Talk:Wave function/Archive 11

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explanation for the curious

How can it be that there is such disagreement about "measurement" here?

In a nutshell, the kind of measurement that is mathematically analyzed above in detail measures all comers by interacting materially and substantially with them. It is a measurement in a more obvious and thorough sense than the one that Dirac used. It emphasizes Heisenberg uncertainty for this reason. The kind of "measurement", to use the term that is used, for example in discussions of "collapse", and was used by Dirac, rejects any but the desired eigenstate, and passes unscathed the desired eigenstate effectively by selection rather than substantial interaction. Only the desired eigenstate is "measured". All other comers are absorbed into the side walls of the device, or by some like mechanism. In this way Heisenberg uncertainty is manifest in virtually infinite error in the rejected particles: 'no result because of no passage' may be counted as infinite error. There is still some room for finite error for the particles that pass, but it is manageable. It comes under Dirac's heading of "sorting apparatus, such as slits". Dirac, 4th edition (1958), pp. 11–12: "A state of a system may be defined as an undisturbed motion that is restricted by as many conditions or data as are theoretically possible without mutual interference or contradiction. In practice the conditions could be imposed by a suitable preparation of the system, consisting perhaps in passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being left undisturbed after the preparation." An arrangement of choppers would count here as 'sorting apparatus'.Chjoaygame (talk) 06:53, 17 March 2016 (UTC)

On thinking about it overnight, I see my Heisenberg story just above is wrong. I just wrote the just above off the top of my head. The proper story is of course as follows. The method considered by Dirac produces a beam of systems, each system in a pure state. If the pure state degree of freedom considered is the momentum in the direction of the beam, that has been produced by an arrangement of choppers, the value of the momentum will be narrowly defined by the choppers, nearly an eigenstate. But the position in the same direction will be practically undefined, with a practically infinite uncertainty. It is the position that is the conjugate variable to the momentum that goes into the uncertainty product.Chjoaygame (talk) 22:22, 17 March 2016 (UTC)

Heisinberg is correct. Narrowly defined by the choppers, nearly an eigenstate is possible indeed.

On pages 132–133, Shankar (2nd edition, 1980/1994) reads: "In our earlier discussion on how to produce well-defined states |ψ⟩ for testing quantum theory, it was observed that the measurement process could itself be used as a preparation mechanism: if the measurement of Ω on an arbitrary, unknown initial state given a result ω, we are sure we have the state |ψ⟩ =  |ω⟩. ... To prepare a state for studying quantum theory then, we take an arbitrary initial state and filter it by a sequence of compatible measurements till it is down to a unique, known vector. Any nondegenerate operator, all by itself, is a "complete set.""Chjoaygame (talk) 23:39, 17 March 2016 (UTC)

Shankar is incorrect. But he, on the other hand assumes the artifact of quantum ideal measurement using arbitrarily ultra-soft photons with precise energy. These do not exist because the quantum states producing them would have to have infinite lifetime. (The actual lifetime of these states provide an other means of deriving Bohrs formula of above.)

Shankar isn't talking about "arbitrarily ultra-soft photons with precise energy." He is talking about "measurement" by selection in the same sense as do L&L, Cohen-Tannoudji et al., and Dirac.Chjoaygame (talk) 21:14, 18 March 2016 (UTC)

On page 235, Cohen-Tannoudji, Diu & Laloë (2nd edition, 1973/1977) write: "Similarly, we can construct devices, intended to prepare a quantum system, in such a way that they only allow the passage of one state, corresponding to a particular eigenvalue of each of the observables of the complete set chosen."Chjoaygame (talk) 23:51, 17 March 2016 (UTC)

Incorrect in general for reasons stated.

The correct treatment is found in section 44 or Landau and Lifshitz with the result that

$|v'_{x}-v_{x}|\Delta p_{x}\approx \hbar /\Delta t,$

where

${\begin{aligned}&\Delta p_{x}&{\text{is uncertainty in measured value of momentum,}}\\&\Delta t&{\text{is duration of measurement,}}\\&v_{x}&{\text{is velocity of particle }}{\textit {before}}{\text{ measurement,}}\\&v'_{x}&{\text{is velocity of particle }}{\textit {after}}{\text{ measurement.}}\\\end{aligned}}$

Read also section 7 as a preparation. Chjoaygame, why do you chose not to quote L&L this time? They are the ones treating this. YohanN7 (talk) 08:53, 18 March 2016 (UTC)

People often talk about "simultaneous" measurement. They usually don't mean it literally and exactly. More precisely they usually mean compatible serial measurement, not envisaging literal simultaneity. One might say instead perhaps 'joint measurement', or 'coexistent measurement'. But 'simultaneous' seems the customary term. The idea is that each system, on its way from oven to anti-oven (= detector), passes through several sorting devices in tandem (meaning 'at length', not in parallel). These discussions, if referring to several degrees of freedom, in many cases seem more about idealized scenarios than exactly and practically real ones. In practice, it is not easy to chain together too many slits, polarizers, choppers, beam-splitters, prisms, Stern–Gerlach magnets, calcite crystals, and what not. The key concept is commutativity.Chjoaygame (talk) 09:09, 18 March 2016 (UTC)

No one is contradicting L&L. The relevant devices considered by the writers I cite are designed to make one aspect of the degree of freedom rather closely defined, while the conjugate aspect has huge latitude of error.Chjoaygame (talk) 09:15, 18 March 2016 (UTC)

Page 5 of the 3rd edition of L&L (Pergamon 1977): "We shall now formulate the meaning of a complete description of a state in quantum mechanics. Completely described states occur as a result of the simultaneous measurement of a complete set of physical quanti­ties."Chjoaygame (talk) 09:27, 18 March 2016 (UTC)

This is written before reading your last addition:

You are mixing in irrelevancies, and do not lecture on what people mean. I know what they mean. My original objection is that you cannot

• Obtain a complete measurement an arbitrary state precisely at any given time

AND

• Expect a repeated measurement to yield the same result.

Note the AND. It is capitalized for you so that you surely note it. In particular, you cannot use the reasoning to define quantum states more "precisely". This is what you did in the article, and this is what found profoundly misleading.

Regarding L&L. Among the math, they, in several places, take note that non-repeatability of precise momentum measurements is fundamental to quantum mechanics. (Don't take this as a literal quote. I don't have my book in my lap atm.) I dare say that L&L are more reliable than Dirac, especially when it comes to isolated one-liners, since they do the math and Landau was just as clever as Dirac, if not cleverer. The book has also been revised into modern times. YohanN7 (talk) 08:53, 18 March 2016 (UTC)

This is written after reading your last addition:

Now you try to make it as if you were right al the time. L&L has gone from "do not apply" to be "maybe true, I don't care because I am right anyway". YohanN7 (talk) 09:33, 18 March 2016 (UTC)

---

I have now left the building. I intended to do that when I retracted my first reply to let you have your final word. Now I must do it. It takes too much of my time and is too frustrating. The article is all yours. YohanN7 (talk) 09:40, 18 March 2016 (UTC)

<Greiner, W. Quantum Mechanics: An Introduction, 4th edition, Springer, Berlin, ISBN 3-540-67458-6> p. 438: "Axiom 1: As a result of the measurement of an observable, only one of the eigenvalues of the corresponding operator can be found. After the measurement, the system occupies that state which corresponds to the measured eigenvalue."

Pure states and von Neumann's projection postulate

A state of a system may be defined as an undisturbed motion that is restricted by as many conditions or data as are theoretically possible without mutual interference or contradiction. In practice, the conditions could be imposed by a suitable preparation of the system, consisting perhaps of passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being undisturbed after preparation.

When we measure a real dynamical variable ξ, the disturbance involved in the act of measurement causes a jump in the state of the dynamical system. From physical continuity, if we make a second measurement of the same dynamical variable ξ immediately after the first, the result of the second measurement must be the same as that of the first. Thus after the first measurement has been made, there is no indeterminacy in the result of the second.

von Neumann (1932) makes his famous projection postulate on pp. 200–201:

( P.) The probability that in the state φ the quantities with the operators R₁, ... ,R_l take on values from the respective intervals I₁ ... I_l is

|| E₁(I₁) ... E_l(I_l) φ ||²

where E₁(λ), ..., E_l(λ) are the resolutions of the identity belonging to R₁, ... ,R_l respectively.

...we postulate P. for all commuting R₁, ... ,R_l . Then the E₁(I₁), ..., E_l(I_l) commute, and therefore E₁(I₁) ... E_l(I_l) is a projection (THEOREM 14. in II. 4.), and the probability in question becomes

P  =  || E₁(I₁) ... E_l(I_l) φ ||²  =  (E₁(I₁) ... E_l(I_l) φ , φ)

According to Weinberg on page 26:

...Thus if a system is in a state represented by a wave function ψ, and we make a measurement that puts the system in any one of a set of states represented by orthonormal wave functions ψ_n (which may or may not be energy eigenfunctions) then the probability that the system will be found to be in a particular state represented by the wave function ψ_m is

$P(\psi \rightarrow \psi _{m})=\left|\int \psi _{m}^{*}\psi \right|^{2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1.5.18)$

This can be taken as the fundamental interpretive postulate of quantum mechanics.

Weinberg in 2013 is using the traditional term "measurement" as Dirac used it in 1958, to express the von Neumann projection postulate.

Greiner, W. (2001), Quantum Mechanics: An Introduction, 4th edition, Springer, Berlin, ISBN 3-540-67458-6

P. 438:

Axiom 1: As a result of the measurement of an observable, only one of the eigenvalues of the corresponding operator can be found. After the measurement, the system occupies that state which corresponds to the measured eigenvalue.

P. 469:

Now we ask how measurement influences the system. Let the measurement have the result q_l. Immediately after the measurement, we assume it to be repeated. If the measurements are to make physical sense, we have to claim that the experimental result does not change: the second measurement also has to result in the value q_l.

Greiner in 2001 is using the traditional term "measurement" as Dirac used it in 1958, to express the von Neumann projection postulate.

Page 5 of the 3rd edition of L&L (Pergamon 1977):

We shall now formulate the meaning of a complete description of a state in quantum mechanics. Completely described states occur as a result of the simultaneous measurement of a complete set of physical quanti­ties.

L&L in 1977 used the traditional term "measurement" as Dirac used it in 1958, to express the meaning of a complete description of a state.

On page 235, Cohen-Tannoudji, Diu & Laloë (2nd edition, 1973/1977) write:

Similarly, we can construct devices, intended to prepare a quantum system, in such a way that they only allow the passage of one state, corresponding to a particular eigenvalue of each of the observables of the complete set chosen.

These authors do not here use the traditional term "measurement" for their preparation procedure, though they rely on devices which prepare by selecting or filtering.

The traditional term "measurement", as used by Dirac, Greiner, Landau & Lifshitz, and Weinberg, is, regrettably, an oft-encountered abuse of language. A better term would be 'purification by selection or filtration', or Dirac's "sorting apparatus".

This proposed better term is not original research. For example, Park & Band in 1992 wrote:

P. 658:

It is not unusual to find excellent didactical presentations of the quantal algorithm embedded in a kind of structureless philosophical void which offers the student no clue as to the empirical significance of the formalism. ... We believe that the student of quantum theory can successfully weather the above-mentioned and other common affronts to his intellect if at some point in the educational process, preferably early, he is introduced to the preparation-measurement format of experimental science and then immediately taught the particular manner in which quantum theory copes with physical problems by employing that framework.

P. 659:

Nevertheless, to this day the preparation process is still often misnamed "measurement" ...

P. 662:

A further selection may be imposed ... We call this an almost "pure" state because the result of any subsequent momentum measurement has an almost foregone conclusion. ... When we have completed this selective preparation of a pure state, we have, according to conventional wisdom, already performed the measurement. ... Conventional discussions based on the von Neumann mathematical scheme stopped at the pure state preparation and asserted that the measurement act had left the particle in the pure state corresponding to the result of the measurement. This is really nothing worse than a semantic error.

Perhaps this may be useful.Chjoaygame (talk) 13:25, 30 March 2016 (UTC)

""""measurement"""" as state preparation ???

As I have repeatedly written in this talk page, the word 'measurement' in quantum mechanics is a trap for young players. I have advocated 'observation' instead simply because it is less fraught with prejudice, and more likely to get the reader to have a critical attitude.

Without prejudice, the following quote is from a textbook that is free on the Internet, by J.D. Cresser, talking about the so-called 'von Neumann projection postulate':

This postulate is almost stating the obvious in that we name a state according to the information that we obtain about it as a result of a measurement. But it can also be argued that if, after performing a measurement that yields a particular result, we immediately repeat the measurement, it is reasonable to expect that there is a 100% chance that the same result be regained, which tells us that the system must have been in the associated eigenstate. This was, in fact, the main argument given by von Neumann to support this postulate. Thus, von Neumann argued that the fact that the value has a stable result upon repeated measurement indicates that the system really has that value after measurement.

This is referring to the same kind of """measurement""" as considered above as cited from Dirac, Cohen-Tannoudji et al., Shankar, Feynman, and page 5 (but not necessarily section 44) of L&L. I think the essence of this kind of """measurement""" is that it works by selection rather than by interaction. This contrasts with what Pauli cited above calls 'measurement of the second kind', which tells about the state before measurement, and which involves 'kicking' the system.

Without prejudice, it my be useful to cite von Neumann. Page 380 of the Princeton English translation says:

... Therefore, each measurement on a state is irreversible , unless the eigenvalue of the measured quantity (i.e., this quantity in the given state) has a sharp value, in which case the measurement does not change the state at all.

Though not the only ideas to consider, these cannot be just dismissed as irrelevant, mistaken, or nonsensical.Chjoaygame (talk) 23:13, 19 March 2016 (UTC)

Drilling into the details of measurement is indeed off topic for this article. There are other articles for these things. The way you are going, the article will just be another article on quantum states or even QM itself.

A few weeks ago I intended to stay away, yet I have been silly enough to engage with you on this talk page. This will be my last reply. M∧Ŝc²ħεИ_τlk 23:36, 19 March 2016 (UTC)

bras and kets

Without prejudice, here, in Baaquie, B.E. (2013), The Theoretical Foundations of Quantum Mechanics, Springer, New York, ISBN 978-1-4614-6223-1, on page 222, one may find the following:

The initial state |ψ⟩ is the “history state” of the transition amplitude, and the final state ⟨χ| is its “destiny state.” In quantum theory both the history state and the destiny state can be independently specified.

We may consider the possible worth of this, in the light of the above cited statements of Feynman, of L&L, and of Auletta, G., Fortunato, M., & Parisi, G..Chjoaygame (talk) 23:39, 19 March 2016 (UTC)

Also we may consider another writer, Schwartz, M.D. (2014), Quantum Field Theory and the Standard Model, Cambridge University Press, Cambridge UK, ISBN 978-1-107-03473-0, here on page 56:

We can write such inner products as ⟨f;t_f|i;t_i⟩, where |i;t_i⟩ is the initial state we start with at time t_i and ⟨f;t_f| is the final state we are interested in at some later time t_f.

Also on page 70:

The state |i⟩ is the initial state at t = −∞ and ⟨f| is the final state at t = +∞.

Perhaps others may turn up? Chjoaygame (talk) 00:30, 20 March 2016 (UTC)

Schwinger distinguished bras and kets as creation and destruction symbols. He went contary to the usual custom of seeing kets as creation and bras as destruction symbols, but this choice is allowed by the symmetry of the theory. He wrote (as reported by Englert in a posthumously assembled text):

... certainly ⟨a'|, symbolizing a creation act, cannot be equated to |a'⟩, representing an act of destruction, (reading L → R)."<Schwinger, J. (2001). Quantum Mechanics: Symbolism of Atomic Measurements, edited by B.-G. Englert, Springer, Berlin, ISBN 3-540-41408-8, p. 38.>

Chjoaygame (talk) 04:03, 22 March 2016 (UTC)

Not qualifying as a reliable source is this:

|Ψ⟩ represents a system in the state Ψ and is therefore called the state vector. The ket can also be interpreted as the initial state in some transition or event. The bra ⟨ | represents the final state or the language in which you wish to express the content of the ket | ⟩.

Chjoaygame (talk) 05:24, 20 March 2016 (UTC)

A deprecatory view of the proposed preparation/observation interpretation of kets/bras is given by Peres, A. (2002), Quantum Theory: Concepts and Methods, Kluwer, New York, ISBN 0-792-33632-1, on page 77:

... (... also fruitless attempts to attribute different physical meanings to the two types of vectors, such as preparation states and observation states).

Evidently, opinion is not uniform.Chjoaygame (talk) 02:29, 20 March 2016 (UTC)

a representation defined by a complete set of bras

In the 4th edition (1958), on page 5, Dirac writes:

The problem we must now consider is how to fit in these ideas with the known facts about the resolution of light into polarized components and the recombination of these components.

On page 53:

To set up a representation in a general way, we take a complete set of bra vectors, i.e. a set such that any bra can be expressed linearly in terms of them (as a sum or an integral or possibly an integral plus a sum). These bras we call the basic bras of the representation. They are sufficient, as we shall see, to fix the representation completely.

The beam of systems passes through the representation's characteristic quantum analyzer and comes out in its several output channels with the appropriate weighting or probability distribution. The representation's basic bras denote the several output channels, with a detector in each. The analyzer has resolved the beam into components for the representation. If there were no detectors in the output channels to fix the systems as bras, they would still be coherently available as kets for interference or re-assembly into a replica of the original beam from the source.

The analyzer acts as the representation's resolution of the identity:

$\sum _{\xi ^{\prime }}|\xi ^{\prime }\rangle \langle \xi ^{\prime }|=1$

and

$\int |\xi ^{\prime }\rangle d\xi ^{\prime }\langle \xi ^{\prime }|=1$.

Chjoaygame (talk) 16:40, 20 March 2016 (UTC)

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