Validation of QM Local Prod Calc Pauli Matrices

[FrediFizzx]

The work for this wasn't so tremendous after adopting John Reed's method that he used for the Quaternions. The other way I was trying was a lot of work. Here is a PDF of the Mathematica code.

download/QMprod_calc.pdf

And here is a plot of the result



So more proof that the product calculation is correct for the local QM model using QM itself. No big surprise here as we already knew it would work from the QM to GA mapping.
.

[FrediFizzx]

For those that might be interested that have Mathematica, here is a link to download the notebook file,

www.sciphysicsforums.com/spfbb1/download/QMprod_calc.nb
.

[FrediFizzx]

Here is a sample of the correlation data to 9 digit accuracy. The first number is the correlation and the second number is -a.b.

Code: Select all

0.017949549,0.017949549
-0.0938793443,-0.0938793443
-0.789596145,-0.789596145
0.922477832,0.922477832
-0.322036654,-0.322036654
0.395230537,0.395230537
0.757630627,0.757630627
-0.640295221,-0.640295221
-0.327395507,-0.327395507
0.300092385,0.300092385
-0.209833365,-0.209833365
0.70517569,0.70517569
-0.764990807,-0.764990807
-0.474730157,-0.474730157
-0.868153599,-0.868153599
-0.869267941,-0.869267941
-0.602666268,-0.602666268
-0.322720546,-0.322720546
0.0232923634,0.0232923634
0.875370634,0.875370634
0.723617988,0.723617988
0.934730789,0.934730789
-0.331005444,-0.331005444
-0.379023637,-0.379023637
-0.792449733,-0.792449733
-0.441063469,-0.441063469
0.108057849,0.108057849
-0.256954126,-0.256954126
0.627745488,0.627745488
-0.294752727,-0.294752727
-0.398463393,-0.398463393
-0.185674157,-0.185674157
-0.825926328,-0.825926328
0.550036913,0.550036913
0.698602655,0.698602655
-0.992675411,-0.992675411
0.0975430222,0.0975430222
0.790740101,0.790740101
-0.763338655,-0.763338655
0.207484344,0.207484344
0.629799273,0.629799273
0.391404568,0.391404568
-0.0685358598,-0.0685358598
-0.771001405,-0.771001405
0.834117616,0.834117616
-0.995934318,-0.995934318
0.119162173,0.119162173
-0.257385749,-0.257385749
-0.377966588,-0.377966588
-0.192731443,-0.192731443
0.65459252,0.65459252
0.803735122,0.803735122
0.259108379,0.259108379
0.719862624,0.719862624
0.596531103,0.596531103
-0.433634313,-0.433634313
-0.643214111,-0.643214111
0.488317607,0.488317607
0.511413062,0.511413062
-0.663298959,-0.663298959
0.00692383577,0.00692383577
0.365534934,0.365534934
-0.0776455309,-0.0776455309
0.405618687,0.405618687
-0.346737972,-0.346737972
0.0713019485,0.0713019485
-0.723454809,-0.723454809
0.751693929,0.751693929
-0.499445488,-0.499445488
-0.21352609,-0.21352609
-0.785223159,-0.785223159
0.574765814,0.574765814
0.626914433,0.626914433
0.427915922,0.427915922
0.391288536,0.391288536
-0.473157665,-0.473157665
0.307498883,0.307498883
0.663627824,0.663627824
-0.160323667,-0.160323667
-0.0908786081,-0.0908786081
-0.944245989,-0.944245989
0.795248871,0.795248871
-0.766409105,-0.766409105
0.887035994,0.887035994
0.835185484,0.835185484
-0.444994419,-0.444994419
-0.0743864029,-0.0743864029
0.176077951,0.176077951
0.98978712,0.98978712
-0.187589697,-0.187589697
-0.16518859,-0.16518859
0.673625877,0.673625877
-0.864551724,-0.864551724
-0.585735659,-0.585735659
0.374909262,0.374909262
-0.239559603,-0.239559603
-0.811515798,-0.811515798
-0.167552397,-0.167552397
-0.274433247,-0.274433247
0.0548776858,0.0548776858
-0.441253908,-0.441253908
0.731850946,0.731850946
-0.20050776,-0.20050776
-0.141226352,-0.141226352
0.462598928,0.462598928
-0.70000302,-0.70000302
-0.0976652,-0.0976652
-0.756053038,-0.756053038
0.952066912,0.952066912
-0.428566922,-0.428566922
0.132003068,0.132003068
0.286085275,0.286085275
-0.354791548,-0.354791548
-0.0400083341,-0.0400083341
0.166708562,0.166708562
0.686989706,0.686989706
-0.881401398,-0.881401398
0.861895095,0.861895095
0.750834892,0.750834892
0.384457025,0.384457025
-0.326053825,-0.326053825
-0.736869687,-0.736869687
-0.0322222404,-0.0322222404
0.670766959,0.670766959
-0.00721312412,-0.00721312412
-0.599191891,-0.599191891
0.700372963,0.700372963
-0.643265624,-0.643265624
-0.178860902,-0.178860902
0.278945755,0.278945755
0.314571006,0.314571006
-0.0386327407,-0.0386327407
-0.491505303,-0.491505303
0.497310796,0.497310796
0.851954894,0.851954894
0.94284716,0.94284716
0.614649227,0.614649227
0.221957956,0.221957956
-0.519308438,-0.519308438
-0.643143314,-0.643143314
-0.590759742,-0.590759742
0.281922788,0.281922788
0.659773504,0.659773504
0.941360026,0.941360026
0.185575738,0.185575738
-0.561635376,-0.561635376
0.573138888,0.573138888
-0.697103988,-0.697103988
0.228181539,0.228181539
0.244474401,0.244474401
0.308398048,0.308398048
-0.754627216,-0.754627216
0.110541838,0.110541838
0.217793894,0.217793894
0.545836733,0.545836733


.

[FrediFizzx]

I also did a check on the A and B +/-1 data. Out of 10,000 trials I get,

++ = 2517
+- = 2497
-+ = 2515
-- = 2471
total = 10,000

That is pretty close to the QM prediction of 25 percent for each one. I'm sure it gets even better for more trials.
.

[Joy Christian]

I also did a check on the A and B +/-1 data. Out of 10,000 trials I get,

++ = 2517
+- = 2497
-+ = 2515
-- = 2471
total = 10,000

That is pretty close to the QM prediction of 25 percent for each one. I'm sure it gets even better for more trials.
.

They can't be exactly 25%. Quantum mechanical probabilities for them vary sinusoidally, as cos^2 and sin^2 of the angles.

For exact expressions of the quantum probabilities, see Eqs. (2) to (5) of this paper: https://arxiv.org/pdf/1405.2355.pdf.

***

[Heinera]

I assume he has just just used many different angles and accumulated, so these numbers are averages. Of course a lot of things will generate these numbers, not only the QM correlations (for example, random coin tosses on both sides). It would be more interesting if he could post the distribution for just a specific set of angles, e.g. a=0 and b=22.5 (degrees).

[FrediFizzx]

I assume he has just just used many different angles and accumulated, so these numbers are averages. Of course a lot of things will generate these numbers, not only the QM correlations (for example, random coin tosses on both sides). It would be more interesting if he could post the distribution for just a specific set of angles, e.g. a=0 and b=22.5 (degrees).

Those numbers are the totals for how many times I get A, B to be ++, etc. out of 10,000 trials with random 3D vectors. I don't understand what you are asking for with the "distribution"? What good would that do?
.

[Heinera]

Do you mean that A and B are random in your simulation?

[FrediFizzx]

Do you mean that A and B are random in your simulation?

Of course A and B are random +/-1 because a, b and s are all random.
.

[Heinera]

Do you mean that A and B are random in your simulation?

Of course A and B are random +/-1 because a, b and s are all random.
.

So I thought. Can you run your simulation again with A and B set to two specific values, A = 0 and B = 22.5 (degrees), if it's not too much effort? Just curious.

[FrediFizzx]

Do you mean that A and B are random in your simulation?

Of course A and B are random +/-1 because a, b and s are all random.
.

So I thought. Can you run your simulation again with A and B set to two specific values, A = 0 and B = 22.5 (degrees), if it's not too much effort? Just curious.

I suppose you mean the vectors a = 0 and b = 22.5 degrees. A and B are the +/-1 outputs. I might try to play around with fixed vectors. But what is it you want to see with those settings?
.

[Heinera]

I suppose you mean the vectors a = 0 and b = 22.5 degrees. A and B are the +/-1 outputs. I might try to play around with fixed vectors. But what is it you want to see with those settings?
.

Yes, that's what I meant.

What I want to see with those settings is the same as you posted for the random settings, i.e. the numbers for the four combinations.

[FrediFizzx]

I suppose you mean the vectors a = 0 and b = 22.5 degrees. A and B are the +/-1 outputs. I might try to play around with fixed vectors. But what is it you want to see with those settings?
.

Yes, that's what I meant.

What I want to see with those settings is the same as you posted for the random settings, i.e. the numbers for the four combinations.

OK, got it. Now, I suppose those angles are going to be in the x-y plane if z is the propagation axis? The distribution of ++, etc. is not going to change because s is still random.
.

[Heinera]

OK, got it. Now, I suppose those angles are going to be in the x-y plane if z is the propagation axis? The distribution of ++, etc. is not going to change because s is still random.
.

Yes about the x-y plane.

If the distribution is not going to change then your model is completely at odds with QM. Se the post by Joy above.

[FrediFizzx]

OK, got it. Now, I suppose those angles are going to be in the x-y plane if z is the propagation axis? The distribution of ++, etc. is not going to change because s is still random.
.

Yes about the x-y plane.

If the distribution is not going to change then your model is completely at odds with QM. Se the post by Joy above.

Ok, so you finally cough up what you really want to see. Yeah, the distribution will change. It will take me a while to get this figured out. a = 0 is no good because that locks a.s to 0.
.

[Joy Christian]

OK, got it. Now, I suppose those angles are going to be in the x-y plane if z is the propagation axis? The distribution of ++, etc. is not going to change because s is still random.
.

Yes about the x-y plane.

If the distribution is not going to change then your model is completely at odds with QM. Se the post by Joy above.

Ok, so you finally cough up what you really want to see. Yeah, the distribution will change. It will take me a while to get this figured out. a = 0 is no good because that locks a.s to 0.

a = 22.5 degrees and b = 45.0 degrees will give the same probabilities.

***

[FrediFizzx]

Yes about the x-y plane.

If the distribution is not going to change then your model is completely at odds with QM. Se the post by Joy above.

Ok, so you finally cough up what you really want to see. Yeah, the distribution will change. It will take me a while to get this figured out. a = 0 is no good because that locks a.s to 0.

a = 22.5 degrees and b = 45.0 degrees will give the same probabilities.

***

When a = b I'm getting ++ and -- = 0 and +- and -+ ~= 1/2. So far so good. My old age laziness is kickin' in again for getting the proper x and y coordinates. Well, 45 degrees is easy 0.707 and 0.707.
.

[Heinera]

Cos and sin will give you the coordinates.

[FrediFizzx]

Cos and sin will give you the coordinates.

So what are the probabilities for 22.5 degrees?
.

[Heinera]

So what are the probabilities for 22.5 degrees?
.

Do you need to know that in order to run the simulation?