Spectral Decomposition of the State Space

[Lynch101]

TL;DR Summary

Trying to get a better understanding of the above.

I'm looking to check my understanding of the information below and ultimately get a better understanding of it.

The observable in the SG experiment is the spin of the particle, not the flash on the photo plate. More generally, an observable of a system is constructed from a spectral decomposition of the state space of the measured system, not the state space of the system performing the measurement. The flash of the photo plate would be better described as a pointer.

Is spectral decomposition a mathematical procedure?
Does "the state space of the measured system" refer to the possible values that the system could take, when measured?

 

[Demystifier]

Is spectral decomposition a mathematical procedure?
Does "the state space of a the measured system" refer to the possible values that the system could take, when measured?

Yes and yes.

 

[Lynch101]

Yes and yes.

Thank you.

Am I right in thinking spectral decomposition is a mathematical procedure applied to matrices with the entries of the resulting matrix representing the possible observable values? Or is that not close at all?

 

[Morbert]

Thank you.

Am I right in thinking spectral decomposition is a mathematical procedure applied to matrices with the entries of the resulting matrix representing the possible observable values? Or is that not close at all?

What I said above was not quite right. It would be better to say the measured system has some state space $\mathcal{H}$. The system is a particle and we are interested in an observable like the spin-x of the particle. We would projectively decompose the identity operator on the state space into the possible measurement results. The two measurement results are spin-up and spin-down and we write the projective decomposition as $$|\uparrow_x\rangle\langle\uparrow_x| + |\downarrow_x\rangle\langle\downarrow_x|$$If we then add the eigenvalues (the values of the possible measurement results) as coefficients of their respective projector, we get $$s_{\uparrow_x}|\uparrow_x\rangle\langle\uparrow_x| + s_{\downarrow_x}|\downarrow_x\rangle\langle\downarrow_x|$$ This last expression is the spectral decomposition of the observable we are interested in measuring. The important point is observables belong to the system being measured rather than the system doing the measurement.

 

[PeterDonis]

Is spectral decomposition a mathematical procedure?

Yes. See my post in the previous thread in response to what you quoted in the OP of this one:

https://www.physicsforums.com/threads/nature-physics-on-quantum-foundations.1045477/post-6812120

In the case of the SG experiment, the "spectral decomposition" of spin is a way of mathematically describing the experiment that, while it simplifies the math, leaves out all the complications that I described in that post, which are important if you want to understand what is actually, physically going on in the experiment.

 

[PeterDonis]

The important point is observables belong to the system being measured rather than the system doing the measurement.

As I noted in my post in the previous thread in response to yours, the flash on the photo plate is an observable of the system being measured (the particle): it is a measurement of its momentum (by measuring which output beam of the SG magnet the particle came out in). The function of the SG magnet is to entangle the particle's momentum with its spin so that its measured momentum can be used to deduce its spin.

 

[Morbert]

As I noted in my post in the previous thread in response to yours [...]

I'll respond in that thread.

 

[Demystifier]

Am I right in thinking spectral decomposition is a mathematical procedure applied to matrices with the entries of the resulting matrix representing the possible observable values?

You are essentially right. A minor technical subtlety is that operators in infinite dimensional Hilbert spaces are not exactly matrices so spectral decomposition is mathematically a bit more complicated, but in most physics applications one can ignore it and pretend that one still works with "infinite matrices".

 

[Lynch101]

Thanks Morbert!

What I said above was not quite right. It would be better to say the measured system has some state space $\mathcal{H}$. The system is a particle and we are interested in an observable like the spin-x of the particle.

Is the state space and/or the observables sometimes written in matrix format?

We would projectively decompose the identity operator on the state space into the possible measurement results. The two measurement results are spin-up and spin-down and we write the projective decomposition as $$|\uparrow_x\rangle\langle\uparrow_x| + |\downarrow_x\rangle\langle\downarrow_x|$$

Am I right in saying that spin-up and spin-down are the "observables"?

Do they correspond then to the "flashes on the photo screen" i.e. the observable referred to as spin-up corresponds to a flash on the detector plate of the measurement device, which would be considered a "spin-up measurement"?

If we then add the eigenvalues (the values of the possible measurement results) as coefficients of their respective projector, we get $$s_{\uparrow_x}|\uparrow_x\rangle\langle\uparrow_x| + s_{\downarrow_x}|\downarrow_x\rangle\langle\downarrow_x|$$ This last expression is the spectral decomposition of the observable we are interested in measuring. The important point is observables belong to the system being measured rather than the system doing the measurement.

Are the probabilities of measurement outcomes calculated from this?

 

[PeterDonis]

Am I right in saying that spin-up and spin-down are the "observables"?

Do they correspond then to the "flashes on the photo screen" i.e. the observable referred to as spin-up corresponds to a flash on the detector plate of the measurement device, which would be considered a "spin-up measurement"?

See the discussion in the other thread linked to in posts #5 and #7. The short answer is "it's more complicated than that".

 

[Lynch101]

See the discussion in the other thread linked to in posts #5 and #7. The short answer is "it's more complicated than that".

Thanks Peter, I saw that and understand the clarification. "Correspond" might be the wrong word in that case.

My understanding is that the spin-up "observable" (in the mathematics) relates to the flash on the screen from which we conclude the measurement was spin-up. It represents it (in some way).

Would that be a reasonable understanding?

 

[PeterDonis]

My understanding is that the spin-up "observable" (in the mathematics) relates to the flash on the screen from which we conclude the measurement was spin-up. It represents it (in some way).

I would say that the experiment is set up so that the flash on the screen gives information about the spin observable via a chain of inferences (how many inferences you want to put in the chain depends on how detailed you want to get about the processes that take place to produce the flash).

 

[Morbert]

Am I right in saying that spin-up and spin-down are the "observables"?

Spin would be the observable (say spin in the x direction). Spin-up and spin-down would be the values spin can take, in the same way the top number on a rolled die is an observable of the die, and 1-6 are the possible values this observable can take.

Observables can be represented as matrices but they don't have to be.

Do they correspond then to the "flashes on the photo screen" i.e. the observable referred to as spin-up corresponds to a flash on the detector plate of the measurement device, which would be considered a "spin-up measurement"?

Are the probabilities of measurement outcomes calculated from this?

There is a correspondence in the sense that quantum mechanics will report probabilities for the different values of spin that will correspond to the relative frequencies of the positions of flashes on the detector plate. The significance of this correspondence will be interpretation-dependent.

 

[Lynch101]

Observables can be represented as matrices but they don't have to be. There is a correspondence in the sense that quantum mechanics will report probabilities for the different values of spin that will correspond to the relative frequencies of the positions of flashes on the detector plate. The significance of this correspondence will be interpretation-dependent.

For all practical purposes, when we talk about experimental verification of the predictions of QM, are the flashes on the detector plate what either confirms (or disconfirms) the predictions? Is there also some statistical analysis involved?

 

[Morbert]

For all practical purposes, when we talk about experimental verification of the predictions of QM, are the flashes on the detector plate what either confirms (or disconfirms) the predictions? Is there also some statistical analysis involved?

Experimental data is used to validate quantum mechanics, and statistical analysis of the data is central to relating data to theory.

But more broadly, quantum mechanics has also been validated through its essential contribution to various industries. The nanoscale structure of the chips in your computer, the electronic structure of the molecules in your pharmaceuticals, the photonics in your transcievers, all owe their effectiveness to the application of quantum physics by researchers.

 

[PeterDonis]

are the flashes on the detector plate what either confirms (or disconfirms) the predictions? Is there also some statistical analysis involved?

Since the predictions of QM for such experiments are of probabilities, statistical analysis has to be involved in comparing the predictions with experimental results.

 

[Lynch101]

Since the predictions of QM for such experiments are of probabilities, statistical analysis has to be involved in comparing the predictions with experimental results.

Is it just a case of comparing the relative frequency of the flashes on the detector plate to the predictions of QM?

Just to outline my basic thinking (wrt position).
1) QM makes statistical predictions about position.
2) Experiments are run to test the predictions.
3) Experimental observations are made of flashes on a detector plate.
4) Statistical analysis of those flashes is performed.
5) The statistical analysis of the flashes is compared to the predictions of QM.

The above would take the flashes on the screen as a proxy for the position of the quantum system, which is then compared to the predictions (of position) made by QM. Is that the case, that the flashes are taken as a proxy for the position of the quantum system, for the purpose of comparison to the predictions?

 

[PeterDonis]

Is it just a case of comparing the relative frequency of the flashes on the detector plate to the predictions of QM?

Hasn't this already been answered?

 

[Lynch101]

Hasn't this already been answered?

I think it's been stated that there is a "correspondence" not necessarily that FAPP, the flash on the detector plate is a direct proxy for the position of the quantum system.

It's been mentioned that there are assumptions about whether or not it is a direct proxy for position, I just wanted to make sure I wasn't missing a key piece of information that statistical analysis is also involved in that aspect.