Understanding Interference in Quantum Ray States

[Varon]

A ray in Hilbert State is pure quantum state. All interference terms are supposed to be there. How does the interference work, between what basis vector, components or properties?

 

[Varon]

What I'm asking is simply this, there is supposed to be interferences in quantum coherence. I want to know which part in Hilbert Space is interfering?

 

[Varon]

It's like the basis vectors have phase.. and the phase of each basis vector can interfere with one another, really? So to create superposition, you have to make each basis interfere?

Doesn't anyone understand what I'm talking about?

 

[SpectraCat]

It's like the basis vectors have phase.. and the phase of each basis vector can interfere with one another, really? So to create superposition, you have to make each basis interfere?

Doesn't anyone understand what I'm talking about?

This is my understanding:

The Hilbert space is abstract, and does not really have the property of interference. Those come out when you project the pure quantum states into (for example) the position representation.

So, in the vector space, a superposition is just a weighted sum of basis vectors. What we observe as interference is the consequence of projecting the superposition into a given representation.

 

[Varon]

This is my understanding:

The Hilbert space is abstract, and does not really have the property of interference. Those come out when you project the pure quantum states into (for example) the position representation.

So, in the vector space, a superposition is just a weighted sum of basis vectors. What we observe as interference is the consequence of projecting the superposition into a given representation.

In the double slit experiment, the single electron can interfere with itself. I wonder how Hilbert space model this interference.

Btw. Pls. try to clarify what kind of Copenhagenists are you so we won't go into endless confusion if you are just basing it on your own theory. I want to be able to distinguish the variants of the Copenhagen Interpretation whether it is Bohr's, Heisenberg's, Einstein's, Wheeler's, Wigner's, etc. In other word, pls. identify your affiliation distinctly. If you are not of your affiliation, pls. meditate on it first, then let us know.

 

[SpectraCat]

In the double slit experiment, the single electron can interfere with itself. I wonder how Hilbert space model this interference.

Btw. Pls. try to clarify what kind of Copenhagenists are you so we won't go into endless confusion if you are just basing it on your own theory. I want to be able to distinguish the variants of the Copenhagen Interpretation whether it is Bohr's, Heisenberg's, Einstein's, Wheeler's, Wigner's, etc. In other word, pls. identify your affiliation distinctly. If you are not of your affiliation, pls. meditate on it first, then let us know.

I definitely don't have my own theory .. I go by standard Q.M., and as I just posted in the other thread, I am more of an Instrumentalist than a follower of the CI. [EDIT: I am happy to use Hilbert space representations as tools because they are useful abstractions that give correct results about experimental measurements when properly used within the framework of QM. ]

Hilbert spaces are unconnected to any interpretation .. they are purely mathematical abstractions. Furthermore, as I said, to the best of my knowledge, interference is not a property possessed by vectors in Hilbert spaces, so there is no "modeling" of interferences in Hilbert spaces.

The interference in the double-slit experiment arises because the electron becomes entangled with the measuring apparatus, so that it is in a superposition of two states, each representing one of the possible outcomes. If you want to now represent each of those states in an abstract Hilbert space, fine .. call them |1> and |2>. In that case the superposition is just |1> + |2>. Again, the question of interference doesn't arise until you project those abstract states back into the lab frame somehow (i.e. the position representation).

I believe that this is the interpretation of the double slit experiment given in standard QM.

 

[Varon]

I definitely don't have my own theory .. I go by standard Q.M., and as I just posted in the other thread, I am more of an Instrumentalist than a follower of the CI. [EDIT: I am happy to use Hilbert space representations as tools because they are useful abstractions that give correct results about experimental measurements when properly used within the framework of QM. ]

Hilbert spaces are unconnected to any interpretation .. they are purely mathematical abstractions. Furthermore, as I said, to the best of my knowledge, interference is not a property possessed by vectors in Hilbert spaces, so there is no "modeling" of interferences in Hilbert spaces.

The interference in the double-slit experiment arises because the electron becomes entangled with the measuring apparatus, so that it is in a superposition of two states, each representing one of the possible outcomes. If you want to now represent each of those states in an abstract Hilbert space, fine .. call them |1> and |2>. In that case the superposition is just |1> + |2>. Again, the question of interference doesn't arise until you project those abstract states back into the lab frame somehow (i.e. the position representation).

I believe that this is the interpretation of the double slit experiment given in standard QM.

Ok. So you are a pragmatist/instrumentalist.

Anyway. In the general concept of Quantum Coherence, there is interference.. so what's interfering in quantum coherence?

Hope other Copenhagenists (not affiliated with the pragmatic approach) but more of ontology, etc. can share too.

 

[SpectraCat]

Ok. So you are a pragmatist/instrumentalist.

Anyway. In the general concept of Quantum Coherence, there is interference.. so what's interfering in quantum coherence?

Hope other Copenhagenists (not affiliated with the pragmatic approach) but more of ontology, etc. can share too.

The wavefunction is interfering in quantum coherence, and the wavefunction is the position representation of the pure quantum state from the Hilbert space.

 

[Varon]

The wavefunction is interfering in quantum coherence, and the wavefunction is the position representation of the pure quantum state from the Hilbert space.

What do you think of the following, how the interference is modeled in Hilbert Space which you deny:

http://www.ipod.org.uk/reality/reality_decoherence.asp

"Now here is the absolutely key point: every component eigenstate has an associated phase (this was considered back in The Quantum Casino). It is this phase which gives the wavefunction its "wavelike" character (in complex space, remember). In order for the components to combine together correctly to produce a superposition state, they must be in the same phase (must be coherent). This is what happens in the double-slit experiment: interference components possessing the same phase combine to produce the interference effects."

 

[SpectraCat]

What do you think of the following, how the interference is modeled in Hilbert Space which you deny:

http://www.ipod.org.uk/reality/reality_decoherence.asp

"Now here is the absolutely key point: every component eigenstate has an associated phase (this was considered back in The Quantum Casino). It is this phase which gives the wavefunction its "wavelike" character (in complex space, remember). In order for the components to combine together correctly to produce a superposition state, they must be in the same phase (must be coherent). This is what happens in the double-slit experiment: interference components possessing the same phase combine to produce the interference effects."

Ok I agree with most of that, but what does it have to do with Hilbert space representations? I guess I still don't understand what you are asking, but maybe this will help. The complex phases of the different components come from the projection into a given eigenbasis:

Consider for example the arbitrary state |a>. Now suppose that we want to represent that as a superposition in some complete orthonormal eigenbasis |s> (where s is an indexing variable over all the elements of the eigenbasis). You can use the completeness relation to extract the expansion coefficients as follows:

[EDIT: Sorry .. tex formatting is buggy ... can't fix it .. will try again]

The c_s expansion coefficients are in general complex, and contain the information about the complex phases of the basis states. They determine which specific interference pattern will be observed when a given superposition is projected into a wavefunction representation. Is that what you were looking for?

 

[Varon]

Ok I agree with most of that, but what does it have to do with Hilbert space representations? I guess I still don't understand what you are asking, but maybe this will help. The complex phases of the different components come from the projection into a given eigenbasis:

What do you mean what it has got to do with Hilbert space representations. Isn't it the Eigenstates and Eigen vectors are located in Hilbert Space?

 

[SpectraCat]

What do you mean what it has got to do with Hilbert space representations. Isn't it the Eigenstates and Eigen vectors are located in Hilbert Space?

Yes, you can represent it that way, but that description presents interference as being a property of the wavefunction, exactly as I have done.

To follow up my previous post, the information that leads to interference effects in the wavefunction is encoded in the expansion coefficients of the superposition in the Hilbert space representation. In the Hilbert space it is just a way of choosing a representation for the quantum state .. it is still a just a vector though.

 

[Varon]

Ok I agree with most of that, but what does it have to do with Hilbert space representations? I guess I still don't understand what you are asking, but maybe this will help. The complex phases of the different components come from the projection into a given eigenbasis:

Consider for example the arbitrary state |a>. Now suppose that we want to represent that as a superposition in some complete orthonormal eigenbasis |s> (where s is an indexing variable over all the elements of the eigenbasis). You can use the completeness relation to extract the expansion coefficients as follows:

[EDIT: Sorry .. tex formatting is buggy ... can't fix it .. will try again]

The c_s expansion coefficients are in general complex, and contain the information about the complex phases of the basis states. They determine which specific interference pattern will be observed when a given superposition is projected into a wavefunction representation. Is that what you were looking for?

You sure the interferences is only in position representation, and not on momentum, spin or charge representation?

Btw.. are you a student of physics or a graduate or just an enthusiast? Are you a novice? I'm asking because I don't want other novices to further create more confusion for me.

Anyway what do you think of the following (continuation of what the author in the website said above):

"What happens to a quantum particle in the real world is that each of its component states gets entangled (separately) with different aspects of its environment. As seen in the page on Quantum Entanglement, when particles become entangled you have to consider them as one single, entangled state (you use the tensor product to calculate the resultant state). So each component of our quantum particle forms separate entangled states. The phases of these states will be altered. This destroys the coherent phase relationships between the components. The components are said to decohere."

 

[SpectraCat]

You sure the interferences is only in position representation, and not on momentum, spin or charge representation?

Who said anything about "only" the position representation? Of course there the relationships between the complex phases will manifest somehow in other representations, and it might even be natural to call them interferences .. I am not used to thinking in the momentum representation, so I can't say for sure without further consideration. However, the phenomenon normally referred to as interference is a property of the wavefunction representation in position space.

Btw.. are you a student of physics or a graduate or just an enthusiast? Are you a novice? I'm asking because I don't want other novices to further create more confusion for me.

I am a professor of Physical Chemistry.

Anyway what do you think of the following (continuation of what the author in the website said above):

"What happens to a quantum particle in the real world is that each of its component states gets entangled (separately) with different aspects of its environment. As seen in the page on Quantum Entanglement, when particles become entangled you have to consider them as one single, entangled state (you use the tensor product to calculate the resultant state). So each component of our quantum particle forms separate entangled states. The phases of these states will be altered. This destroys the coherent phase relationships between the components. The components are said to decohere."

I think that it is a good description of the phenomenon of decoherence, and does a good job of explaining why quantum effects are not generally observable for macroscopic systems.

 

[Varon]

Yes, you can represent it that way, but that description presents interference as being a property of the wavefunction, exactly as I have done.

To follow up my previous post, the information that leads to interference effects in the wavefunction is encoded in the expansion coefficients of the superposition in the Hilbert space representation. In the Hilbert space it is just a way of choosing a representation for the quantum state .. it is still a just a vector though.

But isn't it that "wave function" is exactly the same thing as the "state vector" in Hilbert space? as in:

http://en.wikipedia.org/wiki/Wave_function#Definition

In quantum entanglement. The phases are entangled.. but how can the phases be entangled when the state vector is not yet collapsed to the basis vector where the phase information is located?