Hilbert space in QM

[geordief]

I am learning  that Hilbert space is .very central to QM.

Does that mean that every aspect (,or attribute?) of a quantum system (I think one has to use the term "system" rather than "object") exists in something like it's own dimension?

Edited September 29, 2022 by geordief

[studiot]

2 hours ago, geordief said:

I am learning  that Hilbert space is .very central to QM.

Does that mean that every aspect (,or attribute?) of a quantum system (I think one has to use the term "system" rather than "object") exists in something like it's own dimension?

Nothing fancy about Hilbert spaces.

They are just ordinary cartesian x,y,z...   'spaces' that have an  infinite count of dimensions.

So the space is not ordinary physical space  -  it is phase space, which is a fancy way of saying that it has as many  dimensions as necessary to draw a 'graph' in.

 

[exchemist]

2 hours ago, studiot said:

Nothing fancy about Hilbert spaces.

They are just ordinary cartesian x,y,z...   'spaces' that have an  infinite count of dimensions.

So the space is not ordinary physical space  -  it is phase space, which is a fancy way of saying that it has as many  dimensions as necessary to draw a 'graph' in.

 

Nice explanation +1. Regarding @geordief's question about QM entities and dimensions, I suppose eigenstates being orthogonal means each state a QM entity can be in is  in a different dimension, doesn't it? 

[uncool]

5 hours ago, studiot said:

They are just ordinary cartesian x,y,z...   'spaces' that have an  infinite count of dimensions.

Bit more than that; don't forget the inner product. I'd say that that's the central aspect of Hilbert spaces.

7 hours ago, geordief said:

Does that mean that every aspect (,or attribute?) of a quantum system (I think one has to use the term "system" rather than "object") exists in something like it's own dimension?

Depends on what you mean by "aspect" or "attribute"; I'd also be careful about what you mean when you say "something like its own dimension".

If you mean that each particle (and simplifying to quantum mechanics, where there is no particle creation or annihilation), then kind-of, but not really. To explain further, and in some formality: the state space for a combination of particles comes from the (completion of the) tensor product of the state spaces of the individual particles, not a direct sum. Sorry, but I don't really have a less technical explanation for what that means.

3 hours ago, exchemist said:

Regarding @geordief's question about QM entities and dimensions, I suppose eigenstates being orthogonal means each state a QM entity can be in is  in a different dimension, doesn't it? 

I'd say that each eigenstate a QM "entity" can be in can be thought of somewhat separately, yes (until you get some time evolution that mixes between eigenstates). To my recollection (which is admittedly rusty), the use of "state" can be somewhat ambiguous; it can either be used to mean a single eigenstate (of an unspecified operator), or can mean an arbitrary mixture. In the latter case, I'd say that the answer to your question is "no"; for example, the the states |a>, (3/5) |a> + (4/5) |b>, and |b> cannot really be thought of as in "different dimensions".

[studiot]

15 minutes ago, uncool said:

Bit more than that; don't forget the inner product. I'd say that that's the central aspect of Hilbert spaces.

You always add something worthwhile to a thread.  +1

Yes they can have quite  a few extra properties

[exchemist]

38 minutes ago, uncool said:

Bit more than that; don't forget the inner product. I'd say that that's the central aspect of Hilbert spaces.

Depends on what you mean by "aspect" or "attribute"; I'd also be careful about what you mean when you say "something like its own dimension".

If you mean that each particle (and simplifying to quantum mechanics, where there is no particle creation or annihilation), then kind-of, but not really. To explain further, and in some formality: the state space for a combination of particles comes from the (completion of the) tensor product of the state spaces of the individual particles, not a direct sum. Sorry, but I don't really have a less technical explanation for what that means.

I'd say that each eigenstate a QM "entity" can be in can be thought of somewhat separately, yes (until you get some time evolution that mixes between eigenstates). To my recollection (which is admittedly rusty), the use of "state" can be somewhat ambiguous; it can either be used to mean a single eigenstate (of an unspecified operator), or can mean an arbitrary mixture. In the latter case, I'd say that the answer to your question is "no"; for example, the the states |a>, (3/5) |a> + (4/5) |b>, and |b> cannot really be thought of as in "different dimensions".

Ah yes, mixtures of states. That doesn't fit the idea of clean separate dimensions, indeed.

Well I hope @geordief gets something out this at least.  It seems to me important to stress that Hilbert space is an abstract mathematical concept and one should not think of these "dimensions" in the loose way that the word is often employed in sci-fi, denoting a series of alternative universes to ours or anything like that.    

[geordief]

11 minutes ago, exchemist said:

Ah yes, mixtures of states. That doesn't fit the idea of clean separate dimensions, indeed.

Well I hope @geordief gets something out this at least.  It seems to me important to stress that Hilbert space is an abstract mathematical concept and one should not think of these "dimensions" in the loose way that the word is often employed in sci-fi, denoting a series of alternative universes to ours or anything like that.    

Well I have only just gained any understanding  of this area at all and so any new knowledge of the model  is quite a boost.

I think I may have already gleaned from the little I have read that even the ,4 spatiotemporal  dimensions  can be modeled in this Hilbert phase space .

 

Maybe this use of Hilbert  space to model reality  gives another sense of what dimensions are,but  would I be right to feel that the 4 spatiotemporal  dimensions are  a different beast to the  other eigenstates(if that is the term) even if the Hilbert space model treats them the same?

[MigL]

11 hours ago, geordief said:

Does that mean that every aspect (,or attribute?) of a quantum system (I think one has to use the term "system" rather than "object") exists in something like it's own dimension?

The 'model' facilitates calculations and predictions.
Don't confuse the 'model' with 'reality'.

[geordief]

1 hour ago, MigL said:

The 'model' facilitates calculations and predictions.
Don't confuse the 'model' with 'reality'.

Well I only have a sketchy understanding  of parts of the quantum model itself .Perhaps I was wondering whether  the model was treating these "dimensions "  in more or less the same way as we treat dimensions in the macro world or at least in a very similar way.

(Dimensions in the macro world are also a model,aren't they?The macro world and the micro world are ,in reality the same place-there is only one universe**)

 

**unless that can be disputed😀

Edited September 30, 2022 by geordief

[studiot]

12 hours ago, exchemist said:

It seems to me important to stress that Hilbert space is an abstract mathematical concept and one should not think of these "dimensions" in the loose way that the word is often employed in sci-fi, denoting a series of alternative universes to ours or anything like that.    

 

9 hours ago, MigL said:

The 'model' facilitates calculations and predictions.
Don't confuse the 'model' with 'reality'.

 

Great advice, +1

 

Please remember that in Physics we work from the premise that we 'observe'  such and such a phenomenon and then try to develop theory to explain, model, or place that observation in.

Usually Physics borrows from Mathematics to do this.

But Mathematics works the other way round. It starts with a theoretical model or mathematical structure and doesn't care whether there are any physical applications or not.

Hilbert space is one such mathematical construct. Don'r forget there are many different Hilbert spaces.

Like all mathematical 'spaces' it comprises a collection of several sets.

Being linear, it has one or more sets of mathematical vectors, a set of coefficients, a set of rules.

But Nature is under no obligation to follow these, it is up to us to choose the most suitable HS for our purposes ie the one that most closely matches our needs.

 

A very simple HS would be:-

Vector sets)   a set of forces, a set of displacements.

Coefficient set)  The set of real numbers.

Rule set)   Rules include the inner product 'work'  = force times displacement. Being linear means we can add up the inner products of work and that the coefficients scale the work ie twice the force or twice the displacement leads to twice the work.

Note although very simple, this space includes at least one transfinite set.

[swansont]

8 hours ago, geordief said:

Well I only have a sketchy understanding  of parts of the quantum model itself .Perhaps I was wondering whether  the model was treating these "dimensions "  in more or less the same way as we treat dimensions in the macro world or at least in a very similar way.

They are similar (analogous), in that vectors in each dimension are orthogonal to vectors in a different dimension. So if you take a dot product you get zero - there is no way to represent a vector in one dimension as a linear combination of other vectors in the orthogonal directions. This concept applies to the eigenstates mentioned above, and why one might sometimes think of them as dimensions, but for eigenfunctions instead of vectors. 

[geordief]

25 minutes ago, swansont said:

They are similar (analogous), in that vectors in each dimension are orthogonal to vectors in a different dimension. So if you take a dot product you get zero - there is no way to represent a vector in one dimension as a linear combination of other vectors in the orthogonal directions. This concept applies to the eigenstates mentioned above, and why one might sometimes think of them as dimensions, but for eigenfunctions instead of vectors. 

What would be the maximum  number of these eigenfunctions for any particular system?

 

[swansont]

5 minutes ago, geordief said:

What would be the maximum  number of these eigenfunctions for any particular system?

 

It's infinite in many systems. Legendre polynomials, which show up in the solution for the hydrogen atom, are one example.

https://people.iith.ac.in/bpriyo/Ananya.pdf