Isn't "eigensubspace" in Postulate 5 really just everyday space?

[BohmianRealist]

From: https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics

In the section, "Effect of measurement on the state" (aka Postulate 5), it says:

Therefore the state vector must change as a result of measurement, and collapse onto the eigensubspace associated with the eigenvalue measured.

If this is just a fancy way of talking about our very own space in which we live (ie, in which an observation has been made), then why not just say so?

 

[PeroK]

From: https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics

In the section, "Effect of measurement on the state" (aka Postulate 5), it says:

If this is just a fancy way of talking about our very own space in which we live (ie, in which an observation has been made), then why not just say so?

The eigensubspace in question is a subspace of quantum states, and depends on the observable.

 

[BohmianRealist]

"a subspace of quantum states" sounds very much like Hilbert space (which of course doesn't depend on the observable) to me.

I take it that you would answer "no" to the question as posed. In that case, I would like to know more about how this kind of "sub"-space would fit into the next (and last) postulate, "Time evolution of a system".

 

[PeroK]

The Hilbert space depends on the system. The state after measurement depends on what is observed. It must be an eigenstate of the observable and different observables have different eigenstates. All these states are part of the same Hilbert space.

Time evolution is what happens to the state before or after measurement. In general, after measurement the state evolves away from the eigenstate; unless the observable is compatible with the energy observable. I.e. that the operator representing the observable commutes with the Hamiltonian.

Ultimately, you are probably going to need a better grasp of linear algebra to digest this stuff.

 

[BohmianRealist]

You used the word "space" exactly once. Then, you kept using the word "state", either by itself or in combination with the prefix eigen-. My question was about space, plain and simple. You seem to want to veer the discussion away from that idea for some reason.

I just have no earthly clue what the phrase "eigensubspace" is supposed to denote (if it isn't the everyday space in which we live).

 

[PeterDonis]

I just have no earthly clue what the phrase "eigensubspace" is supposed to denote

It means the subspace of the total Hilbert space that contains state vectors which correspond to the measured eigenvalue (i.e., are eigenvectors of the measurement operator with that eigenvalue). In simplified thought experiments, it is assumed that only one state vector is an eigenvector of the measurement operator with the given eigenvalue, but in the general case that's not true; there will be a whole continuous range of state vectors that all have that property. That continuous range of state vectors is the eigensubspace referred to.

 

[PeroK]

I just have no earthly clue what the phrase "eigensubspace" is supposed to denote (if it isn't the everyday space in which we live).

That's why you need to learn some linear algebra. Vector spaces, linear operators and eigenthings etc.

 

[BohmianRealist]

"It means the subspace of the total Hilbert space"

...seems to logically contradict the ideas of "after measurement" and "collapsing onto" contained in the phrase...

"and collapse onto the eigensubspace associated with the eigenvalue measured"

In other words, a Hilbert space by definition contains [pure] quantum states. I'm quite sure that measurements have nothing to do with projecting backward onto the Hilbert space. I'm pretty sure they have to do with projecting forward somewhere else. I've always thought that the just mentioned "somewhere else" is the everyday space in which a given observation was made (ie, via some kind of experimental setup).

 

[PeterDonis]

"It means the subspace of the total Hilbert space"

...seems to logically contradict the ideas of "after measurement" and "collapsing onto" contained in the phrase...

"and collapse onto the eigensubspace associated with the eigenvalue measured"

Before measurement, the state vector could be any vector in the entire Hilbert space.

After measurement, the state vector has to be somewhere in the subspace of the Hilbert space that contains all the state vectors that are eigenvectors of the measurement operator, with eigenvalue equal to the measured eigenvalue.

That is what the language you refer to is saying.

I'm quite sure that measurements have nothing to do with projecting backward onto the Hilbert space.

Since "projecting backward onto the Hilbert space" is meaningless, you are correct here.

I'm pretty sure they have to do with projecting forward somewhere else.

Here, however, you are incorrect, since "projecting forward somewhere else" is also meaningless. The "projection" in question is not "backward" or "forward". It is just what I described above.

I've always thought that the just mentioned "somewhere else" is the everyday space in which a given observation was made (ie, via some kind of experimental setup).

Then you thought incorrectly.

 

[BohmianRealist]

Unless we can get to the final postulate, "Time evolution of a system", I am afraid you are making self-identical claims regarding the mathematical machinery of linear operators over the [non-physical] abstraction called Hilbert space. The statements you are making concerning "subspaces" are all taking place at precisely the same point in time, and so there is not any possibility of time evolution inherent in the phrase, "somewhere in the subspace of the Hilbert space".

The very idea of the passage of physical time in the Schrödinger equation implies the existence of a physical space wherein the [pure] quantum states [of Hilbert space] are somehow intermixed and interacting with each other. (The precise nature of this interaction is not important at the moment. It might arise due to the maintenance of some symmetry of the entire system, for instance.)

So now, if the collapse due to a measurement does in fact project onto some kind of concrete space (such as Euclidean space) within which it makes sense to start making mathematically provable claims about physical reality, then (and only then) can we get to the entire purpose (time evolution of the entire system as governed by the Schrödinger equation) of the realm of thought known as quantum mechanics.

 

[PeterDonis]

The very idea of the passage of physical time in the Schrödinger equation implies the existence of a physical space wherein the [pure] quantum states [of Hilbert space] are somehow intermixed and interacting with each other.

No, it doesn't. The Schrödinger Equation evolution takes place in Hilbert space. The link with actual physical space is through the separate QM postulates about measurement.

if the collapse due to a measurement does in fact project onto some kind of concrete space (such as Euclidean space)

It doesn't. You have already been told this.

 

[PeterDonis]

The OP question has been answered. Continuing to repeat the answer is pointless. Thread closed.