Equations for Entanglement and the Double Slit Experiment: A Layman's Guide

[renkma]

I know this is a strange request; I certainly hope not unwelcomed here. I am not a physicist or a student of such, I'm just a humble tattoo artist in need of two equations - as I have a client that wants these equations designed into a piece:

1) the/an equation explaining the entanglement of particles

2) the/an equation explaining the outcome of the double slit experiment, when observed and the "wave" becomes particle.

I am attaching two equations I took screen shots of in a youtube video of a talk on these subjects, and although it was a professor of physics giving the talk, I don't want to take JUST his word for it that these equations are THE equations.

http://brainpanpublishing.com/entangled%20particles%20equasion.JPG
http://brainpanpublishing.com/observed%20to%20particle%20equasion.JPG

So would all of you very understanding and kind (and amazing) physicists out there please either agree that my attached equations are correct, or post the proper equations?

I very much appreciate everyone's help.

RE
New Graffiti

 

[Geofleur]

To make them into equations, you'll want there to be a "$|\Psi \rangle =$" on the left hand sides. So,

$| \Psi \rangle = \left( | \downarrow \uparrow \rangle + | \uparrow \downarrow \rangle \right ) / \sqrt{2}$

and

$| \Psi \rangle = \left( \Psi_U | D_U \rangle + \Psi_L |D_L \rangle \right) / \sqrt{2}$.

They look right to me.

 

[Nugatory]

To make them into equations, you'll want there to be a "$|\Psi \rangle =$" on the left hand sides. So,
$| \Psi \rangle = \left( | \downarrow \uparrow \rangle + | \uparrow \downarrow \rangle \right ) / \sqrt{2}$
and
$| \Psi \rangle = \left( \Psi_U | D_U \rangle + \Psi_L |D_L \rangle \right) / \sqrt{2}$.

They look right to me.

Pretty much (although the second works better if the magnitudes of $\Psi_U$ and $\Psi_D$ are equal to one).

The first equation describes the state of a pair of spin-entangled particles. The second isn't about entanglement; it's the general expression for the polarization of a single photon.

 

[renkma]

Nugatory, I know the 2nd isn't entanglement.
2nd is apparently the equation that describes the wave / particle measurement problem - when observed during the double slit experiment ... is a particle rather than wave.
That is the 2nd equation I'm looking for.

And thanks Geofleur for your help.

 

[ddd123]

Nugatory, I know the 2nd isn't entanglement.
2nd is apparently the equation that describes the wave / particle measurement problem - when observed during the double slit experiment ... is a particle rather than wave.
That is the 2nd equation I'm looking for.

And thanks Geofleur for your help.

The double slit result is

$| \psi |^2 = | \psi_1 |^2 + | \psi_2|^2 + 2 | \psi_1 | | \psi_2 | \cos( \phi_1 - \phi_2 )$

 

[Geofleur]

ddd123, that equation is for calculating the probability from the amplitude but it might be better to keep the wave function expression so that this equation goes together better with the one for entanglement. Remember, it's a tattoo we're talking about here. It's supposed to look nice.

 

[ddd123]

ddd123, that equation is for calculating the probability from the amplitude but it might be better to keep the wave function expression so that this equation goes together better with the one for entanglement. Remember, it's a tattoo we're talking about here. It's supposed to look nice.

That is the part where the particle detection law actually shows up, though. The wavefunction form is very implicit and can represent other things like polarization which has nothing to do with the double slit...

 

[Geofleur]

I see your point. Maybe it's better to stick with the amplitude squared after all...

 

[ddd123]

I think he should ask the client which one he wants.

 

[cgk]

First one (with the arrows) is much better (with the psi= on the left hand side, as Geofleur said in post #2). Everyone will understand the first one, it is effectively screaming "entanglement! That is me! looook at me, I'm entangled!", and it does not need any further definitions. Second one does not do that to me..., in fact, without looking it up, I'm not sure what D_U and D_L are supposed to be... and quantum mechanics is my day job.EDIT: you need to properly typeset the right angle brackets, though. They are supposed to look like in #2, not like "larger than" signs as in your original screenshots. That is important.

 

[Geofleur]

What if we write the double slit result like this:

$\langle \Psi | \Psi \rangle = \langle \Psi_R | \Psi_R \rangle + \langle \Psi_L | \Psi_L \rangle + 2Re \langle \Psi_R | \Psi_L \rangle$

Then at least R and L can stand for "right slit" and "left slit", the $\Psi$ would be capital in both equations, and the bra-ket notation would be used in both equations. Also the interference still shows up clearly in the "overlap" term.

 

[ddd123]

I still prefer the subscripts 1, 2 instead of L, R but it's nicer now.

 

[Heinera]

I feel that we are still lacking some important physical information here, e.g. where on the body are these formulas supposed to be tattooed? Space might be an issue.

 

[renkma]

I love that so many got involved in this - thank you all so much!

I do have to ask though, why is the "result of the double slit" equations all look so different from each other, and from the one I had initially found. I am really hoping for some agreement on this one.

The placement is inside the bicep for the double slit (represents to my client the "law of attraction" - unobserved, remains wave - observed, becomes a reality)
The placement for entanglement is center chest (with two roses - to my client, it represents her heart - her soulmate - I think it is a very romantic notion)
Space is not really an issue if the equations are reasonably short. I don't have room for a textbook LOL

Because my client is a woman, I do definitely want the font to be beautiful for these tattoos. I've found some very decorative math and tech fonts to help me with design.

Thank you all so much.

Renee

 

[Demystifier]

I do have to ask though, why is the "result of the double slit" equations all look so different from each other, and from the one I had initially found. I am really hoping for some agreement on this one.

Just as with ordinary english language, with mathematical language one can say nearly the same thing in many different ways. And not all ways are equally "beautiful". In the double-slit case, the one in post #11 is probably the most beautiful so far.

 

[ddd123]

I am really hoping for some agreement on this one

We all agree!

I've found some very decorative math and tech fonts to help me with design.

I strongly suggest you post the definitive version, since some fonts might make an equation look wrong.

 

[dipole]

What if we write the double slit result like this:

$\langle \Psi | \Psi \rangle = \langle \Psi_R | \Psi_R \rangle + \langle \Psi_L | \Psi_L \rangle + 2Re \langle \Psi_R | \Psi_L \rangle$

Then at least R and L can stand for "right slit" and "left slit", the $\Psi$ would be capital in both equations, and the bra-ket notation would be used in both equations. Also the interference still shows up clearly in the "overlap" term.

Named functions should not be italicized. This is more correct.

$\langle \Psi | \Psi \rangle = \langle \Psi_R | \Psi_R \rangle + \langle \Psi_L | \Psi_L \rangle + 2\textrm{Re} \langle \Psi_R | \Psi_L \rangle$

 

[Geofleur]

Ah, right - the "Re" looks better that way!

 

[bahamagreen]

What a delightful thread; you know the ghost of Hilbert is watching this and grinning... :)

 

[renkma]

We all agree!
I strongly suggest you post the definitive version, since some fonts might make an equation look wrong.

I will. Now, thanks to everyone's help here, I have my two equations and I will definitely post them with the fonts for everyone to look over for me ...

Ah, right - the "Re" looks better that way!

Geofleur - thanks so much. You've been such a huge help.

All of you - I'm just amazed at the response and assistance I received. I just love physicists!

And lastly, I will actually post the designs once they are finished ... AND the tattoos once they are done. :)

 

[renkma]

Well, here are the finished equations ready for my client's tattoo. She loves them; didn't want them fancy. How do they look to everyone? I went over them several times to make sure I didn't muck them up. If anyone sees any errors, please let me know.

Thanks all.

 

[Geofleur]

The top equation is missing a vertical bar to the left of the first term in parentheses: It has $\downarrow \uparrow \rangle$ and it should have $|\downarrow \uparrow \rangle$. Think of the symbols $| \rangle$ as boxes that you can put labels on. In other words, one of your boxes is missing a side!

 

[kith]

Thanks for starting this nice thread! I don't know how open for change you still are, but I'll give my aesthetic considerations anyway.

I don't like the many vertical elements in the first equation. When I write this equation, I write it as $|\Psi \rangle = \frac{1}{\sqrt{2}} \left( |\uparrow \downarrow \rangle + |\downarrow \uparrow \rangle \right)$.

But for the tatoo, I think I would drop the $\frac{1}{\sqrt{2}}$ completely. $|\Psi \rangle = |\uparrow \downarrow \rangle + |\downarrow \uparrow \rangle$ is much more beautiful to me and it isn't wrong. It uses a different convention which makes calculations more inconvenient, but it captures the essence of entanglement just like the other one. Arguably even more because of its simpler form. Also your second equation seems to already use the same convention.

 

[renkma]

Thanks Geofleur - glad I posted before tattooing an error (Oh, that would be so horrible!), but before I hit Photoshop to correct my entanglement, may I ask ... do you agree with Kith? I agree that Kith's equation is prettier without all the vertical what-have-you. But I need to see how many I can get to agree before going with it.

And BTW, thanks Kith.

Everyone, please weigh in on what you think - I would very much appreciate it.

(50% off tattoos for any of you zany physicists if you ever make it to SW Canada - LOL)

 

[Demystifier]

Thanks Geofleur - glad I posted before tattooing an error (Oh, that would be so horrible!), but before I hit Photoshop to correct my entanglement, may I ask ... do you agree with Kith? I agree that Kith's equation is prettier without all the vertical what-have-you. But I need to see how many I can get to agree before going with it.

And BTW, thanks Kith.

Everyone, please weigh in on what you think - I would very much appreciate it.

I also agree that the kith's version of entanglement is better.

In addition, I would like to praise your customer for choosing entanglement and two-slit experiment as two best representatives of the essence of quantum mechanics.

 

[ddd123]

Thanks Geofleur - glad I posted before tattooing an error (Oh, that would be so horrible!), but before I hit Photoshop to correct my entanglement, may I ask ... do you agree with Kith? I agree that Kith's equation is prettier without all the vertical what-have-you. But I need to see how many I can get to agree before going with it.

And BTW, thanks Kith.

Everyone, please weigh in on what you think - I would very much appreciate it.

(50% off tattoos for any of you zany physicists if you ever make it to SW Canada - LOL)

Of course, that equation is also correct. It's not a matter of opinion :p

 

[renkma]

No ddd123, the correctness of equations isn't a "matter of opinion" in the way you are implying (I'm not so thick as to think that way), but there most certainly can be errors within an equation perhaps due to a typo, or even through ignorance. In such instances I would rather ask as many people (in a position to know) as possible if they all agree in the correctness of said equation than permanently tattoo something wrong on someone's body. Case in point ... the funny ad on TV where the tattoo artist tattoos "No Regerts" on her client because she is distracted.

Anyway ...
Thank you Demystifier, for your thumbs up.

Anyone else ... have a "helpful" contribution?

 

[gmax137]

what if you multiply thru by root 2:

$\sqrt{2} |\Psi \rangle = |\uparrow \downarrow \rangle + |\downarrow \uparrow \rangle$

or

$|\Psi \rangle \sqrt{2} = |\uparrow \downarrow \rangle + |\downarrow \uparrow \rangle$

Maybe not, it's more confusing than I hoped that way

 

[ddd123]

No ddd123, the correctness of equations isn't a "matter of opinion" in the way you are implying (I'm not so thick as to think that way), but there most certainly can be errors within an equation perhaps due to a typo, or even through ignorance. In such instances I would rather ask as many people (in a position to know) as possible if they all agree in the correctness of said equation than permanently tattoo something wrong on someone's body. Case in point ... the funny ad on TV where the tattoo artist tattoos "No Regerts" on her client because she is distracted.

Right, but we are a physics forum, so it'd be like asking if "regrets" is correct on an English language forum. I was just reassuring you, that it's correct is so obvious that people aren't going to trail the thread with "yes"es.

To recap, given the various contributions, I also agree that the nicest choice is

$| \Psi \rangle = | \uparrow \downarrow \rangle + | \downarrow \uparrow \rangle$

 

[Demystifier]

$| \Psi \rangle = | \uparrow \downarrow \rangle + | \downarrow \uparrow \rangle$

A minor improvement:
$| \Psi \rangle = |\! \uparrow \downarrow \rangle + |\! \downarrow \uparrow \rangle$
(I reduced the space between "|" and the arrow. In Latex this is done by \! command, which often makes equations look better.)

 

[ddd123]

Aha I was trying to do that, thanks.

 

[renkma]

So far I like the look of Kith's version. I see it's gotten a couple of thumbs up. I would really like to see a couple more thumbs up before taking it into Photoshop, and then tattooing it into skin. Hopefully a few more might see this thread and weigh in.

Best all ...

But for the tatoo, I think I would drop the $\frac{1}{\sqrt{2}}$ completely. $|\Psi \rangle = |\uparrow \downarrow \rangle + |\downarrow \uparrow \rangle$ is much more beautiful to me and it isn't wrong. It uses a different convention which makes calculations more inconvenient, but it captures the essence of entanglement just like the other one. Arguably even more because of its simpler form. Also your second equation seems to already use the same convention.

 

[kith]

@renkma, you can send people a notification by writing an "@" in front of their name. So if you want the opinions of other people who have commented earlier in your thread, you can call them like this: @Geofleur @Nugatory @cgk @dipole

Also you should include Demystifier's suggestion (as you already did in your first photoshop version). The excess of space between the horizontal bar and the arrow isn't intentional, it is an artifact of using simplistic typesetting.

 

[atyy]

I too like kith's suggestion of dropping the 1/2, and Demystifier's version (post #30) of kith's suggestion looks good to me.

 

[renkma]

Excellent. I'm going with the double slit experiment equation I originally posted in #21 ,
and the following from post #30 (originally Kith's suggestion) for my entanglement.

A minor improvement:
$| \Psi \rangle = |\! \uparrow \downarrow \rangle + |\! \downarrow \uparrow \rangle$
(I reduced the space between "|" and the arrow. In Latex this is done by \! command, which often makes equations look better.)

Thank you everyone for getting involved and helping me out. I so appreciate it. I will post the tattoos when they are completed.