QM: Measurement Operation & Operators Explained

[alexepascual]

Like all beginners to QM I'm really confused about the measurement operation. I understand that measurement is simply a dot product with an "operator" and the result is one of the operator's eigenvalues.

Now my question is what exactly is an operator? If someone could explain what physical entity is an operator in the following situations that would help me understand this better.

1. A particle moving at some velocity hits a wall/detector. Till it hits the detector its position is described by a combination of position eigenstates. Once it hits the detector its position becomes a single eigenvalue. Is the wall an operator here?
2. A particle moves through an SG apparatus. Till it passes through the apparatus its spin state is a combination of two eigenstates (in some axis). Once it moves through the magnetic field its spin state becomes one of the eigenstates. Is the magnetic field an operator here?

To all posters on this thread:
I thing svnaras has asked very simple questions and you all have failed to give a simple answer. svnaras must be even more confused than when he first started the thread.
I tried to provide an answer in two oportunities but had some problems submitting my post and lost the text in both oportunities. So I am a little frustrated about these technical dificulties and feel somewhat reluctand to try again.
The questions svnaras asked can be answered according to the orthodox interpretation concentrating more on how the math works than anything else. I'll try to give an answer even shorter (and necessarily incomplete) than my previous answers:
(1) An operator that represents an observable (which can be represented as a matrix for discreet variables) is used to obtain a weighted average of the possible values that the measurement may yield. The weights are the probabilities which are encoded in the state vector. To get the average you multiply (inner product) the matrix by the state vector on both sides. This gives you a single (real) number.
(2)An act of measurement is not described using the observable's operator. It is usually described using a projector. The projector in its simplest expression is a matrix with all zeros except one element in the diagonal that has a 1. To get the final state, you multiply the state vector by the projection operator and get the the new state vector. If you want to get the actual value (momentum, position etc.) that this state vector represents, you can use the procedure I described in (1) using the new state vector.
(3) The wall is not an operator. The collapse that happened at the wall can be represented using the projector operator.
(4) In SG the magnetic field only allows you to select a particular basis. Each orientation in space is a different basis (frame of reference). The magneti field does not collapse the state vector to one of the eigenvalues (you can re-combine the beams). See Feynman Lectures on Physics volume III. If you put an obstacle in one of the paths, then this is a measurement that can be represented using a projection operator. If both pathsare open, instead of a projector, you got an identity matrix. If you put an obstacle in the lower path, then you this kills the lower eigenstate. So you put a zero in the diagonal's lower right.
So, going back to your question, the magnetic field could be seen as a change of basis unitary operator.

 

[ytuab]

ytuab, in BI, electrons don't have charge. If charge were a property of particles in BI you would get the "ultraviolet catastrophe" from all of the radiation they would emit while being accelerated by the pilot wave.

Mmm... Thanks for telling me this thing.

So the word " illusion" was used.....

 

[Fredrik]

In addition to Peter Holland's 1993 textbook 'The Quantum Theory of Motion' that I referred to in my last post, there are also several more modern treatments including Duerr and Teufel's recent "Bohmian mechanics" book (2009)

Which one is better if you're only willing to buy one? (They're about 90 USD each, so I'd rather not buy both). Demystifier recommended Holland's book to me in February, but maybe the other one didn't even exist back then.

 

[zenith8]

Which one is better if you're only willing to buy one? (They're about 90 USD each, so I'd rather not buy both). Demystifier recommended Holland's book to me in February, but maybe the other one didn't even exist back then.

Hi Fredrik,

Holland's book is an exhaustively detailed presentation of the whole theory - essentially recalculating every result in standard QM from this new perspective. If you don't mind the excessive detail, it's great for the non-relativistic stuff. It's less good for the relativistic stuff (which wasn't that well developed back then anyway) and I would say his treatment of spin is misguided, but never mind.

Duerr and Teufel's book is alright, but there is far too much mathematics and not enough physics. And their chippiness about the fact that everyone ignores them even though they are obviously right might be annoying to some people. I know I do this too in my posts here, but it's a carefully studied pose to provoke people into a reaction :shy: - I can get away with doing this in an internet forum but they probably shouldn't do it in a book. Despite the length of the book, their choice of topics is also much more limited than Holland (they don't do any relativistic stuff at all).

So if you only buy one of them - well, neither of them are perfect but I would still recommend Holland (even though it's 16 years older).

You could also read Bohm + Hiley's 'The Undivided Universe' from the same year as Holland, but I wouldn't bother yet (they don't have the patience to bother with boring details, the style is a bit annoying, and they mix in far too much speculative nutter stuff to make it a good introductory textbook).

If you just want a decent summary, Towler's lecture course is good. Obviously he lacks the detail of a proper textbook but he manages to pack a surprising amount in (he doen't get very far into the relativistic theory either).

Antony Valentini is apparently writing a comprehensive textbook that should be out sometime in the next few years. This won't help you at the moment obviously but it will be the one to read, I'm sure. His recent historical study "Quantum Theory at the Crossroads: reconsidering the 1927 Solvay Conference" (2009) - also available online - was a revelation to me regarding the historical context.

A final decent option might be reading some of the review articles. There is a comprehensive list of Bohm/pilot-wave references with links on Towler's http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" (Click 'Further Reading' in the right hand column).

 

[Fredrik]

Thank you. I have added Holland's book to my Amazon cart, and it will be included in my next order.

That book by Valenti sounds interesting, but it doesn't sound like it will be suitable as an introductory text. This is from his web page:

His central tenet is that quantum theory is not fundamental but merely describes a statistical equilibrium state, which the universe happens to be in at the present time. He has extended the de Broglie-Bohm pilot-wave formulation of quantum theory to nonequilibrium distributions outside the domain of standard quantum physics, and is searching for evidence of the nonequilibrium breakdown of quantum mechanics in the early universe.Antony is completing a book that re-examines modern physics (including quantum gravity, black holes, cosmology, inflation, quantum information and computation) from a pilot-wave and general hidden-variables viewpoint.

 

[JK423]

I`ve read some of the Towler`s lectures and quite a few posts from zenith8 and demystifier who seem to be experts in BM. The explanation that BM gives is very satisfactory plus the fact that the famous paradoxes of CI vanish. The fact that its a non-relativistic theory doesn`t "annoy" me much because only a few people actually work on this theory. So its logical to be slowly developed!
I just wonder what would the today top-experts physicists say to me, if i asked them "why haven't you, as far, worked on Bohmian mechanics" ?
They would just say "Noone else did, so i didnt bother" or they would point me a few arguments which so that BM theory is unsatisfactory and not worth to work on it?

I`m an undergraduate in 4rth year, still in my early steps. I want to be a theoritical physicist in the future, and i really don't want to ignore theories that seem to have a future.
I just can't understand why everyone (even the best physicists) ignore it! There must be reason!

 

[zenith8]

I`ve read some of the Towler`s lectures and quite a few posts from zenith8 and demystifier who seem to be experts in BM. The explanation that BM gives is very satisfactory plus the fact that the famous paradoxes of CI vanish. The fact that its a non-relativistic theory doesn`t "annoy" me much because only a few people actually work on this theory. So its logical to be slowly developed! I just wonder what would the today top-experts physicists say to me, if i asked them "why haven't you, as far, worked on Bohmian mechanics" ? They would just say "Noone else did, so i didnt bother" or they would point me a few arguments which so that BM theory is unsatisfactory and not worth to work on it?

I`m an undergraduate in 4rth year, still in my early steps. I want to be a theoritical physicist in the future, and i really don't want to ignore theories that seem to have a future.

Good lad! May I heartily encourage you..

I just can't understand why everyone (even the best physicists) ignore it! There must be reason!

From what I remember Towler's lecture 7 - 'Why does nobody like pilot-wave theory?' or whatever it's called - was about precisely this point. I think at the end - the answer is simply misunderstanding and peer pressure. No technical objection has ever been sustained - it all comes from Bohr and Heisenberg's insistence on their own infallibility which was widely believed and accepted for decades - despite the fact that their arguments were merely circular and not definitive. The de Broglie-Bohm solution is so simple and stems so obviously from the basic formalism of QM that acceptance of it is tantamount to claiming that Bohr, Heisenberg et al were idiots. Nobody can ever believe that people so widely revered could have been so wrong.

Notice that this is so even if de Broglie-Bohm turns out not to be literally 'true'. Bohr and Heisenberg wrote a great deal about what was possible in QM - and about that they were about as mistaken as it is possible to be.

Remember also that only for about the last ten years has it been possible to say you're interested in de Broglie-Bohm and still keep your job. Even now I have to say that there are people in my department who are aware that I am working on this, and conclude that I must have taken leave of my senses.

Courage, mon ami! (the number of Bohm papers per year in the literature has been increasing steadily since about 1990).

The fact that its a non-relativistic theory doesn`t "annoy" me much because only a few people actually work on this theory.

Actually, it is relativistic (or can be made to be so) - see Towler Lecture 5. The only problem is that it disagrees with some of the 'metaphysics' of relativity, not the physics. The relativistic Bohmian theories are in complete accord with experiment. One could say therefore that the problem lies in the 'interpretation' of relativity (though most people of course are unaware that it even has one!)