Re: Some questions about Quantum Mechanics

[Brent Meeker]

On 3/16/2025 1:51 PM, Alan Grayson wrote:

1) What necessitates the use of complex numbers (whereas in GR only real numbers are used)?

QM exhibits interference so it must have wave-like phases that can add and subtract.  It predicts probabilities which must be positive numbers.  So one way to do this mathematically is to have probability amplitudes, Psi, that are the "square root" of probabilities, Psi*Psi (where * denotes the Hermitian conjugate), that have phases so they can interfere.  Then the dynamics are linear in the Psi.

2) What necessitates the postulates that some, but presumably not all operators are non commuting?

3) With respect to 2), why is the non commuting difference i*h (or i*hbar)?

It is conjugate pairs that fail to commute.  See attached.

Brent

[Alan Grayson]

What above waves in spacetime? What's waving, and do they interfere with each other? AG 

[Alan Grayson]

What ABOUT waves in spacetime, that is gravitational waves? What's waving, and do they interfere with each other? AG 

[Brent Meeker]

That would be gravitational waves and what's waving is the gravitational potential, i.e. local curvature of the metric.  And yes they interfere, although I can't think of any direct measurement of that.

Brent

[Alan Grayson]

As I recall, the gravitational potential is the negative derivative of the gravitational force, but if there's no force in GR, how is that measured?  AG

[John Clark]

On Thu, Mar 20, 2025 at 1:07 AM Alan Grayson <agrays...@gmail.com> wrote:

> As I recall, the gravitational potential is the negative derivative of the gravitational force, but if there's no force in GR, how is that measured?  AG

The gravitational potential energy of an object at a particular location is related to how much slower time runs there compared to infinity where spacetime is flat, let's call that factor X. The weaker the gravitational field is the closer  X comes to be equal to 1. On Earth's surface gravity is so weak X is 0.9999999997, so you can simplify the very complex tensor equations in General Relativity to the simple Newtonian equations you learned in high school and you get an excellent approximation. 

However when gravity starts to get strong X starts to approach zero, and at a Black Hole it is zero, so in General Relativity the Newtonian idea of energy, and not just gravitational potential energy, becomes problematic.  General Relativity doesn't consider energy and momentum to be two separate things, there is only the "stress–energy–momentum tensor", it is a complete description of how much stuff there is and how it is moving and flowing and exerting pressure or tension.  But the result is when gravity becomes very large and space-time has become very curved you can't point to a spot and ask how much gravitational potential energy is there at that point because the question has become meaningless . 

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

2br

[Alan Grayson]

Is the gravity wave detected by measuring the vibrating change of the distance between the two separated locations of the detector? If so, how is this a variation of spacetime, instead of just a measurement of spacial differences? AG

[John Clark]

On Thu, Mar 20, 2025 at 11:47 AM Alan Grayson <agrays...@gmail.com> wrote:

> Is the gravity wave detected by measuring the vibrating change of the distance between the two separated locations of the detector? If so, how is this a variation of spacetime, instead of just a measurement of spacial differences? AG

 

LIGO detected the peak to peak displacement that a gravitational wave caused, it does not detect the RMS power in a wave, that's why LIGO's ability to detect gravitational waves only decreases by a factor of 1/r not by 1/r^2 as conventional telescopes that use light or any form of  electromagnetic waves do. However gravitational waves with enormously longer wavelengths can and have been detected by variations in time, not space, by  using pulsars, a.k.a. neutron stars. They detected a gravitational wave "hum" with wavelengths light-years long that were almost certainly caused by the millions of merging supermassive Black Holes, each being billions of times more massive than the sun, that have occurred since the Big Bang.

The NANOGrav 15 yr Data Set: Constraints on Supermassive Black Hole Binaries from the Gravitational Wave Background

It is also been propose that when Thorium 229 nuclear clocks are perfected they could be used to detect gravitational waves. 

Breakthrough promises new era of ultra precise nuclear clocks

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

92t

[Alan Grayson]

On Friday, March 21, 2025 at 6:51:18 AM UTC-6 John Clark wrote:

On Thu, Mar 20, 2025 at 11:47 AM Alan Grayson <agrays...@gmail.com> wrote:

> Is the gravity wave detected by measuring the vibrating change of the distance between the two separated locations of the detector? If so, how is this a variation of spacetime, instead of just a measurement of spacial differences? AG

 

LIGO detected the peak to peak displacement that a gravitational wave caused,

What exactly is waving, space or spacetime, or something else? Variations of time? Time of what? AG

 

it does not detect the RMS power in a wave, that's why LIGO's ability to detect gravitational waves only decreases by a factor of 1/r not by 1/r^2 as conventional telescopes that use light or any form of  electromagnetic waves do. However gravitational waves with enormously longer wavelengths can and have been detected by variations in time, not space, by  using pulsars, a.k.a. neutron stars. They detected a gravitational wave "hum" with wavelengths light-years long that were almost certainly caused by the millions of merging supermassive Black Holes, each being billions of times more massive than the sun, that have occurred since the Big Bang.

The NANOGrav 15 yr Data Set: Constraints on Supermassive Black Hole Binaries from the Gravitational Wave Background

It is also been propose that when Thorium 229 nuclear clocks are perfected they could be used to detect gravitational waves. 

Breakthrough promises new era of ultra precise nuclear clocks

John K Clark    See what's on my new list at  Extropolis2

[John Clark]

On Sun, Mar 23, 2025 at 4:07 AM Alan Grayson <agrays...@gmail.com> wrote:

> What exactly is waving, space or spacetime,

Spacetime. So if you detect a variation in space caused by a gravitational wave then you can use Einstein's equations to figure out what the variation in time must've been, and if you detect a variation in time you can figure out what the variation in space must be. As Einstein's teacher Hermann Minkowski said: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of both will retain an independent reality".

John K Clark    See what's on my new list at  Extropolis2

idr

[Alan Grayson]

Interesting. I keep thinking of spacetime as one thing which is measured, but that's really not the case. Generally speaking, what are the wave lengths of those space variation waves, and how are they measured? AG 

2

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[John Clark]

On Sun, Mar 23, 2025 at 11:18 AM Alan Grayson <agrays...@gmail.com> wrote:

> Generally speaking, what are the wave lengths of those space variation waves, and how are they measured? 

 

LIGO is able to measure the distance between two mirrors 2 1/2 miles apart to an accuracy of  1/10,000 the width of a proton. And you need that sort of accuracy if you want to detect gravitational waves. They achieve this astounding level of precision by measuring the interference effects between two laser beams.   

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

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> What exactly is waving, space or spacetime,

Spacetime. So if you detect a variation in space caused by a gravitational wave then you can use Einstein's equations to figure out what the variation in time must've been, and if you detect a variation in time you can figure out what the variation in space must be. As Einstein's teacher Hermann Minkowski said: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of both will retain an independent reality".

Interesting. I keep thinking of spacetime as one thing which is measured, but that's really not the case. AG 

2

[Alan Grayson]

On Sunday, March 23, 2025 at 10:52:00 AM UTC-6 John Clark wrote:

On Sun, Mar 23, 2025 at 11:18 AM Alan Grayson <agrays...@gmail.com> wrote:

> Generally speaking, what are the wave lengths of those space variation waves, and how are they measured? 

 

LIGO is able to measure the distance between two mirrors 2 1/2 miles apart to an accuracy of  1/10,000 the width of a proton. And you need that sort of accuracy if you want to detect gravitational waves. They achieve this astounding level of precision by measuring the interference effects between two laser beams.   

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

So they measure an interference pattern. How do they know it's a gravitational wave? AG

[John Clark]

On Sun, Mar 23, 2025 at 2:36 PM Alan Grayson <agrays...@gmail.com> wrote:

>> LIGO is able to measure the distance between two mirrors 2 1/2 miles apart to an accuracy of  1/10,000 the width of a proton. And you need that sort of accuracy if you want to detect gravitational waves. They achieve this astounding level of precision by measuring the interference effects between two laser beams.   

> So they measure an interference pattern. How do they know it's a gravitational wave? AG

LIGO is L shaped with each leg being 2 1/2 miles long, theory says gravitational waves should shrink the distance between one leg at the same time it's expanding the distance in the other leg, nothing else could do that. And to make sure they have two identical facilities, one in Louisiana and the other in Oregon, if it's a gravitational wave then the two detectors should measure the same thing at almost the same time because gravitational waves move at the speed of light, any slight delay between the two can help them figure out the direction the gravitational wave is coming from.    

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

e3b

[John Clark]

Correction: the two LIGO installations are in Louisiana and Washington state. not Oregon as I originally said. 

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

n0w

e3b

[Alan Grayson]

On Sunday, March 23, 2025 at 1:06:06 PM UTC-6 John Clark wrote:

Correction: the two LIGO installations are in Louisiana and Washington state. not Oregon as I originally said. 

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

What do you mean by L shaped, if there are two separate detectors? AG 

[John Clark]

On Sun, Mar 23, 2025 at 3:14 PM Alan Grayson <agrays...@gmail.com> wrote:

On Sunday, March 23, 2025 at 1:06:06 PM UTC-6 John Clark wrote:

Correction: the two LIGO installations are in Louisiana and Washington state. not Oregon as I originally said. 

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

What do you mean by L shaped, if there are two separate detectors? AG  

There’s an L-shaped detector in Louisiana, and another one in Washington state, Both Detectors have two arms and both arms are 2 1/2 miles long.    John K Clark

n0w

e3b

On Sun, Mar 23, 2025 at 2:47 PM John Clark <johnk...@gmail.com> wrote:

On Sun, Mar 23, 2025 at 2:36 PM Alan Grayson <agrays...@gmail.com> wrote:

 

>> LIGO is able to measure the distance between two mirrors 2 1/2 miles apart to an accuracy of  1/10,000 the width of a proton. And you need that sort of accuracy if you want to detect gravitational waves. They achieve this astounding level of precision by measuring the interference effects between two laser beams.   

> So they measure an interference pattern. How do they know it's a gravitational wave? AG

LIGO is L shaped with each leg being 2 1/2 miles long, theory says gravitational waves should shrink the distance between one leg at the same time it's expanding the distance in the other leg, nothing else could do that. And to make sure they have two identical facilities, one in Louisiana and the other in Oregon, if it's a gravitational wave then the two detectors should measure the same thing at almost the same time because gravitational waves move at the speed of light, any slight delay between the two can help them figure out the direction the gravitational wave is coming from.    

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

e3b

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[Brent Meeker]

There are also four other gravitational wave detectors that work on the same principle:

GEO600 has an arm-length of 600m and is located in Ruthe, Germany (near Hanover). It is operated by the Max Planck Institute for Gravitational Physics in Hanover. The GEO collaboration consists of 250 members and is part of the LIGO scientific collaboration.

LIGO is an abbreviation for “Laser Interferometer Gravitational Wave Observatory”. The LIGO project has two interferometric detectors in the United States, “Advanced LIGO”, each with arm lengths of 4 kilometres. They are located in Hanford, Washington, and Livingston, Louisiana. A third detector of the LIGO network has been under construction in India in 2020.

Virgo is a European detector with an arm-length of 3 kilometres near Pisa in Italy. The initial detector, Virgo, was upgraded to achieve 10 times higher sensitivity and started operation as Advanced Virgo in 2017.

KAGRA (Kamioka Gravitational Wave Detector) is an underground gravitational wave detector in the Kamioka mine in Japan. It uses cyrogenic technology, which means that the mirrors are cooled down to 20 Kelvin in order to reduce thermal noise. The detector has been in operation since 2020.

https://www.einstein-online.info/en/spotlight/gw_detectors/

Brent

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[Alan Grayson]

On Monday, March 17, 2025 at 9:45:50 PM UTC-6 Brent Meeker wrote:

That's the definition of conjugate pairs, that they don't commute. I notice that E and t are also considered conjugate pairs, but since t is a parameter in QM and not an operator, how can that be intelligible? AG 

[Brent Meeker]

On 3/29/2025 8:40 AM, Alan Grayson wrote:

On Monday, March 17, 2025 at 9:45:50 PM UTC-6 Brent Meeker wrote:

On 3/16/2025 1:51 PM, Alan Grayson wrote:

1) What necessitates the use of complex numbers (whereas in GR only real numbers are used)?

QM exhibits interference so it must have wave-like phases that can add and subtract.  It predicts probabilities which must be positive numbers.  So one way to do this mathematically is to have probability amplitudes, Psi, that are the "square root" of probabilities, Psi*Psi (where * denotes the Hermitian conjugate), that have phases so they can interfere.  Then the dynamics are linear in the Psi.

2) What necessitates the postulates that some, but presumably not all operators are non commuting?

3) With respect to 2), why is the non commuting difference i*h (or i*hbar)?

It is conjugate pairs that fail to commute.  See attached.

Brent

That's the definition of conjugate pairs, that they don't commute.

No it's not.  They are variables related by a Fourier transform.

I notice that E and t are also considered conjugate pairs, but since t is a parameter in QM and not an operator, how can that be intelligible? AG 

I've posted it before.  In quantum mechanics energy and the time per unit change of a variable are conjugate variables. So they satisfy an Heisenberg uncertainty relation, often written [math]\Delta E \Delta t \geq \hbar[math\] . This is sloppy though and not quite right. What is right is given any operator A and the Hamiltonian H defining the time evolution of A, then [math]\Delta A \Delta H \geq \frac{1}{2} \hbar [d<A>/dt][math\] .

Brent

Brent

[Alan Grayson]

On Saturday, March 29, 2025 at 4:10:44 PM UTC-6 Brent Meeker wrote:

On 3/29/2025 8:40 AM, Alan Grayson wrote:

      On Monday, March 17, 2025 at 9:45:50 PM UTC-6 Brent Meeker wrote:

              On 3/16/2025 1:51 PM, Alan Grayson wrote:

1) What necessitates the use of complex numbers (whereas in GR only real numbers are used)?

QM exhibits interference so it must have wave-like phases that can add and subtract.  It predicts probabilities which must be positive numbers.  So one way to do this mathematically is to have probability amplitudes, Psi, that are the "square root" of probabilities, Psi*Psi (where * denotes the Hermitian conjugate), that have phases so they can interfere.  Then the dynamics are linear in the Psi.

2) What necessitates the postulates that some, but presumably not all operators are non commuting?

3) With respect to 2), why is the non commuting difference i*h (or i*hbar)?

It is conjugate pairs that fail to commute.  See attached.

Brent

That's the definition of conjugate pairs, that they don't commute.

No it's not.  They are variables related by a Fourier transform.

I googled conjugate operators and this is what I got. No mention of Fourier transform.

  

In the context of linear algebra and operator theory, the "conjugate" of an operator, often referred to as the Hermitian conjugate or adjoint, is obtained by taking the complex conjugate of each element in the operator's matrix representation and then transposing the matrix. 

AG

I notice that E and t are also considered conjugate pairs, but since t is a parameter in QM and not an operator, how can that be intelligible? AG 

I've posted it before.  In quantum mechanics energy and the time per unit change of a variable are conjugate variables. So they satisfy an Heisenberg uncertainty relation, often written [math]\Delta E \Delta t \geq \hbar[math\] . This is sloppy though and not quite right. What is right is given any operator A and the Hamiltonian H defining the time evolution of A, then [math]\Delta A \Delta H \geq \frac{1}{2} \hbar [d<A>/dt][math\] .

What is the domain and range of the operator you call "Time per unit change of a variable"? AG

Brent