Wave function -- Why is there an imaginary part?

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Discussion Overview

The discussion revolves around the nature of the wave function in quantum mechanics, specifically addressing the presence of an imaginary part and its implications. Participants explore the relationship between complex numbers and wave functions, touching on both quantum and classical mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question the necessity of an imaginary part in the wave function, suggesting that if waves are real, the presence of complex numbers is puzzling.
  • Others argue that the squared modulus of the wave function, which represents a probability distribution, is what holds physical significance, while the wave function itself lacks direct physical meaning.
  • It is noted that ordinary waves can also be represented using complex numbers, indicating that this is not exclusive to quantum mechanics.
  • One participant emphasizes that complex numbers, including imaginary numbers, are as "real" as other mathematical constructs like vectors and matrices, challenging the notion of their "imaginary" status.
  • A technical perspective is provided, suggesting that the requirement for continuous transformations between quantum states necessitates the use of imaginary numbers, drawing parallels to the utility of negative numbers in certain contexts.
  • Another participant points out that classical mechanics can also utilize complex wave functions, suggesting that the use of complex numbers is not limited to quantum mechanics.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation and significance of the imaginary part of the wave function. There is no consensus on whether the imaginary component is essential or merely a mathematical convenience.

Contextual Notes

Some arguments depend on the definitions of "real" and "imaginary," and the discussion does not resolve the implications of using complex numbers in different physical contexts.

preitiey
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If wave is a real concept, then why we have a complex(imaginary) part associated with the wave function?
 
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It is the squared modulus of the wave function (= a probability distribution) that is of physical interest, since it predicts the likelihood that physical observables assume certain values. The wave function itself cannot be measured and, as far as I know, does not have a physical meaning.
 
preitiey said:
If wave is a real concept, then why we have a complex(imaginary) part associated with the wave function?

An imaginary number is not imaginary. That's just a word someone dreamt up one day! Complex numbers are just as "real" as vectors, matrices and continuous functions.
 
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Technically its got to do with the requirement for continuous transformations between quantum states. If a system is in a state and one second later is in another state then we reasonably expect that after half a second it went through some state in getting there. It turns out if you require that then so called imaginary numbers are required:
http://www.scottaaronson.com/democritus/lec9.html

BTW they are no more or less imaginary than say negative numbers. You can't point to a negative number of ducks for example. But if you owe someone two ducks then saying you have -2 ducks is very convenient. Same with imaginary numbers. You can't point to square root -1 (doubly so since you can't even point to -1 of anything) but in modelling some things its very convenient to introduce it - QM being a good example.

Thanks
Bill
 
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Classical mechanics can also be described in terms of complex wave functions, so they aren't inherently quantum mechanical things.
 

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