of the world ‘the great pond’ in his religious text Contemplations.

Another 17th century reference to a stretch of sea being called the pond is found in a pamphlet entitled Time’s Alterations, 1642: 

Well met, my Lord It seems that you have taken flight over the great Pond, pray what newes in England?  The text is a record of a conversation between two courtiers of Charles I, Lord John Finch and Sir Francis Windebank. Although the text above has been taken by some as the first use of the name the great pond for the Atlantic, this isn’t correct. Neither Finch nor Windebank ever travelled to America. They fell out of favour at the English court and fled to Holland and France. The pond they referred to was probably the English Channel.

Before we get to the Pond we need to take a diversion to an earlier name for the Atlantic – The Herring Pond. This is first used in John Dunton’s Letters from New England, 1686:

Tomorrow if a gale presents we saile on for a new-world (for soe they call America): at my first arrival I’le send an account of the wonders I met on the Great Herring-Pond.

Over time, the Great Herring Pond became shortened to the Great Pond and later the Pond. The first use of this shorter form is found in the Royal Gazette, which was a pro-British newspaper printed in New York by James Rivington in 1780:

Then Jack was sent across the Pond To take her in the rear, Sir.

It may well be that ‘the Pond’ was used colloquially for some time before Rivington was writing but we have no evidence of it. Nevertheless the term ‘across the pond’ has been with us for longer than we might have expected. 

The history of “Across the pond” in printed materials

Trend of across the pond in printed material over time

Related phrases and meanings

American origin

Transport and travel

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Acronyms

Across the board

Across the pond

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Act the giddy goat

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Gary Martin

Writer and researcher on the origins of phrases and the creator of the Phrase Finder website. Over the past 26 years more than 700 million of his pages have been downloaded by readers. He is one of the most popular and trusted sources of information on phrases and idioms. 

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5.1: The Difference Between Relativistic and Non-Relativistic Quantum Mechanics

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• Niels Walet
• University of Manchester

One of the key points in particles physics is that special relativity plays a key role. As you all know, in ordinary quantum mechanics we ignore relativity. Of course people attempted to generate equations for relativistic theories soon after Schrödinger wrote down his equation. There are two such equations, one called the Klein-Gordon and the other one called the Dirac equation.

The structure of the ordinary Schrödinger equation of a free particle (no potential) suggests what to do. We can write this equation as

𝐻̂𝜓=12𝑚𝑝⃗2𝜓=𝑖ℏ∂∂𝑡𝜓.$$\hat{H}\psi =\frac{1}{2m}{\overrightarrow{p}}^2\psi =i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi .$$

This is clearly a statement of the non-relativistic energy-momentum relation, 𝐸=12𝑚𝑣2$E={\displaystyle \frac{1}{2}}m{v}^2$, since a time derivative on a plane wave brings down a factor energy. Remember, however, that 𝑝⃗$\overrightarrow{p}$ as an operator also contains derivatives, 

𝑝⃗=ℏ𝑖∇⃗.$$\overrightarrow{p}=\frac{\hslash }{i}\overrightarrow{\mathrm{\nabla }}.$$

A natural extension would to use the relativistic energy expression,

𝐻̂𝜓=𝑚2𝑐4+𝑝⃗2𝑐2‾‾‾‾‾‾‾‾‾‾‾√𝜓=𝑖ℏ∂∂𝑡𝜓.(5.1.1)$$\begin{array}{}\text{(5.1.1)}& \hat{H}\psi =\sqrt{m^2{c}^4+{\overrightarrow{p}}^2{c}^2}\psi =i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi .\end{array}$$

This is a nonsensical equation, unless we specify how to take the square root of the operator. The first attempt to circumvent this problem, by Klein and Gordon, was to take the square of Equation 5.1.1$\text{5.1.1}$ to generate the Klein-Gordon Equation (Equation 5.1.2$\text{5.1.2}$):

Klein-Gordon Equation

(𝑚2𝑐4+𝑝⃗2𝑐2)𝜓=−ℏ2∂2∂𝑡2𝜓.(5.1.2)$$\begin{array}{}\text{(5.1.2)}& \left(m^2{c}^4+{\overrightarrow{p}}^2{c}^2\right)\psi =-{\hslash }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\psi .\end{array}$$

This is an excellent equation for spin-less particles or spin one particles (bosons), but not to describe fermions (half-integer spin), since there is no information about spin is in this equation. This needs careful consideration, since spin must be an intrinsic part of a relativistic equation!

Dirac realized that there was a way to define the square root of the operator. The trick he used was to define four matrices 𝛼$\alpha$, 𝛽$\beta$ that each have the property that their square is one, and that they anticommute,

2𝛼𝑖𝛼𝑖𝛽𝛽𝛼𝑖𝛽+𝛽𝛼𝑖𝛼𝑖𝛼𝑗+𝛼𝑗𝛼𝑖=𝐼,=𝐼=0=0𝑖≠𝑗.(5.1.3)(5.1.4)(5.1.5)(5.1.6)$$\begin{array}{}\text{(5.1.3)}& 2{\alpha }_i{\alpha }_i& =I,& \text{(5.1.4)}& \beta \beta & =I\text{(5.1.5)}& {\alpha }_i\beta +\beta {\alpha }_i& =0\text{(5.1.6)}& {\alpha }_i{\alpha }_j+{\alpha }_j{\alpha }_i& =0i\ne j.\end{array}$$

This then leads to an equation that is linear in the momenta – and very well behaved:

Dirac Equation

(𝛽𝑚𝑐2+𝑐𝛼⃗⋅𝑝⃗)Ψ=𝑖ℏ∂∂𝑡Ψ$$(\beta m{c}^2+c\overrightarrow{\alpha }\cdot \overrightarrow{p})\mathrm{\Psi }=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }$$

Note that the minimum dimension for the matrices in which we can satisfy all conditions is 4$4$, and thus Ψ$\mathrm{\Psi }$ is a four-vector! This is closely related to the fact that these particles have spin.

Let us investigate this equation a bit further. One of the possible forms of 𝛼𝑖${\alpha }_i$ and 𝛽$\beta$ is

𝛼𝑖=(0𝜎𝑖𝜎𝑖0),𝛽=(𝐼00−𝐼),$${\alpha }_i=\left(\begin{array}{cc}0& {\sigma }_i\\ {\sigma }_i& 0\end{array}\right),\beta =\left(\begin{array}{cc}I& 0\\ 0& -I\end{array}\right),$$

where 𝜎𝑖${\sigma }_i$ are the two-by-two Pauli spin matrices

𝜎1=(0110),𝜎2=(0𝑖−𝑖0),𝜎3=(100−1).$${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

(These matrices satisfy some very interesting relations. For instance

𝜎1𝜎2=𝑖𝜎3,𝜎2𝜎1=−𝑖𝜎3,𝜎2𝜎3=𝑖𝜎1,$${\sigma }_1{\sigma }_2=i{\sigma }_3,{\sigma }_2{\sigma }_1=-i{\sigma }_3,{\sigma }_2{\sigma }_3=i{\sigma }_1,$$

etc. Furthermore 𝜎2𝑖=1${\sigma }_i^2=1$.)

Once we know the matrices, we can try to study a plane-wave solution (i.e., free particle):

Ψ(𝑥⃗,𝑡)=𝑢(𝑝⃗)𝑒𝑖(𝑝⃗⋅𝑥⃗−𝐸𝑡)/ℏ.$$\mathrm{\Psi }(\overrightarrow{x},t)=u(\overrightarrow{p})e^{i(\overrightarrow{p}\cdot \overrightarrow{x}-Et)/\hslash }.$$

(Note that the exponent is a “Lorentz scalar”, it is independent of the Lorentz frame!).

If substitute this solution we find that 𝑢(𝑝⃗)$u(\overrightarrow{p})$ satisfies the eigenvalue equation

⎛⎝⎜⎜⎜⎜𝑚𝑐20𝑝3𝑐𝑝1𝑐+𝑖𝑝2𝑐0𝑚𝑐2𝑝1𝑐−𝑖𝑝2𝑐−𝑝3𝑐𝑝3𝑐𝑝1𝑐+𝑖𝑝2𝑐−𝑚𝑐20𝑝1𝑐−𝑖𝑝2𝑐−𝑝3𝑐0−𝑚𝑐2⎞⎠⎟⎟⎟⎟⎛⎝⎜⎜⎜⎜𝑢1𝑢2𝑢3𝑢4⎞⎠⎟⎟⎟⎟=𝐸⎛⎝⎜⎜⎜⎜𝑢1𝑢2𝑢3𝑢4⎞⎠⎟⎟⎟⎟.$$\left(\begin{array}{cccc}m{c}^2& 0& p_{3}c& p_{1}c-i{p}_{2}c\\ 0& m{c}^2& p_{1}c+i{p}_{2}c& -p_{3}c\\ p_{3}c& p_{1}c-i{p}_{2}c& -m{c}^2& 0\\ p_{1}c+i{p}_{2}c& -p_{3}c& 0& -m{c}^2\end{array}\right)\left(\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right)=E\left(\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right).$$

The eigenvalue problem can be solved easily, and we find the eigenvalue equation

(𝑚2𝑐4+𝑝2𝑐2−𝐸2)2=0$$(m^2{c}^4+p^2{c}^2-E^2{)}^2=0$$

which has the solutions 𝐸=±𝑚2𝑐4+𝑝2𝑐2‾‾‾‾‾‾‾‾‾‾‾√$E=\pm \sqrt{m^2{c}^4+p^2{c}^2}$. The eigenvectors for the positive eigenvalues are

⎛⎝⎜⎜⎜⎜10𝑝3𝑐/(𝐸+𝑚𝑐2)(𝑝1𝑐−𝑖𝑝2𝑐)/(𝐸+𝑚𝑐2)⎞⎠⎟⎟⎟⎟, and ⎛⎝⎜⎜⎜⎜01(𝑝1𝑐+𝑖𝑝2𝑐)/(𝐸+𝑚𝑐2)−𝑝3𝑐/(𝐸+𝑚𝑐2)⎞⎠⎟⎟⎟⎟,$$\left(\begin{array}{c}1\\ 0\\ p_{3}c/(E+m{c}^2)\\ (p_{1}c-i{p}_{2}c)/(E+m{c}^2)\end{array}\right)\text{, and }\left(\begin{array}{c}0\\ 1\\ (p_{1}c+i{p}_{2}c)/(E+m{c}^2)\\ -p_{3}c/(E+m{c}^2)\end{array}\right),$$

with similar expressions for the two eigenvectors for the negative energy solutions. In the limit of small momentum the positive-energy eigenvectors become

⎛⎝⎜⎜⎜⎜1000⎞⎠⎟⎟⎟⎟, and ⎛⎝⎜⎜⎜⎜0100⎞⎠⎟⎟⎟⎟,$$\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right)\text{, and }\left(\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right),$$

and seem to denote a particle with spin up and down. We shall show that the other two solutions are related to the occurrence of anti-particles (positrons).

Just as photons are the best way to analyze (decompose) the electro-magnetic field, electrons and positrons are the natural way way to decompose the Dirac field that is the general solution of the Dirac equation. This analysis of a solution in terms of the particles it contains is called (incorrectly, for historical reasons) “second quantisation”, and just means that there is a natural basis in which we can say there is a state at energy 𝐸$E$, which is either full or empty. This could more correctly be referred to as the “occupation number representation” which should be familiar from condensed matter physics. This helps us to see how a particle can be described by these wave equations. There is a remaining problem, however!

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