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The Heisenberg's uncertainty principle states that a particle cannot have a precise value of its position and conjugate momentum simultaneously.

If these uncertainties are intrinsic properties of a state why is the word 'simultaneously' important? Is this important only for those states which have non-trivial time-dependence? For trivial time-dependence i.e., energy eigenstates $\Delta x$ and $\Delta p_x$ are fixed in time and it appears that one can measure them at separate times. But this is not true for other states. Am I correct?

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Just look at the formal version of the Heisenberg uncertainty principle: $$ \sigma_x(\psi) \sigma_p(\psi) \geq \hbar/2,$$ where $\sigma_A(\psi) = \sqrt{\langle\psi\vert A^2\vert \psi\rangle - \langle \psi \vert A\rvert \psi\rangle^2}$ is the standard deviation of an operator $A$ for the state $\vert \psi \rangle$.

When we say a state has a "well-defined" or "precise" value of the observable $A$, we mean it is an eigenstate. It is straightforward to check that $\sigma_A(\psi) = 0$ in an eigenstate. So no state can be both an eigenstate of $x$ and of $p$, since that would mean $0\geq \hbar / 2$, which is clearly false.

The "simultaneously" means precisely that: A (time-dependent) state $\psi(t)$ may be an eigenstate of $x$ in one instance, and an eigenstate of $p$ in another, but it is impossible that $\psi(t_0)$ for any fixed $t_0$ is both.

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The word "simultaneously" is there because you can obtain a precise measurement of an particle's position and afterward obtain a precise measurement of the momentum. But you can't do both at the same time. Further, when you make the second measurement (the measurement of the momentum), you disturb the object's position, so it in fact no longer has a definite position, and the position measurement you had made at first no longer contains any meaningful information about the particles position.

For a concrete example, consider a particle in the ground state of the quantum harmonic oscillator. In this state, $\Delta x$ and $\Delta p$ are both non-zero and constant in time, so neither the position nor momentum is known. Now if you measure the position, the particle's wave function collapses to a delta function, and you observe a precise position for the particle. But now when you measure the momentum, the wavefunction collapses into a plane wave state, which has infinite extent. So while you obtain a precise value for the momentum, you have lost all the information about the particle's position.

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    $\begingroup$ Everything you describe here is correct, but it seems to be more a description of how projective measurements work, and not about the uncertainty principle. You've hidden the application of the uncertainty principle behind the assertion that a $\delta$-function doesn't have a definite momentum, essentially. $\endgroup$ – ACuriousMind Aug 24 '17 at 16:34
  • $\begingroup$ @ACuriousMind I didn't talk about the uncertainty principle because I assumed they already knew it. I was trying to address why simultaneity was important. To do this, I gave a counter-example, where at $\Delta x$ from one time was zero and a $\Delta p$ from another time was zero. I think they understood that the uncertainty principle applies when $\Delta x$ and $\Delta p$ are considered simultaneously, and therefore it applies to constant states even when $\Delta x$ and $\Delta p$ aren't considered simultaneously, but they had a hard time thinking about states that change with time. $\endgroup$ – Brian Moths Aug 24 '17 at 16:52
  • $\begingroup$ @ACuriousMind Accordingly, I gave an example of a state which had a time dependence (owing to the fact that it is being measured), and therefore had varying $\Delta x$'s and $\Delta p$'s, such that a non-simultaneous product came out to be zero. This shows why the word "simultaneous" is necessary, and that he is correct that time-varying states play an important role. $\endgroup$ – Brian Moths Aug 24 '17 at 16:55
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In principle it holds that when you decide to measure the momentum the more uncertain of the position and vice versa but surely this leads to there is a simultaneity in these states but this principle means we are unable to measure the two occuring variables without interference. it doesn't detract from the actual physical occurrence of the two, we just ain't able to measure the two simultaneously in principle. Well that's what I think it means

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protected by Qmechanic Aug 25 '17 at 5:30

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