Why amplitude squared gives probability / Schrodinger Equation

[ArjenDijksman]

Yes I didn't parse it this way. As you just explained it though its just composite quantum probabilities (projected onto a subspace and renormalized) and has nothing to say about why square amplitudes.

Probability is amplitude squared because we project the total state-space (N*N separate basis states |A,A>, |A,B> ,|B,A>, |B,B>) on the detectable subspace (N separate events |A,A>, |B,B>).

 

[vitruvianman]

An extra info:

Let's say in the famous double-slit exp., a photon (when only one of the slits is opened) has a probability A(s,t) which is the complex "probability" number z, to come the the point t from the photon source s. And the probability A(t,p) which is the complex number w. These are the possibilities of coming to the point t from the source s, and after that, to the point p (on the screen) lastly.

Multiplying is surely so:
$$|zw|^2=|z|^2|w|^2=|A(s,t)|^2|A(t,p)|^2=|A(s,t)A(t,p)|^2$$

But if the photon has 2 different options we should add the probabilities and here apperas the characteristics of quantum mechanics. We should square the amplitude of w+z, and here comes a correction term, $\cos\theta$. The sum will be:
$$|z+w|^2=|w|^2+|z|^2+2|w||z|\cos\theta$$
$\theta$ is the angle between w and z points on the Argand plane. After a quick (and very basic) geometric calculation you'll find out why the correction term is $2|w||z|\cos\theta$ , because of Pythagoras theorem :)

 

[jambaugh]

Probability is amplitude squared because we project the total state-space (N*N separate basis states |A,A>, |A,B> ,|B,A>, |B,B>) on the detectable subspace (N separate events |A,A>, |B,B>).

No. That won't do it. The projection is linear and one linearly renormalizes when projecting. "You caint get thare frum here!"

It is clear that the squaring comes from going from mode vectors to the operators
|A> ----> |A><A|
which is the proper domain to speak of boolean observables and their expectation values (probabilities are expectation values of logical bits which are in turn observables of a given physical system.)

More properly said the square-rooting comes from representing mode operators (density operators) for sharp modes by just giving the eigen-vectors. The Hilbert space is "the square root" of the operator algebra. Its the operator algebra which is the proper mathematical object in which to represent the physics.

 

[ArjenDijksman]

It is clear that the squaring comes from going from mode vectors to the operators
|A> ----> |A><A|
which is the proper domain to speak of boolean observables and their expectation values (probabilities are expectation values of logical bits which are in turn observables of a given physical system.)

Well, what you say, is just what I mean but stated otherwise. Let me restate it applied to https://www.physicsforums.com/showpost.php?p=2482669&postcount=40": the squaring of the amplitudes (factors of the Hilbert state vectors representing Alice's possible locations) comes from the measurement process which yields probabilities (Bob detecting Alice at a particular location = operating on Alice's state with an operator).

As a non-trivial example: Alice can be at any two places M or N, with respective weights w(M)=3/5 and w(N)=4/5 (choosing these different weights avoids LaTeX script for square roots). For Bob the same. The respective weights for the composite system (Alice,Bob) are thus w(M,M)=9/25, w(M,N)=12/25, w(N,M)=12/25, w(N,N)=16/25.

We can therefore write the Hilbert space state vector |Alice>= 3/5 |M> + 4/5 |N>. But the proper domain to speak of expectation values is when Bob observes the state of Alice → Bob meets Alice at the same place → Bob operates with an operator on |Alice>. That yields prob(M,M)=9/25 and prob(N,N)=16/25 which are the squares of the Hilbert state vector amplitudes.

Greetings,
Arjen

 

[jambaugh]

Well, what you say, is just what I mean but stated otherwise. Let me restate it applied to https://www.physicsforums.com/showpost.php?p=2482669&postcount=40": the squaring of the amplitudes (factors of the Hilbert state vectors representing Alice's possible locations) comes from the measurement process which yields probabilities (Bob detecting Alice at a particular location = operating on Alice's state with an operator).

As a non-trivial example: Alice can be at any two places M or N, with respective weights w(M)=3/5 and w(N)=4/5 (choosing these different weights avoids LaTeX script for square roots). For Bob the same. The respective weights for the composite system (Alice,Bob) are thus w(M,M)=9/25, w(M,N)=12/25, w(N,M)=12/25, w(N,N)=16/25.

We can therefore write the Hilbert space state vector |Alice>= 3/5 |M> + 4/5 |N>. But the proper domain to speak of expectation values is when Bob observes the state of Alice → Bob meets Alice at the same place → Bob operates with an operator on |Alice>. That yields prob(M,M)=9/25 and prob(N,N)=16/25 which are the squares of the Hilbert state vector amplitudes.

Greetings,
Arjen

No. We are not describing the same thing though the math is paralleling. Yes those weights work out to square to probabilities. But you now cannot interpret those weights as rescaled probabilities themselves. Your exposition provides no enlightenment as to why e.g. weights of 3/5|M> + 4i/5|N> express a distinct case from yours above.

In an actual measurement process the quantum (if you want to call her Alice) is observed each trial in either |M> or |N> modes. Those cases where "Bob and Alice are not at the same slot" are not simply discarded trials that didn't yield valid results. Some Bob always sees Alice. In thinking of Bob as the measuring device, there are in fact an array of "Bob"'s at each location announcing "Hey! Alice just bumped into Me!" (For example a cloud chamber is a room of Bobs who so state by condensing a droplet when ionizing Alice passes by.)

I think your exposition provides "false enlightenment". The better interpretation would be to take the Bra's and Ket's to be say arrays of Alice's and Bob's observing Elvis. An Alice announces "Elvis has entered the building!" and a Bob announces "Elvis has left the building!" We then know Elvis was in building 1 when we get a A1,B1 announcement pair represented by |1><1|. We must calibrate our Alices and Bobs by repeating enough Elvis concerts to see which Alice and Bob pairs correlate exactly. (Mathematically this defines the metric on our Hilbert space) That provides the logic of Elvis sightings. Then to express probabilities we weight in-out correlations:

rho = = w1 P1 + w2 P2 + ... w1 |1><1| + w2|2><2| ...
(P = projector = logical predicate about Elvis.)

We start with the projectors and then we may for convenience use eigen-vectors which requires we square-root the weighting system (not just individual weights) as we construct the mathematical space on which the projectors act to define these eigen-vectors. The actual bras and kets are one level of abstraction further removed from the physical system and not more directly representative of it.

It is in the algebra of these predicates where we get the odd quantum behavior because such predicates=projectors don't all commute. PQ != QP. We can mathematically view the details by observing that an eigen-vector of P must be expressed as a linear combination of eigen-vectors of Q with coefficients which are not directly interpretable because we're doing all this in an abstract mathematical structure farther removed from the physics. It is the operators which are most closely connected to the physics and that's our logical starting point. We shouldn't (and as it turns out can't) deconstruct them further physically. Mathematical deconstruction, though necessary for mathematical reasons, is moving farther from the physical representation of what's happening in the lab.

A point here is that we never directly represent the quantum Elvis. The Bra's and Ket's express channels of Elvis behavior not Elvis states. Alice and Bob as symbols refer to parts of our measuring process not directly to the entity being measured.

Now on a tangential note. We can trivially quantify the system (speak of varying the number of instances) to be one or zero. In that extension we can treat the Hilbert space vectors and dual vectors (kets and bras) as respectively system creation actions and system annihilation actions. Then system transitions are expressed via pairs of annihilation-creation actions. This as I see it is the only way to "put a little more meat into them" i.e. give a bit more metaphysical meaning to the bra's and kets. But in standard quantum mechanics these actions are virtual, not physical. You can think of these acts of creation and annihilation rather as boundary crossings and an |a><b| transition channel as our painting a little cut in the system domain where the system instantaneously leaves and re-enters the domain. But here again they do not refer to the system but act as labels for modes by which the quantum leaves or enters this abstract domain of "being".

 

[ArjenDijksman]

No. We are not describing the same thing though the math is paralleling. Yes those weights work out to square to probabilities. But you now cannot interpret those weights as rescaled probabilities themselves. Your exposition provides no enlightenment as to why e.g. weights of 3/5|M> + 4i/5|N> express a distinct case from yours above.

Well, I was discussing Carl's classical analogy, which I don't see as a complete exposition of quantum probabilities. It merely gives insight in the fact that quantum probabilities arise in case of measurement, from weighted coefficients pertaining to the eigenstates of the system. That analogy provides by no means insight in the complex-valued nature of quantum amplitudes and you've raised the relevant objections. On that point, I have further ideas that I mentioned earlier in this thread.

In an actual measurement process the quantum (if you want to call her Alice) is observed each trial in either |M> or |N> modes. Those cases where "Bob and Alice are not at the same slot" are not simply discarded trials that didn't yield valid results. Some Bob always sees Alice. In thinking of Bob as the measuring device, there are in fact an array of "Bob"'s at each location announcing "Hey! Alice just bumped into Me!" (For example a cloud chamber is a room of Bobs who so state by condensing a droplet when ionizing Alice passes by.)

I think your exposition provides "false enlightenment". The better interpretation would be to take the Bra's and Ket's to be say arrays of Alice's and Bob's observing Elvis. An Alice announces "Elvis has entered the building!" and a Bob announces "Elvis has left the building!" We then know Elvis was in building 1 when we get a A1,B1 announcement pair represented by |1><1|. We must calibrate our Alices and Bobs by repeating enough Elvis concerts to see which Alice and Bob pairs correlate exactly. (Mathematically this defines the metric on our Hilbert space) That provides the logic of Elvis sightings. Then to express probabilities we weight in-out correlations:

rho = = w1 P1 + w2 P2 + ... w1 |1><1| + w2|2><2| ...
(P = projector = logical predicate about Elvis.)

We start with the projectors and then we may for convenience use eigen-vectors which requires we square-root the weighting system (not just individual weights) as we construct the mathematical space on which the projectors act to define these eigen-vectors. The actual bras and kets are one level of abstraction further removed from the physical system and not more directly representative of it.

I consider Carl's analogy as a "partial" enlightenment, rather than a "false" enlightenment: it's better to have an analogy than no analogy, in as much as one is aware of its limits. Your "bra/ket announcement pair" interpretation is interesting as well. Is there any literature (or posts) developing that insight?

Cheers,
Arjen

 

[Dunedain]

To be frank you are talking about an equation that works after the fact, and isn't derived from anything other than quantum mechanics, so it's always going to agree with what is predicted in that wave function. It works experimentally that's all you can ask from it, it's not derivable from anything but itself. We can talk interpretation but frankly that's not science that's philosophy, we are stuck with relating equations to experiment, not predicting experiments with equations, at least for now.

It's kind of like asking why c? Why indeed it just is it seems...

 

[carllooper]

To be frank you are talking about an equation that works after the fact, and isn't derived from anything other than quantum mechanics, so it's always going to agree with what is predicted in that wave function. It works experimentally that's all you can ask from it, it's not derivable from anything but itself. We can talk interpretation but frankly that's not science that's philosophy, we are stuck with relating equations to experiment, not predicting experiments with equations, at least for now.

It's kind of like asking why c? Why indeed it just is it seems...

Science and philosophy are not mutually exclusive terms. The equations of quantum mechanics were not developed "after the fact" (as if a function of the facts) but have their roots in creative assumptions regarding the facts. For example, Plank 'discovered' the quantum through speculative assumptions he made regarding experimental results. His equations represented those assumptions (as much as the results). Indeed the assumptions he made were so speculative he could not (initially) see how they could ever fit into the "science" of the time. Plank rejected his own equations because of 'science'. But his equations show that he had (in both fact and philosophy) discovered the quantum. If you look at how Einstein, Heisenberg, Bohr, Shrodinger (etc) approached the development of their equations you will see they also engaged in philosophy when doing so.

I can't see how it would be possible to do otherwise.

Carl

 

[jambaugh]

...
I consider Carl's analogy as a "partial" enlightenment, rather than a "false" enlightenment: it's better to have an analogy than no analogy, in as much as one is aware of its limits.

Fair enough.

Your "bra/ket announcement pair" interpretation is interesting as well. Is there any literature (or posts) developing that insight?

Cheers,
Arjen

Hmmm... I kind of came up with it on the spot based on my understanding of QM. (The announcement part). I've been thinking of the vectors and dual vectors as "kind of like boundary operators" for some time. I see a strong analogy between the duality of operators (as actions) and density operators used with trace to form dual cooperators and the duality of chain and differential form in calculus.

I'm of course greatly influenced by my thesis advisor and mentor David Finkelstein. He has a book "Quantum Relativity" which you might find interesting.

 

[ArjenDijksman]

I'm of course greatly influenced by my thesis advisor and mentor David Finkelstein. He has a book "Quantum Relativity" which you might find interesting.

David Finkelstein, Georgia Tech. I better understand why you emphasize the role of operators. I created http://en.wikiquote.org/wiki/David_Finkelstein" [/I]" and acknowledge that we speak a different quantum dialect: I'm more inclined to put forward analogies, whether mathematical or mechanical, but always experimentally testable. Nevertheless, there are interesting thoughts on your website.

Greetings,
Arjen

 

[carllooper]

... In an actual measurement process the quantum (if you want to call her Alice) is observed each trial in either |M> or |N> modes. Those cases where "Bob and Alice are not at the same slot" are not simply discarded trials that didn't yield valid results. Some Bob always sees Alice. In thinking of Bob as the measuring device, there are in fact an array of "Bob"'s at each location announcing "Hey! Alice just bumped into Me!" (For example a cloud chamber is a room of Bobs who so state by condensing a droplet when ionizing Alice passes by.) ...

Lets suppose we have a Bob array with only two elements (Bob0 and Bob1) and one Alice. And that there are two locations Alice and Bob can occupy (X and Y). For simplicity when Bob is occupying location X, assume Bob0 occupys location X and his spatially correlated partner Bob1 occupys location Y - and vice versa (ie. one oreintation of Bob is upside down with respect to the other orientation)

Holding X,Y stationary (so to speak) there are four possible outcomes:

X: Bob0 Y: Alice, Bob1
X: Bob0,Alice Y: Bob1
X: Bob1 Y: Alice, Bob0
X: Bob1,Alice Y: Bob0

So the probability of anyone of these outcomes remains the joint probability: 1/4.

Now while we can say, for example, that Alice meets, say Bob0, 2 out of 4 times. Or "some Bob always sees Alice" 4 out of 4 times, such statements are conflating otherwise different (ie. inconsistent) worlds.

In other words, the analogy has a little more scope than one might assume.

Carl