# Why are Hilbert spaces used in quantum mechanics?

• I
• Frank Castle
In summary, classical mechanics uses a 6n-dimensional phase space, itself a vector space, to describe the state of a given system at anyone point in time, with the evolution of the state of a system being described in terms of a trajectory through the corresponding phase space. However, in quantum mechanics we instead use Hilbert spaces. What is the intuition for why we use Hilbert spaces? Is it simply due to the non-commutative nature of observables and the fact that we now have operators acting on states to extract observable quantities and also time evolve them, and these operators naturally act on a Hilbert space?Note that 'Hilbert space' is a pretty general concept, every vector space with an inner product (and no 'missing points
Frank Castle
In classical mechanics we use a 6n-dimensional phase space, itself a vector space, to describe the state of a given system at anyone point in time, with the evolution of the state of a system being described in terms of a trajectory through the corresponding phase space. However, in quantum mechanics we instead use Hilbert spaces. What is the intuitive reasoning for why we use Hilbert spaces? Is it simply due to the non-commutative nature of observables and the fact that we now have operators acting on states to extract observable quantities and also time evolve them, and these operators naturally act on a Hilbert space?

Note that 'Hilbert space' is a pretty general concept, every vector space with an inner product (and no 'missing points' e.g. like real numbers but not only the rational ones) is a hilbert space. Classical phase space is also a Hilbert space in this sense where positions and momenta constitute the most useful basis vectors.

In Quantum mechanics, the whole system cannot be described in terms of separate particles each having different positions and momenta. Still, you can define some hilbert space describing the possible states of the particles. Then, one usually choses a basis in this quantum hilbert space as the eigenvectors of an operator of interest.

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thephystudent said:
Note that 'Hilbert space' is a pretty general concept, every vector space with an inner product (and no 'missing points' e.g. like real numbers but not only the rational ones) is a hilbert space. Classical phase space is also a Hilbert space in this sense where positions and momenta constitute the most useful basis vectors.

In Quantum mechanics, the whole system cannot be described in terms of separate particles each having different positions and momenta. Still, you can define some hilbert space describing the possible states of the particles. Then, one usually choses a basis in this quantum hilbert space as the eigenvectors of an operator of interest.

Why do we distinguish between classical phase space and Hilbert spaces then? In all introductions that I've read on quantum mechanics , Hilbert spaces are introduced with little motivation and its also implied that they are a new space (that wasn't present in classical mechanics) in which quantum states exist.

Frank Castle said:
Why do we distinguish between classical phase space and Hilbert spaces then? In all introductions that I've read on quantum mechanics , Hilbert spaces are introduced with little motivation and its also implied that they are a new space (that wasn't present in classical mechanics) in which quantum states exist.

See:

It's so matrix methods can be linked to the functions in Schrodinger equation. The wave-functions of Schrodinger's equation need to form a vector space which naturally leads to a Hilbert space.

If you want something deeper this explains it from a deep analysis of the logical foundations of QM:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Be warned - its what mathematicians call non trivial - meaning its hard.

One of the key results used is Pirons theorem:
https://www.quora.com/What-is-the-significance-of-Pirons-theorem

It shows quantum logic nearly, but not quite, implies a Hilbert Space is required.

Recently a new theorem, called Solers Theorem, gets us even closer:
https://golem.ph.utexas.edu/category/2010/12/solers_theorem.html

Its tantalizingly close to proving Hilbert spaces are required, but its still not there.

Thanks
Bill

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bhobba said:
See:

It's so matrix methods can be linked to the functions in Schrodinger equation. The wave-functions of Schrodinger's equation need to form a vector space which naturally leads to a Hilbert space.

If you want something deeper this explains it from a deep analysis of the logical foundations of QM:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Be warned - its what mathematicians call non trivial - meaning its hard.

One of the key results used is Pirons theorem:
https://www.quora.com/What-is-the-significance-of-Pirons-theorem

It shows quantum logic nearly, but not quite, implies a Hilbert Space is required.

Recently a new theorem, called Solers Theorem, gets us even closer:
https://golem.ph.utexas.edu/category/2010/12/solers_theorem.html

Its tantalizingly close to proving Hilbert spaces are required, but its still not there.

Thanks
Bill

Thanks for all the links, I shall have a read through and get back if I have any further queries.

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thephystudent said:
Classical phase space is also a Hilbert space in this sense where positions and momenta constitute the most useful basis vectors.

Classical phase space is the union (as sets) of a bunch of vector spaces (cotangent spaces), but is not itself a vector space, it is a vector bundle.

vanhees71, Truecrimson and bhobba
George Jones said:
Classical phase space is the union (as sets) of a bunch of vector spaces (cotangent spaces), but is not itself a vector space, it is a vector bundle.

You are absolutely right, I simplified too much. I'll delete my post
EDIT: too late to remove it

Frank Castle said:
Why do we distinguish between classical phase space and Hilbert spaces then?

Phase spaces emphasize the symplectic geometry while Hilbert spaces emphasize the orthogonal geometry. The former leads to Hamilton's equations and the latter gives a statistical distance between two quantum states. IMHO, in the end it is a matter of convenience because quantum mechanics also has a symplectic structure and we can do statistical mechanics on a phase space.

Quantum mechanics can be formulated in a phase space by allowing negative distributions
https://en.wikipedia.org/wiki/Phase_space_formulation

Classical mechanics can be formulated in a complex Hilbert space with no non-commuting observables, I think. I'm not familiar with this approach.
https://en.wikipedia.org/wiki/Koopman–von_Neumann_classical_mechanics

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bhobba
This discussion is a bit over my head, but I was curious about one point. Do "ordinary" vector spaces allow complex values?

bhobba said:
It's so matrix methods can be linked to the functions in Schrodinger equation. The wave-functions of Schrodinger's equation need to form a vector space which naturally leads to a Hilbert space.

So are Hilbert spaces simply introduced such that Schrödinger's wave mechanics and Heisenberg's matrix mechanics are isomorphic within this space? The wave functions need to form a vector space due to the linearity of the Schrödinger equation, hence they should satisfy the vector space axioms?

Let's restrict ourselves to finite dimensional spaces first because of complications in the infinite dimensional case. A finite dimensional Hilbert space is just a vector space with an inner product, which we use to compute the transition probability between two states.

sandy stone said:
This discussion is a bit over my head, but I was curious about one point. Do "ordinary" vector spaces allow complex values?
Sure, you can have vector spaces with any field (in the sense of the mathematical structure, in German called a "Körper") as scalars.

Any ##n##-dimensional complex vector space is equivalent (via a basis) with the vector space ##\mathbb{C}^n## of ##n##-tupels of complex numbers.

If the square of the modulus of the amplitude of the wave function is going to represent a probability distribution then the wave function must be square integrable. That is: it lies in the Hilbert space of square integrable complex valued functions.

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lavinia said:
If the square of the modulus of the amplitude of the wave function is going to represent a probability distribution then the wave function must be square integrable. That is: it lies in the Hilbert space of square integrable complex valued functions.

Is this the main motivation for the usage of Hilbert spaces in quantum mechanics, or are there other motivating qualities?

Well, the most convincing argument for using a mathematical structure in physics is that it works, i.e., it is successful in describing what's objectively and reproducibly observed in Nature, and the Hilbert space is precisely the mathematical structure that is needed to describe matter from quarks and leptons up to neutron stars. That's why it's used.

Zafa Pi
vanhees71 said:
Well, the most convincing argument for using a mathematical structure in physics is that it works, i.e., it is successful in describing what's objectively and reproducibly observed in Nature, and the Hilbert space is precisely the mathematical structure that is needed to describe matter from quarks and leptons up to neutron stars. That's why it's used.
Yes, I understand the reasoning at that level, but I'm struggling a bit with the mathematical reasoning as to why one uses Hilbert spaces in quantum mechanics? Is it simply because the normed vector space structure complies with the linearity of the Schrödinger equation and allows one to construct probability densities and a notion of unit story. Additionally, completeness guarantees that one can use calculus and thus the Schrödinger equation and that wave functions are bounded and their decomposition onto sets of basis vectors are convergent?!

Frank Castle said:
Yes, I understand the reasoning at that level, but I'm struggling a bit with the mathematical reasoning as to why one uses Hilbert spaces in quantum mechanics?

If you want the physical reason and not from QM foundations its got to do with the requirement to have continuous transformations between pure states:
https://arxiv.org/pdf/quant-ph/0101012v4.pdf

Guys - I know I link to that paper a lot. But I really do beieve its very important as far as quantum foundations go. QM is almost pulled out of thin air.

Thanks
Bill

bhobba said:
Guys - I know I link to that paper a lot. But I really do beieve its very important as far as quantum foundations go. QM is almost pulled out of thin air.
I don't think mentioning a paper a lot is a bad thing. Its just that when you mention a paper, you should be careful not to present it as a mainstream opinion in the scientific community if its not.

Its correct that its an interesting view but its far from satisfactory and also I think it introduces more problems than it solves. Assuming that QM is only another kind of probability theory, a mathematical theory applied to physics, just makes the philosphical issues of QM worse than before! Its definitely not a price I'm eager to pay only to derive QM from some axioms! I really prefer the out-of-thin-air QM to this proposal.

P.S.
Also it doesn't treat the continuous observbles.

Frank Castle said:
Is this the main motivation for the usage of Hilbert spaces in quantum mechanics, or are there other motivating qualities?

I am new to this stuff but Leonard Susskind says that unlike classical state space, quantum state space is a vector space - which means that linear combinations of states are also states. So the square integrablility of states is not the only use of Hilbert space. Quantum Mechanics also uses its linear structure.

Hilbert space is the only normed linear space that has an inner product. In Quantum Mechanics, inner products of two states represent transition amplitudes from one quantum state to the next. So these transition amplitudes are orthogonal projections of one state onto another. So quantum Mechanics uses the inner product feature of Hilbert space as well.

Measurements in Quantum Mechanics are linear operators on the vector space of quantum states. For each operator there is an orthonormal basis of " eigen states "(eigen vectors of the operator). The idea of orthonormality only makes sense in a Hilbert space.

These eigen vectors are the possible states that a ensue after a measurement. When measuring a property of a quantum state, the outcome of the measurement is uncertain. It can be any of the eigen states of the linear operator. The square of the probability that the measurement will land in a particular eigen state is the Hilbert space inner product of the quantum state with the eigen state.

So the full suite of features of Hilbert space are used to describe quantum states.

- I wonder if every vector in the Hilbert space is a possible quantum state.

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Frank Castle said:
Yes, I understand the reasoning at that level, but I'm struggling a bit with the mathematical reasoning as to why one uses Hilbert spaces in quantum mechanics? Is it simply because the normed vector space structure complies with the linearity of the Schrödinger equation and allows one to construct probability densities and a notion of unit story. Additionally, completeness guarantees that one can use calculus and thus the Schrödinger equation and that wave functions are bounded and their decomposition onto sets of basis vectors are convergent?!

I wonder if you don't to some extent have it a bit backwards? The basics of the QM formalism was "created" by various physicists during the first couple of decades of the 20th century and not all of them were well versed in "formal" math or eve what we would now consider fairly basic algebra (remember that Heisenberg had to "re-invent" matrix mechanics, he was not taught it at university because at the time it was too obscure). Hence, QM was only "formalized" a bit later by others (e..g. Hilbert himself but also e,g. von Neumann). Hence, I suspect quite a few of the people who did calculations had not idea whatsoever of which space they were using. They just knew that what what they were doing was working since it matched experiments.

Truecrimson, bhobba and PeroK
Shyan said:
Also it doesn't treat the continuous observbles.

There is a way to handle it using RHS's, but not everyone agrees. Assuming the criticisms are valid, and I don't believe they are, then it is a BIG problem.

If you want to pursue it best to start a new thread where I can explain how it's handled. Also most foundational approaches have the issue, so, like I said before, if it can't be handled we are in deep do do.

Thanks
Bill

lavinia said:
.I wonder if every vector in the Hilbert space is a possible quantum state.

It called the strong principle of superposition. Its usually assumed, but it is an assumption.

Thanks
Bill

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bhobba said:
There is a way to handle it using RHS's, but not everyone agrees. Assuming the criticisms are valid, and I don't believe they are, then it is a BIG problem.

If you want to pursue it best to start a new thread where I can explain how it's handled. Also most foundational approaches have the issue, so, like I said before, if it can't be handled we are in deep do do.

Thanks
Bill
My main issues with the proposal are the philosophical ones.(Actually I think there are physical issues too!)
But I'm not much into the business of deriving QM from postulates so probably I don't know enough about different approaches to have a discussion about it, I just take your word for it.

bhobba said:
It called the strong principle of superposition. Its usually assumed, but it is an assumption.

Thanks
Bill

f95toli said:
I wonder if you don't to some extent have it a bit backwards? The basics of the QM formalism was "created" by various physicists during the first couple of decades of the 20th century and not all of them were well versed in "formal" math or eve what we would now consider fairly basic algebra (remember that Heisenberg had to "re-invent" matrix mechanics, he was not taught it at university because at the time it was too obscure). Hence, QM was only "formalized" a bit later by others (e..g. Hilbert himself but also e,g. von Neumann). Hence, I suspect quite a few of the people who did calculations had not idea whatsoever of which space they were using. They just knew that what what they were doing was working since it matched experiments.

That is a good point. Did Hilbert, von Neumann and others arrive at the Hilbert space formalism since it provides a way to consistently unify the (originally distinct) formalisms of wave mechanics and matrix mechanics and thus provides a coherent description of quantum systems that leads to the same observables?

Frank Castle said:
That is a good point. Did Hilbert, von Neumann and others arrive at the Hilbert space formalism since it provides a way to consistently unify the (originally distinct) formalisms of wave mechanics and matrix mechanics and thus provides a coherent description of quantum systems that leads to the same observables?

Of course.

See the link in post 4.

Von Neumann sorted it out at a rigorous level over a number of years beginning from the early days to publishing his famous book.

The other approach was the transformation theory of Dirac published in 1926. It had mathematical issues that caused Von Neumann to despair (read the introduction to hi text - its actually a bit nasty ). The mathematicians however didn't leave it at that and sorted out how to make it legit - but it took some of the greatest mathematicians of the 20th century such as Grothendieck.

Thanks
Bill

## 1. Why are Hilbert spaces used in quantum mechanics?

Hilbert spaces are used in quantum mechanics because they provide a mathematical framework for describing the behavior of quantum systems. These spaces are uniquely suited for representing the states of quantum particles, allowing for calculations of various physical quantities such as energy, momentum, and position.

## 2. What makes Hilbert spaces different from other mathematical spaces?

Hilbert spaces are different from other mathematical spaces because they are complete, meaning that they contain all possible limits of convergent sequences. This property is crucial in quantum mechanics, as it allows for the accurate representation of the continuous and probabilistic nature of quantum systems.

## 3. Can other mathematical spaces be used in place of Hilbert spaces in quantum mechanics?

While other mathematical spaces can be used to describe quantum systems, Hilbert spaces are the most commonly used and have proven to be the most effective. They have a number of important properties that make them well-suited for quantum mechanics, including the ability to describe complex states and the existence of a natural inner product.

## 4. How are Hilbert spaces related to the uncertainty principle?

Hilbert spaces are intimately related to the uncertainty principle in quantum mechanics. The uncertainty principle states that certain physical quantities, such as position and momentum, cannot be known simultaneously with high precision. Hilbert spaces allow for the calculation of the uncertainty in these quantities through the use of operators, which represent physical observables in quantum mechanics.

## 5. Are Hilbert spaces used in all areas of quantum mechanics?

Yes, Hilbert spaces are used in all areas of quantum mechanics, from basic calculations of particle states to advanced theories such as quantum field theory. They are an essential mathematical tool for understanding and predicting the behavior of quantum systems, and have been widely accepted and used since the early development of quantum mechanics in the early 20th century.

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