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I have read different proofs of the uncertainty principle. My questions are:

  1. The principle depends on a theory of physics (quantum mechanics). Correct?
  2. Given the theory, mathematics is used to come up with the inequality $\Delta x \Delta p\ge\displaystyle\frac{\hbar}{2}$. Correct?

So if the 2 statements are correct, then the uncertainty principle is not a mathematical proof, but an inequality derived from a theory of physics. Correct?

So if one asserts the physical theory of quantum mechanics, then the uncertainty principle inequality follows. It has been verified through experiment and observation which supports the physical theory.

My point is that the uncertainty principle is not a mathematical proof at all. It is one of the mathematical expressions of the theory. Just like the equations for the general theory of relativity are the mathematical expression for that theory.

The universe did not have to behave this way, but it does. Another universe could not behave this way and the uncertainty principle inequality would not apply to that universe. Correct?

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  • $\begingroup$ You make assumptions in every mathematical proof. That doesn't prevent them from being mathematical proofs. $\endgroup$
    – aquirdturtle
    Commented Nov 5, 2015 at 6:16
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    $\begingroup$ If you multiply the mean width of a pulse $s(t)$ with the mean width of its Fourier transform $F(s(t))$ you will see that $width(s(t))*width(F(s(t))) > k$, where $k$ is a constant. This is the uncertainty principle in its most pure mathematical form. $\endgroup$
    – Energizer777
    Commented Nov 5, 2015 at 7:59
  • $\begingroup$ A mathematical derivation of a physical principle is not always possible, since there's always need for empirical input. But in the case of Heisenberg's uncertainty principle, there's a purely mathematical way of seeing it, namely that: the wavefunctions that we obtain from Schrödinger's equation for position and momentum, are conjugate to one-another, meaning one can be obtained as the fourier transform of the other. This in turn means that if one of your wavefunctions has compact support, its fourier transform cannot, thus translating to complete uncertainty, as in the value can be anything. $\endgroup$
    – Ellie
    Commented Nov 5, 2015 at 9:21
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    $\begingroup$ @Energizer777: Wrong, the quantum mechanical uncertainty relation is far more general than the Fourier-width relationship. $\endgroup$
    – ACuriousMind ♦
    Commented Nov 5, 2015 at 13:50

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I agree with the first half of your question, but I think you have wandered astray in the second part.

The theory of quantum mechanics is based on a number of axioms. A thorough description of these can be found in this paper, though this is a rather greater level of detail than would be useful for most non-physicists. If you construct the theory of quantum mechanics using these axioms then the uncertainty principle is an inevitable result. This is roughly what you say in your statements (1) and (2).

Axioms are assumptions - you can't prove an axiom. However you can experimentally test the theory we get from the axioms and so far quantum mechanics has proven to be the most accurate theory ever discovered. This gives us considerable faith that the axioms represent some fundamental reality rather than just being fantasies of some mad physicist.

Now, your last paragraph asks:

The universe did not have to behave this way, but it does. Another universe could not behave this way and the uncertainty principle inequality would not apply to that universe. Correct?

And the answer is that in any universe where quantum mechnics works physics must obey the uncertainty principle. If you postulate some universe where mechanics is purely classical then of course there would be no uncertainty principle. However it's hard to see how you would construct any such universe. Everything we observe is based ultimtely on QM - take this away and you'd have nothing left.

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  • $\begingroup$ Thanks for your answer. In my question I was trying to separate the math from the physics. The physics is how the universe behaves. The math is the equations and inequalities that physicists and mathematicians come up with to describe/predict the physical behavior. $\endgroup$
    – John Hoebel
    Commented Nov 6, 2015 at 11:19
  • $\begingroup$ @JohnHoebel: we don't know how the universe behaves. The best we can ever do is measure (approximately) how the universe behaves then construct a mathematical model that correctly predicts the observations. We hope that the axioms used in our mathematical model tell us something important about the universe, but we will never know for sure if that is the case. $\endgroup$
    – John Rennie
    Commented Nov 6, 2015 at 11:28
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My understanding is that the Uncertainty principle or relation is a mathematical consequence of the axioms of quantum mechanics.

So in it is an inevitable consequence of the mathematical theory known as quantum mechanics.

What can occur in another universe is a irrelevant to any questions of what can occur here. Uncertainty principle is only a physical fact if one assumes that quantum mechanics is true. That is far from certain. Experimental confirmation is not proof a theory is true, just that it is consistent with observations.
If you are asking what physical facts account for the uncertainty relation, that answer is we don't know since quantum theory is silent on how matter and energy behave at quantum realm. It can be described as a physical limit in that to measure a system one must disturb it.

I hope that helps.

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