I asked this question, where I'm thinking about this joshphysics's answer:

And the gist of what I get is (from the answers and the chatroom) within interpretations of quantum mechanics which say the measurement is non-unitary one can definitely make the case that the derivation cannot be applied to all situations (in fact I feel the application is quite limiting).

However, it seems intuitive to me (after chatroom discussion) with a quasi-classical measuring apparatus and the Copenhagen interpretation enables one to circumvent my objection? (And also enables one to apply it to cases involving a measurement - see related/objection).

Note: The below is a labyrinth of positions one can take on this argument. And I feel there are $2$ ways out (way $3$ or way $4$)of this maze. I'm asking for which way is better? (See Questions for what I mean by "better")

Math and Context

The proof (I questioned)

Let's begin with Joshphysics's answer:

Let a quantum system with Hamiltonian $H$ be given. Suppose the system occupies a pure state $|\psi(t)\rangle$ determined by the Hamiltonian evolution. For any observable $\Omega$ we use the shorthand $$ \langle \Omega \rangle = \langle \psi(t)|\Omega|\psi(t)\rangle. $$ One can show that (see eq. 3.72 in Griffiths QM) $$ \sigma_H\sigma_\Omega\geq\frac{\hbar}{2}\left|\frac{d\langle \Omega\rangle}{dt}\right| $$ where $\sigma_H$ and $\sigma_\Omega$ are standard deviations $$ \sigma_H^2 = \langle H^2\rangle-\langle H\rangle^2, \qquad \sigma_\Omega^2 = \langle \Omega^2\rangle-\langle \Omega\rangle^2 $$ and angled brackets mean expectation in $|\psi(t)\rangle$. It follows that if we define $$ \Delta E = \sigma_H, \qquad \Delta t = \frac{\sigma_\Omega}{|d\langle\Omega\rangle/dt|} $$ then we obtain the desired uncertainty relation $$ \Delta E \Delta t \geq \frac{\hbar}{2} $$ It remains to interpret the quantity $\Delta t$. It tells you the approximate amount of time it takes for the expectation value of an observable to change by a standard deviation provided the system is in a pure state. To see this, note that if $\Delta t$ is small, then in a time $\Delta t$ we have $$ |\Delta\langle\Omega\rangle| =\left|\int_t^{t+\Delta t} \frac{d\langle \Omega\rangle}{dt}\,dt\right| \approx \left|\frac{d\langle \Omega\rangle}{dt}\Delta t\right| = \left|\frac{d\langle \Omega\rangle}{dt}\right|\Delta t = \sigma_\Omega $$

Before I go on about my objection. Let me let you know it's basic premise.

Basic premise

The measurement is cannot be modelled as a unitary transformation. $$ |a \rangle + |b \rangle \to |a \rangle $$ Cannot be done by a unitary transformation and I thought I was vindicated in this post. Which is why the common referral of the measurement as "non-unitary"

The Objection

joshphysics's answer makes use of Heisenberg's equation of motion which rely on unitarity.

$$ {\displaystyle {\begin{aligned}{\operatorname {d} \over \operatorname {d} \!t}A(t)&={i \over \hbar }U^{\dagger }(t)HAU(t)+U^{\dagger }(t)\left({\frac {\partial A}{\partial t}}\right)U(t)+{i \over \hbar }U^{\dagger }(t)A(-H)U(t)\\&={i \over \hbar }U^{\dagger }(t)HU(t)U^{\dagger }(t)AU(t)+U^{\dagger }(t)\left({\frac {\partial A}{\partial t}}\right)U(t)-{i \over \hbar }U^{\dagger }(t)AU(t)U^{\dagger }(t)HU(t)\\&={i \over \hbar }\left(H(t)A(t)-A(t)H(t)\right)+U^{\dagger }(t)\left({\frac {\partial A}{\partial t}}\right)U(t),\end{aligned}}}$$

(the above is part of the derivation I'm referring to. Click on wiki link for full).

I am not aware of a derivation of Heisenberg's equation of motion which is applicable during a non-unitary process (for example the measurement).

See the $2$'nd equation in his (joshphysics) derivation. This clearly relies on "Heisenberg's equation of motion." Further the whole notion of getting a $\Delta t$ is only dimensionally true and not true even as an approximate (if this objection holds).

One way out: is your calling something velocity which has the same dimensions of velocity but is not the time derivative of position. But that is fundamentally wrong! (and abused in that case).

Another way out: is to claim that there is a kind of continuity argument but I was under the impression the measurement is a discontinuous process. So this one will fail.

The third way out: is to limit the physical scope of the applicability of the this derivation. For example, one cannot use this derivation to say the time between $2$ subsequent measurements cannot be taken to $0$. Perhaps it can (pushing it's limitations) tell you some kind of energy cost involved between $2$ subsequent measurements. But that would be at the price of presupposing "the time between $2$ subsequent measurements cannot be taken to $0$" as one would have assume that between 2 measurements unitarity must reign and the time interval cannot be taken to $0$.

So far so good?

I was on board with the third way out. In fact, any interpretation of quantum mechanics which uses the language "collapse of the wave function." I am under the impression will be sympathetic to my misunderstanding to say the least. Taken from wiki, The existence of the wave function collapse is required in

  1. the Copenhagen interpretation
  2. the objective collapse interpretations
  3. the transactional interpretation
  4. the von Neumann interpretation in which consciousness causes collapse.

The fourth way out (?)

Now, another user in the chat is of the opinion my objection is non-sensical (at least thats the impression I get) and upon conversing it seems he has a fourth way to bypass my objection!

The fourth way: In this fourth way one stops the basic premise of my objection which is the measurement being a non-unitary process! In fact this is convincingly done with:

"A measurement is an interaction between a measuring apparatus which is (quasi-)classical and a quantum system You can model this, it's in Landau's QM section 7, I don't want to quote the whole thing, but this wave function collapsing business is pretty straightforward there without all the woo"

In fact, he claims to be a Copenhagen-ist but doesn't like the words "collapse of the wave-function."

The fourth way out does manage to circumvent (if not completely destroy) my objection and also enables one to use this derivation in far more many many situations I previously see as unfathomable for example: the time between $2$ measurements cannot be taken to $0$.

In fact, I'm under the opinion the number of physical experiments the $3$'rd way out enables you to do are under $5$ (?). But the $4$'th way is clearly superior in this metric (?). A better question might be where does it not apply?

More about way $4$

For those who are shocked that Landau was on board with way $4$ and (fairly) demanding evidence before investing time for an entire section $7$ of Landau's QM.

See page 6 of the file (page $9$ of .pdf) of this where he argues why one should take way-out $4$:

’...It is in principle impossible ... to formulate the basic concepts of quantum mechanics without using classical mechanics.’ (LL2)

’...The possibility of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy.’ (LL2)

’...the ’classical object’ is usually called apparatus and its interaction with the electron is spoken of as measurement. However, it must be emphasized that we are here not discussing a process ... in which the physicist-observer takes part. By measurement, in quantum mechanics, we understand any process of interaction between classical and quantum objects, occurring apart from and independently of any observer. The importance of the concept of measurement in quantum mechanics was elucidated by N Bohr.’ (LL2)

The quotes go on ... The above is only the beginning (In fact the .pdf is a condensed version of "pure gold")

At some risk (I'm not sure if the person who exposed me to way $4$ would agree this paper does justice to their arguments): https://arxiv.org/abs/1406.5178

To be fair a different version of way $4$ has been asked on this site:

Unitarity and measurement

Where the answer-er response ends with:

"Why not let the final state be in a superposition also, as quantum mechanics requires? The alive cat is not aware of the dead cat, because the Schrödinger equation is linear."

For a non-absurd association of the same words above: https://www.youtube.com/watch?v=gNAw-xXCcM8

(In light of this let me just say my question is not about the plausibility of the Unitary of the measurement but it's about if assumed can one assume this and push the number of physical situations this derivation is applicable to?)

Is a unitary measurement compatible with broadening the usage of the time-energy uncertainity principle?

Also Landau states: "From section $44$ of Landau, the $\Delta E$ in the time-uncertainty relation is the difference "between two exactly measured values of the energy $E +\varepsilon$ at two different instants, and not the uncertainty in the value of the energy at a given instant" (for a system of two parts with energies $E$ and measuring apparatus energy $\varepsilon$)"

I added this part so it didn't seem like I was skipping any logic. I'm sure Landau would have only said that if he felt his book justified the case.


Is this "fourth way" plausible? Can someone tell me an explicit case where if I assume this circumvention is indeed true what are the physical cases the this "quasi-classical measuring apparatus and the Copenhagen interpretation" seem to work for that the original objection stops (way $3$)? (I do mention one but am under-confident if it is right)

P.S: I feel this post was self-contained but to do justice to way $4$ I had to link it off. Please do not hesitate to contact me if you feel this question can be improved? Also vzn I feel your comment is beyond the scope of this question.

Cross-posted on QC.SE

  • $\begingroup$ @MoreAnonymous so you just mean when measurement is taken into consideration? So are you claiming that the calculations in this answer are somehow invalid when measurement is taken into consideration? I don't see how that should be the case. You say "one can definitely make the case that (...)". So what is this case? (also, you might want to tag people when answering in the commants, or they won't be notified of your comment and might not see it) $\endgroup$ – glS Sep 12 '19 at 12:20
  • $\begingroup$ So the derivation makes use of the Heisenberg picture (which in turn makes use of unitary - see wiki derivation). During the measurement some people say it undergoes "non-unitary" evolution. - I feel I have repeated the "Objection" section. Maybe I should ask why is the objection unclear? $\endgroup$ – More Anonymous Sep 12 '19 at 12:23
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    $\begingroup$ agree the question is not clear even after editing. just citing a lot of other links does not nec handle the context. try to state/ask a self-contained question. "my objection"—what is it??? anyway though in strong contrast to many regulars around here think there is lot of validity to questions about interpretations, and urge you+others to look into new experiments shedding new light, esp this latest one only months old Quantum Leaps, Long Assumed to Be Instantaneous, Take Time quantamagazine.org/… $\endgroup$ – vzn Sep 13 '19 at 3:20
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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – ACuriousMind Sep 15 '19 at 10:15
  • $\begingroup$ Bounty refunded and question closed as unclear. There are very many perspectives quoted here, and the blockquotes seem to be used as a mixture of setting off other people's words and highlighting some of your own words. A successful version of this question would focus on the ideas that you're interested in and identify the originators of those ideas in links to sources elsewhere. $\endgroup$ – rob Sep 15 '19 at 15:15