Stranger Fruit

Hamiltonians and Quaternions

The Ask A ScienceBlogger question for this week inspired me to revisit this post I made on April 22 2005.

i-37d148050ba41166ad3baa563910cf23-hamilton.jpeToday in 1827 the Irish mathematician, physicist, and astronomer William Rowan Hamilton presented his Theory of systems of rays a work that brought together mechanics, optics and mathematics and helped in establishing the wave theory of light. In addition, this year marks the bicentenary of Hamilton?s birth.

Hamilton sticks in my mind for two reasons. Firstly, nearly twenty years ago as a sophomore zoology major, I took a course in quantum mechanics and loved it. So much so, that the professor – whose name I unfortunately forget – tried to persuade me to change majors and specialize in QM. I didn?t (obviously) but still have a soft spot for that field. Hamilton was the originator of the Hamiltonian operator (H) featured in this equation that some out there may recognize:

i-5a90975b9531d5f71bb427dcdf66f357-swe.png

Secondly, Hamilton discovered quaternions in 1843. According to the story he told, on October 16 Hamilton was out walking along the Royal Canal in Dublin with his wife when the solution suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Brougham Bridge (now called Broom Bridge). While the original carving no longer exists, there is a plaque which reads”

Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
i2 = j2 = k2 = i j k = -1

& cut it on a stone of this bridge

To this day, mathematicians make pilgrimage to the bridge. I – though no mathematician – have been there. Today, quaternions are in use by computer graphics, control theory, signal processing and orbital mechanics.

Hamilton described the discovery in a letter to his son Archibald. In it, he states:

But on the 16th day of the same month – which happened to be a Monday, and a Council day of the Royal Irish Academy – I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse – unphilosophical as it may have been – to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely,

i2 = j2 = k2 = ijk = -1

which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away. A more durable notice remains, however, on the Council Books of the Academy for that day (October 16th, 1843), which records the fact, that I then asked for and obtained leave to read a Paper on Quaternions, at the First General Meeting of the session: which reading took place accordingly, on Monday the 13th of the November following.

This is why I love Victorian science! There?s an air of joy in the discovery. I?ll end with a quote from Hamilton:

?Time is said to have only one dimension, and space to have three dimensions. ? The mathematical quaternion partakes of both these elements; in technical language it may be said to be ?time plus space?, or ?space plus time?: and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be.?

Comments

  1. #1 Robert P.
    June 10, 2006

    About a dozen years ago, the question occurred to me: “what name did Hamilton give to the function that we call the Hamiltonian” ? So I looked up Hamilton’s 1835 and 1837 papers, and found that he didn’t name it anything in particular, but he did denote it with the symbol “H” !

    He also refers to “kinetic energy” as “living force”, which is a literal translation of Leibniz’ “vis viva”, but which gives some passages in the papers a rather disturbingly vitalistic tone when quoted out of context.

  2. #2 John Lynch
    June 10, 2006

    I remember learning it as Hop – the Hamiltonian Operator – rather than just H.

    > which gives some passages in the papers a rather
    > disturbingly vitalistic tone when quoted out of context.

    Better not let ID supporters read the papers then!

  3. #3 Roman Werpachowski
    June 11, 2006

    About a dozen years ago, the question occurred to me: “what name did Hamilton give to the function that we call the Hamiltonian” ? So I looked up Hamilton’s 1835 and 1837 papers, and found that he didn’t name it anything in particular, but he did denote it with the symbol “H” !

    It’s rather obvious that he was counting on other people to name it after him, then. This is still one level of immodesty below the discoverer of the Quantum Hall Effect, von Klitzing, who named a particular combination of physical constants… the von Klitzing constant! I heard him use this name in his lecture.

  4. #4 Davis
    June 12, 2006

    I remember learning it as Hop – the Hamiltonian Operator – rather than just H.

    Probably to distinguish it from the Hamiltonian in the sense of classical mechanics. Which, of course, was the version Hamilton devised (since QM was still a long way off).

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