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Some people claim that the single most important conceptual point in all of quantum mechanics is that "probability amplitudes" (inner products) can be complex, as opposed to strictly positive. But in the earlier theory of classical wave mechanics, an inherent feature of wave amplitudes is that they are always positive. So from this perspective, calling a quantum-mechanical inner product a probability "amplitude" seems like uniquely misleading terminology. What is the historical origin of the term?

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    $\begingroup$ Maybe a question for History of Science and Mathematics? $\endgroup$ – Avantgarde Aug 4 '17 at 4:21
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    $\begingroup$ I'm voting to close this question as off-topic because this is about etymology of a word. $\endgroup$ – Qmechanic Aug 4 '17 at 4:53
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    $\begingroup$ @Qmechanic The community consensus seems to be that history of physics questions are on-topic, so I'm voting to reopen. $\endgroup$ – tparker Aug 4 '17 at 5:35
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    $\begingroup$ @Qmechanic Unilateral closure seems rather too strong for this one, but if it was to be closed it could just as well have been migrated to hsm. $\endgroup$ – Emilio Pisanty Aug 4 '17 at 6:37
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    $\begingroup$ @MoziburUllah and others: On/off-topic is in principle an absolute (as opposed to a relative) notion. If one finds another question that is off-topic, please vote-to-close or flag. $\endgroup$ – Qmechanic Aug 4 '17 at 12:03
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As stated in the comments amplitude is a term used in descriptions of sinusoidal functions, where the y variable is the "amplitude" in various descriptions and the x may be space or time. It is used in describing waves in general, as waves are =modeled with differential equations called wave equations.

ampl

A sinusoidal curve

  1. Peak amplitude

2.Peak-to-peak amplitude

  1. Root mean square amplitude

4.Wave period (not an amplitude)

The usage in physics follows the convention of wave descriptions.You ask:

So from this perspective, calling a quantum-mechanical inner product a probability "amplitude" seems like uniquely misleading terminology. What is the historical origin of the term?

The historical reason comes from the interference effects seen in the probability density distributions from quantum mechanical systems, clearly seen as interference patterns in double slit experiments, characteristic of wave solutions.

dblsslit

Particularly the single electron or photon at a time, established that the sinusoidal solutions of the quantum mechanical equations and the rule of using the complex conjugate squared for describing the probability of measuring a particle at an (x,y,z) was a solid prediction, and that the postulates for choosing the quantum mechanical solutions of boundary conditions problems were valid.

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