Why do we need the position operator?

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Discussion Overview

The discussion revolves around the role and necessity of the position operator in quantum mechanics, particularly in relation to the probability density derived from the wavefunction and the expectation value of position. It explores theoretical concepts related to quantum mechanics.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant explains that the probability density |Ψ|² provides the likelihood of finding a particle in a specific location when integrated over a range.
  • Another participant clarifies that the position operator relates to the expectation value of position, distinguishing it from the probability density.
  • A third participant notes that the integrand involving the position operator yields the mean position, further emphasizing the difference between probability density and expectation value.
  • There is a reiteration of the distinction between probability density and expectation value, with one participant expressing understanding after this clarification.

Areas of Agreement / Disagreement

Participants generally agree on the distinction between probability density and expectation value, but there is some contention regarding the implications and necessity of the position operator.

Contextual Notes

The discussion does not resolve the broader implications of the position operator's necessity or its role in quantum mechanics, leaving some assumptions and definitions unaddressed.

Phys12
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As I understand it, |Ψ|2 gives us the probability density of the wavefunction, Ψ. And when we integrate it, we get the probability of finding the particle at whichever location we desire, as set by the limits of the integration. But when we use the position operator, we have integrand Ψ*xΨ dx, which gives us the probability of finding the particle at a given place.

It seems to me as though, both of the above operations are doing the exact same thing. Then why do we have a position operator?
 
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Phys12 said:
which gives us the probability of finding the particle at a given place
No. It gives an expectation value.

One is a probability density, the other is an expectation value. Different things ! Just like a Gauss distribution and the corresponding average are different things.
 
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The latter integrand will give you the mean position.
 
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BvU said:
No. It gives an expectation value.

One is a probability density, the other is an expectation value. Different things ! Just like a Gauss distribution and the corresponding average are different things.
Oh...I see, because it's the *expectation value*, got it. Thank you!
 

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