The Quantum Wave Function Revisited

[Alan Grayson]

Assuming we know all possible results of the measurements of a quantum system, that is, the set of possible eigenvalues, and suppose we also know the associated eigenfunctions, and we write the wf of the system as a linear sum of eigenfunctions each multiplied by a complex constant, is it mathematically assumed, or proven somewhere (perhaps by Von Neumann), that these eigenfunctions are orthogonal and form a basis for the Hilbert space in which they reside? TY, AG

[Brent Meeker]

On 7/29/2025 7:18 AM, Alan Grayson wrote:

Assuming we know all possible results of the measurements of a quantum system, that is, the set of possible eigenvalues, and suppose we also know the associated eigenfunctions, and we write the wf of the system as a linear sum of eigenfunctions each multiplied by a complex constant, is it mathematically assumed, or proven somewhere (perhaps by Von Neumann), that these eigenfunctions are orthogonal and form a basis for the Hilbert space in which they reside? TY, AG --

Yes, that's pretty much it.  The physical system, including the ideal measurement, is modeled by a certain Hilbert space in which the basis states are the eigenfunctions the measurement.  This is implicit in the concept of an ideal measurement as one, which if immediately repeated on the same system, returns the same value again.

Brent

[Alan Grayson]

But is it proven or assumed the eigenfunctions in the sum are basis states which span the space? If proven, where, by whom; if not, then the construct lacks rigor.  AG 

[John Clark]

On Tue, Jul 29, 2025 at 4:12 PM Alan Grayson <agrays...@gmail.com> wrote:

> But is it proven

Yes, but not all proofs are mathematical, some proofs are physical, that's why physicists need to perform experiments.  No mathematician in previous centuries, no matter how intelligent, could've sat in his armchair with a notepad and derived quantum mechanics.  

John K Clark    See what's on my new list at  Extropolis

edd

[Brent Meeker]

It's true by construction that the eigenstates span the Hilbert space.  "The Hilbert space" is the space whose bases are the eigenstates.

Brent

[Alan Grayson]

What does "true by construction" mean? Does that include orthogonality of the basis eigenstates? AG 

[Alan Grayson]

In the UP/DN case of electrons passing through the SG apparatus, the spin might be non-existent, just another quantum number. But the fact that the electrons respond to the magnetic field as they do, in opposite directions, strongly suggests IMO that they have real physical spin. Hence, the idea of modeling them as orthogonal instead of anti-parallel makes no sense. AG

[Bruce Kellett]

What does "true by construction" mean? Does that include orthogonality of the basis eigenstates? AG 

A lot of these things are proved in Dirac's book "The Principles of Quantum Mechanics". For example, the orthogonality of the eigenfunctions of a single operator is proved on page 32 (of my edition). "Two eigenvectors of a real dynamical variable belonging to different eigenvalues are orthogonal".

Any set of linearly independent vectors forms a possible basis of a vector space if the number of linearly independent vectors equals the dimension of the space (The linearly independent vectors need not form a mutually orthogonal set, as long as they are linearly independent.)

Bruce

[Russell Standish]

On Tue, Jul 29, 2025 at 04:42:13PM -0700, Alan Grayson wrote:

Eigenvectors of a hermitian operator are orthogonal (or can be
trivially orthogonalised in the case of multiple linearly independent
eignvectors with the same eigenvalue).

>
> What does "true by construction" mean? Does that include orthogonality of the
> basis eigenstates? AG 
>

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[Russell Standish]

On Wed, Jul 30, 2025 at 10:01:00AM +1000, Bruce Kellett wrote:
>
> Any set of linearly independent vectors forms a possible basis of a vector
> space if the number of linearly independent vectors equals the dimension of the
> space (The linearly independent vectors need not form a mutually orthogonal
> set, as long as they are linearly independent.)
>

True for finite spaces, not necessarily true of infinite
spaces. Consider the space of polynomials, then the linear span of
xⁿwhen n is even also has dimension ℵ₀, but doesn't span the whole
space.

It becomes even trickier with unbounded linear operators (such as d/dx
used for momentum), of course. The answer is that for physical
situations, only well-behaved operators are used, or sometimes rather
well-studied operators (like d/dx) in place of their physical bounded
equivalents as a computational convenience, whilst ignoring the
incovenient nuances.

[Alan Grayson]

On Tuesday, July 29, 2025 at 6:01:15 PM UTC-6 Bruce Kellett wrote:

On Wed, Jul 30, 2025 at 9:42 AM Alan Grayson <agrays...@gmail.com> wrote:

On Tuesday, July 29, 2025 at 5:33:40 PM UTC-6 Brent Meeker wrote:

On 7/29/2025 1:12 PM, Alan Grayson wrote:

On Tuesday, July 29, 2025 at 2:04:31 PM UTC-6 Brent Meeker wrote:

On 7/29/2025 7:18 AM, Alan Grayson wrote:

Assuming we know all possible results of the measurements of a quantum system, that is, the set of possible eigenvalues, and suppose we also know the associated eigenfunctions, and we write the wf of the system as a linear sum of eigenfunctions each multiplied by a complex constant, is it mathematically assumed, or proven somewhere (perhaps by Von Neumann), that these eigenfunctions are orthogonal and form a basis for the Hilbert space in which they reside? TY, AG --

Yes, that's pretty much it.  The physical system, including the ideal measurement, is modeled by a certain Hilbert space in which the basis states are the eigenfunctions the measurement.  This is implicit in the concept of an ideal measurement as one, which if immediately repeated on the same system, returns the same value again.

Brent

But is it proven or assumed the eigenfunctions in the sum are basis states which span the space? If proven, where, by whom; if not, then the construct lacks rigor.  AG 

It's true by construction that the eigenstates span the Hilbert space.  "The Hilbert space" is the space whose bases are the eigenstates.

What does "true by construction" mean? Does that include orthogonality of the basis eigenstates? AG 

A lot of these things are proved in Dirac's book "The Principles of Quantum Mechanics". For example, the orthogonality of the eigenfunctions of a single operator is proved on page 32 (of my edition). "Two eigenvectors of a real dynamical variable belonging to different eigenvalues are orthogonal".

Thanks for the reference. I recently bought that book! AG 

[Brent Meeker]

It means you find the eigenstates and then those are the basis of your Hilbert space.  The eigenstates are automatically orthogonal.

Brent

[Brent Meeker]

On 7/29/2025 4:47 PM, Alan Grayson wrote:

On Tuesday, July 29, 2025 at 2:22:50 PM UTC-6 John Clark wrote:

On Tue, Jul 29, 2025 at 4:12 PM Alan Grayson <agrays...@gmail.com> wrote:

> But is it proven

Yes, but not all proofs are mathematical, some proofs are physical, that's why physicists need to perform experiments.  No mathematician in previous centuries, no matter how intelligent, could've sat in his armchair with a notepad and derived quantum mechanics.  

John K Clark    See what's on my new list at  Extropolis

In the UP/DN case of electrons passing through the SG apparatus, the spin might be non-existent,

Electrons are spin 1/2 particles. But as I've noted several times, you don't send electrons, or any charged particles, thru an SG.

 just another quantum number. But the fact that the electrons respond to the magnetic field as they do, in opposite directions, strongly suggests IMO that they have real physical spin. 

Silver atoms are used because they have a big magnetic moment.

Hence, the idea of modeling them as orthogonal instead of anti-parallel makes no sense. AG

Spin UP and spin DN are orthogonal in Hilbert space.  In 3-space they are anti-parallel.

Brent

[Alan Grayson]

Naive question; how do those silver atoms know they're in Hiibert space, and not 3-space.? AG

[Bruce Kellett]

On Wed, Jul 30, 2025 at 3:03 PM Alan Grayson <agrays...@gmail.com> wrote:

Naive question; how do those silver atoms know they're in Hiibert space, and not 3-space.? AG

The silver atoms are not 'in' Hilbert space! Hilbert space is an abstract mathematical construct used to describe the quantum behaviour of particular systems; in this case, the quantum behaviour of spin one-half particles.

Bruce

[John Clark]

On Tue, Jul 29, 2025 at 7:47 PM Alan Grayson <agrays...@gmail.com> wrote:

> In the UP/DN case of electrons passing through the SG apparatus, the spin might be non-existent, just another quantum number. But the fact that the electrons respond to the magnetic field as they do, in opposite directions, strongly suggests IMO that they have real physical spin.

Electrons certainly have physical spin, but do they have "real" physical spin, that is to say do they always have one and only one physical spin state? You send electrons through a magnetic field that has been oriented in some RANDOM direction and note that some of the electrons swerve in that direction and you decide, for no particular reason, to call that direction "up". If you send those up-electrons through a similar magnetic field they will swerve in that same direction with 100% certainty.

However if you send those up-electrons through a magnetic field oriented in the orthogonal direction, which you decide to call the left/right direction, and then send them through the original up magnetic field again you'll find that the electrons no longer behave in the way they did before, instead 50% of them will go up but 50% will go down. That is very strange but that is nevertheless a fact.   

 > Hence, the idea of modeling them as orthogonal instead of anti-parallel makes no sense. AG

You feel that the idea is ridiculous and you're not alone, some very intelligent people have been saying that about quantum mechanics for about a century now, but nature does not share their opinion. And it's important to remember that being ridiculous is not the same as being paradoxical.  

[Brent Meeker]

Naive indeed!  They're in both.

Brent

[Alan Grayson]

On Wednesday, July 30, 2025 at 12:26:25 PM UTC-6 Brent Meeker wrote:

Naive indeed!  They're in both.

Brent

LOL. Or "in" neither. But Clark denies the SG silver atoms are 3D. He also denies that many of my questions about mathematical proofs about Hilbert spaces are non existent, and must be resolved experimentally, such as orthogonality of the basis vectors. But as BK indicated, the proofs I am seeking reside in Dirac's book on quantum mechanics (which I recently purchased). AG

[John Clark]

On Thu, Jul 31, 2025 at 2:06 AM Alan Grayson <agrays...@gmail.com> wrote:

> Naive question; how do those silver atoms know they're in Hiibert space, and not 3-space.? AG

As Brent says, they're in both. Over many years mathematicians have invented thousands of different mathematical objects, a physicist's job is to figure out, with the help of experiments, which physical traits should be symbolized by which mathematical objects. If you want to make the best prediction possible about what an electron (or a neutral silver atom) is going to do in real 3-D space you're going to need complex 2-D Hilbert space. 

> LOL.

No, I simply do not believe you are laughing out loud over Brent's comment. 

> But Clark denies the SG silver atoms are 3D

And no, Clark does NOT deny "the SG silver atoms are 3D".

John K Clark    See what's on my new list at  Extropolis

jxq

[Brent Meeker]

On 7/30/2025 11:12 PM, Alan Grayson wrote:

On Wednesday, July 30, 2025 at 12:26:25 PM UTC-6 Brent Meeker wrote:

Naive indeed!  They're in both.

Brent

LOL. Or "in" neither. But Clark denies the SG silver atoms are 3D. 

I thought he wrote the atoms were in 3-space.  But not in Hilbert space.  The thing to keep in mind that 3-space and Hilbert space are both mathematical constructs that capture some aspects of reality.

He also denies that many of my questions about mathematical proofs about Hilbert spaces are non existent, and must be resolved experimentally, such as orthogonality of the basis vectors. But as BK indicated, the proofs I am seeking reside in Dirac's book on quantum mechanics (which I recently purchased). AG

The whole point of the mathematical constructs is that you can use them to make inferences and, the extent your construct fits reality, your inferences will match reality.

Brent

[Alan Grayson]

On Thursday, July 31, 2025 at 5:06:53 AM UTC-6 John Clark wrote:

On Thu, Jul 31, 2025 at 2:06 AM Alan Grayson <agrays...@gmail.com> wrote:

> Naive question; how do those silver atoms know they're in Hiibert space, and not 3-space.? AG

As Brent says, they're in both. Over many years mathematicians have invented thousands of different mathematical objects, a physicist's job is to figure out, with the help of experiments, which physical traits should be symbolized by which mathematical objects. If you want to make the best prediction possible about what an electron (or a neutral silver atom) is going to do in real 3-D space you're going to need complex 2-D Hilbert space. 

> LOL.

No, I simply do not believe you are laughing out loud over Brent's comment. 

My LOL refers to the fact that Brent allows the SG silver atoms to "reside" in Hilbert and 3d space, whereas BK denies both, and yet they're both correct in some sense.  AG 

[Alan Grayson]

On Thursday, July 31, 2025 at 12:37:20 PM UTC-6 Brent Meeker wrote:

On 7/30/2025 11:12 PM, Alan Grayson wrote:

On Wednesday, July 30, 2025 at 12:26:25 PM UTC-6 Brent Meeker wrote:

Naive indeed!  They're in both.

Brent

LOL. Or "in" neither. But Clark denies the SG silver atoms are 3D. 

I thought he wrote the atoms were in 3-space.  But not in Hilbert space. 

I thought it was the opposite, in Hilbert but not in 3-space. But he did state several times that my questions about Hilbert space could only be resolved by experiments, not mathematical proofs. I was sure this was mistaken, and it was. AG