# So you think you understand Quantum Mechanics

This post is prompted by the mul­ti­tude of posts and com­ments here us­ing quan­tum this and that in an ar­gu­ment (quan­tum dice, quan­tum im­mor­tal­ity, quan­tum many wor­lds...). But how does one know if they un­der­stand the con­cept they use? In school a stu­dent would have to write a test and get graded. It strikes me as a rea­son­able thing to do here, as well: let peo­ple test their un­der­stand­ing of the ma­te­rial so that they can cal­ibrate their es­ti­mate of their knowl­edge of the topic. This is an at­tempt to do just that.

Let’s look at one of the very first ex­per­i­ments demon­strat­ing that in the micro­scopic world things are usu­ally quan­tized: the Stern-Ger­lach ex­per­i­ment, in which mea­sured an­gu­lar mo­men­tum is shown to take dis­crete val­ues. The gist of the ex­per­i­ment is that in a vary­ing mag­netic field the tidal force on a mag­net is not perfectly bal­anced and so the mag­net moves to­ward or away from the denser field, de­pend­ing on the ori­en­ta­tion of its poles. This is in­tu­itively clear to any­one who ever played with mag­nets: the de­gree of at­trac­tion or re­pul­sion de­pends on the rel­a­tive ori­en­ta­tion of the mag­nets (North pole re­pels North pole etc.). It is less ob­vi­ous that this effect is due to the spa­tially vary­ing mag­netic field den­sity, but it is nonethe­less the case.

In the ex­per­i­ment, one mag­net is large (the S-G ap­para­tus it­self) and one is small (a silver atom in­jected into the mag­netic field of the large mag­net). The ex­per­i­ment shows that an un­ori­ented atom sud­denly be­comes al­igned ei­ther along or against the field, but not in any other di­rec­tion. It’s like a com­pass nee­dle that would only be able to point North and South (and po­ten­tially in a few other di­rec­tions) but not any­where in be­tween.

If nec­es­sary, please read through the more de­tailed de­scrip­tion of the ex­per­i­ment on Wikipe­dia or in any other source be­fore at­tempt­ing the fol­low­ing ques­tions (usu­ally called med­i­ta­tions in the idiosyn­cratic lan­guage used on this fo­rum).

Med­i­ta­tion 1. When ex­actly does the atom al­ign it­self? As soon as it en­ters the field? At some ran­dom mo­ment as it trav­els through the field? The in­stance it hits the screen be­hind the field? In other words, in the MWI pic­ture, when does the world split into two, one with the atom al­igned and one with the atom anti-al­igned? In the Copen­hagen pic­ture, does the mag­netic field mea­sure the atom spin, and if so, when, or does the screen do it?

Hint. Con­sider whether/​how you would tell these cases apart ex­per­i­men­tally.

Med­i­ta­tion 2. Sup­pose you make two holes in the screen where the atoms used to hit it, then merge the atoms into a sin­gle stream again by ap­ply­ing a re­verse field. Are the atoms now un­al­igned again, or 5050 al­igned/​anti-al­igned or some­thing else?

Hint. What’s the differ­ence be­tween these cases?

Med­i­ta­tion 3. Sup­pose that in­stead of the re­vers­ing field in the above ex­per­i­ment you keep the first screen with two holes in it, and put a sec­ond screen (with­out any holes) some­where be­hind the first one. What would you ex­pect to see on the sec­ond screen and why? Some pos­si­ble an­swers: two equally bright blobs cor­re­spond­ing to al­igned and anti-al­igned atoms re­spec­tively; the in­terfer­ence pat­tern from each atom pass­ing through both holes at once, like in the dou­ble-slit ex­per­i­ment; a nar­row sin­gle blob in the cen­ter of the sec­ond screen, as if the atoms did not go through the first part of the ap­para­tus at all; a spread-out blob with a max­i­mum at the cen­ter, like you would ex­pect from the clas­si­cal atoms.

Hint. Con­sider/​re­con­sider your an­swer to the first two ques­tions.

Med­i­ta­tion 4. Sup­pose you want to an­swer M1 ex­per­i­men­tally and use an ex­tremely sen­si­tive ac­celerom­e­ter to see which way each atom is deflect­ing be­fore it hits the screen by mea­sur­ing the re­coil of the ap­para­tus. What would you ex­pect to ob­serve?

Hint. Con­sider a similar setup for the dou­ble-slit ex­per­i­ment.

This test is open-book and there is no time limit. You can con­sult any sources you like, in­clud­ing text­books, re­search pa­pers, your teach­ers, pro­fes­sional ex­per­i­men­tal or the­o­ret­i­cal physi­cists, your fel­low LWers, or the im­mor­tal soul of Niels Bohr through your lo­cal medium. If you have ac­cess to the Stern-Ger­lach ap­para­tus in your physics lab, feel free to perform any ex­per­i­ments you may find helpful. As they say, if you are not cheat­ing, you are not try­ing hard enough.

By the way, if any­one wants to sup­ply the pic­tures to make the setup for each ques­tion clearer, I’d be more than happy to in­clude them in this post. If any­one wants to turn the med­i­ta­tions into polls, please do so in the com­ments.

Foot­note: not post­ing this in Main, be­cause I’m not sure how much in­ter­est there is here for QM ques­tions like this.

• Thanks for do­ing this! (I don’t think these are prop­erly called “med­i­ta­tions”, though: “Re­search shows that you’re much more likely to re­mem­ber use­ful info if you try to solve the prob­lem your­self be­fore read­ing the solu­tion. Suc­ceed or fail, the im­por­tant thing is to have tried first.” I think in this case, the pri­mary point isn’t to re­mem­ber the cor­rect an­swer once you post it, but to see how far off our own were, to cor­rect our con­fi­dence in our un­der­stand­ing of QM.)

Okay, I do not place much con­fi­dence in any of the fol­low­ing, and to do it prop­erly I’d prob­a­bly have to spend far more time on this than I can spare, but I guess it’s still use­ful to find out how wrong I’ll be...

My men­tal image is the fol­low­ing stark sim­plifi­ca­tion: I think of the state at any point in time as two prob­a­bil­ity dis­tri­bu­tions in 2D space (2D = the S-G ap­para­tus viewed from the side, with the atom mov­ing left-to-right and be­ing deflected up/​down), one prob­a­bil­ity dis­tri­bu­tion for the po­si­tion of the atom if it’s al­igned along the field, one for if it’s al­igned against the field. I’m imag­in­ing each dis­tri­bu­tion to be con­cen­trated around a sin­gle point at any point in time, but not nec­es­sar­ily the same point for both dis­tri­bu­tions. By “prob­a­bil­ity dis­tri­bu­tion” I do not mean that I’m ac­tu­ally do­ing mea­sure­ments that have a cer­tain prob­a­bil­ity to come out one way or an­other (though I guess I could), I just mean that I only look at the squared am­pli­tude of the wave func­tion, for­get­ting about the phase, which I’m guess­ing is not rele­vant to the prob­lem (one place where I might be wrong). Be­fore the atom en­ters the S-G ap­para­tus, the two dis­tri­bu­tions are the same. I’m guess­ing that what the mag­netic field does is, it makes the cen­ter of one dis­tri­bu­tion move up and the cen­ter of the other move down; again, this is a place where I might be go­ing wrong, but my rea­son­ing is that this is what seems to be re­quired to get the right be­hav­ior if I put in an atom that is definitely al­igned along/​against the field.

M1: My in­tu­itive way of think­ing about it is that the atom al­igns it­self when the cen­ters of the two dis­tri­bu­tions start mov­ing apart, i.e., in the ap­para­tus. But this is not what you’re talk­ing about. For the MWI “world” to split, we need en­tan­gle­ment with some­thing big, which hap­pens when the atom hits the screen. There­fore, in the Copen­hagen pic­ture, hit­ting the screen is when the mea­sure­ment hap­pens.

M2: I as­sume the prob­a­bil­ity dis­tri­bu­tions are very con­cen­trated, and the holes are big enough to let prac­ti­cally all of the “prob­a­bil­ity mass” through, so that the shape of the dis­tri­bu­tion looks the same af­ter pass­ing the holes, and con­tinue to move in a straight line (un­like in the dou­ble-slit ex­per­i­ment, where af­ter pass­ing the slit, the par­ti­cle seems to move in all di­rec­tions from the slit). Then, the re­verse mag­netic field will make the two blobs come to­gether again, and the atoms are “un­al­igned” (= al­igned in what­ever di­rec­tion each of them was when it en­tered the ap­para­tus … no wait, I guess atoms aren’t spin-1/​2 and I can’t think of their state as be­ing given by an al­ign­ment in a par­tic­u­lar di­rec­tion—I think...; but any­way: I think they are again in what­ever spin state they were be­fore en­ter­ing the ap­para­tus).

M3: “Two equally bright blobs cor­re­spond­ing to al­igned and anti-al­igned atoms re­spec­tively”, be­cause my two blobs of prob­a­bil­ity will just move through the holes undis­turbed and then hit the sec­ond screen just as if the first hadn’t been there.

M4: Alright, this makes me ques­tion whether the men­tal model I’ve been us­ing can be cor­rect, be­cause I’ve as­sumed that no en­tan­gle­ment with the ap­para­tus hap­pens, but I guess for the ac­tion of the ap­para­tus on the atom there prob­a­bly has to be an equal and op­po­site re­ac­tion of the atom on the ap­para­tus, of some form… and I’m not sure how to think of that in the con­text of quan­tum me­chan­ics. I can’t do this one, and won­der whether my an­swers to the oth­ers are wrong be­cause of this.

I hope you’ll post a solu­tion set at some point?

• be­cause I’ve as­sumed that no en­tan­gle­ment with the ap­para­tus hap­pens, but I guess for the ac­tion of the ap­para­tus on the atom there prob­a­bly has to be an equal and op­po­site re­ac­tion of the atom on the ap­para­tus, of some form… and I’m not sure how to think of that in the con­text of quan­tum me­chan­ics.

Yet this en­tan­gle­ment thing is the essence of QM, and a ma­jor con­tention is­sue still.

• Thank you for post­ing your at­tempt! I hope there will be oth­ers. I’d award you one shminux point in ad­di­tion to an up­vote, but they aren’t worth much.

As for M4, check how well you can un­der­stand why in the dou­ble-slit ex­per­i­ment try­ing to spy on the elec­tron ru­ins the in­terfer­ence pat­tern.

• Alright, this makes me ques­tion whether the men­tal model I’ve been us­ing can be cor­rect, be­cause I’ve as­sumed that no en­tan­gle­ment with the ap­para­tus hap­pens, but I guess for the ac­tion of the ap­para­tus on the atom there prob­a­bly has to be an equal and op­po­site re­ac­tion of the atom on the ap­para­tus, of some form...

Here’s my guess of why the en­tan­gle­ment be­tween the atom and the ap­para­tus may not cause de­co­her­ence. (Although it turns out some­thing else does.) First con­sider a 10000-di­men­sional unit ball. If we shift this ball by two units in one of the di­men­sions, it would no longer in­ter­sect at all with the origi­nal vol­ume. But if we were to shift it by 1/​1000 units in each of the 10000 di­men­sions, the shifted ball would still mostly over­lap with the origi­nal ball even though we’ve shifted it by a to­tal of 10 units (be­cause the dis­tance be­tween the cen­ters of the balls is only sqrt(10000)/​1000 = 0.1).

Now con­sider the am­pli­tude blob of the ap­para­tus and the shift pro­vided to it by an al­igned atom as it moves through. The shift is di­vided among all of the di­men­sions of the blob (i.e., all the par­ti­cles of the ap­para­tus), and in each di­men­sion the shift is tiny com­pared to the spread of the blob, so the shifted blob al­most com­pletely over­laps with the origi­nal blob. This means the two pos­si­ble shifted blobs (for al­igned and anti-al­igned atoms) can in­terfere with each other just fine.

(This was me try­ing to an­swer “how would the math have to work if the en­tan­gle­ment be­tween atom and ap­par­tus doesn’t cause de­co­her­ence?” Maybe some­one with bet­ter math skills than me can do the ac­tual math and con­firm this?)

Now what if we add an ac­celerom­e­ter? I have no un­der­stand­ing of the physics of ac­celerom­e­ters so I don’t know if one that can de­tect such a small ac­cel­er­a­tion is even the­o­ret­i­cally pos­si­ble, but if we as­sume that it is, then when an atom passes through the new ap­para­tus (with the ac­celerom­e­ter), its am­pli­tude blob will be shifted a lot in some of the di­men­sions (namely the di­men­sions rep­re­sent­ing the par­ti­cles that make up the ac­celerom­e­ter out­put), which would pre­vent the shifted blobs from in­terfer­ing with each other.

• First con­sider a 10000-di­men­sional unit ball. If we shift this ball by two units in one of the di­men­sions, it would no longer in­ter­sect at all with the origi­nal vol­ume. But if we were to shift it by 1/​1000 units in each of the 10000 di­men­sions, the shifted ball would still mostly over­lap with the origi­nal ball even though we’ve shifted it by a to­tal of 10 units (be­cause the dis­tance be­tween the cen­ters of the balls is only sqrt(10000)/​1000 = 0.1).

Ac­tu­ally no, it doesn’t mostly over­lap. If we con­sider a hy­per­cube of ra­dius 1 (dis­placed along the di­ag­o­nal) in­stead of a ball, for sim­plic­ity, then the over­lap frac­tion is 0.9995^10000 = 0.00673. If we hold the man­hat­tan dis­tance (10) con­stant and let num­ber of di­men­sions go to in­finity, then over­lap con­verges to 0.00674 while eu­clidean dis­tance goes to 0. If we hold the eu­clidean dis­tance (0.1) con­stant in­stead, then over­lap con­verges to 0 (ex­po­nen­tially fast).

For the ball, I calcu­late an over­lap frac­tion of 5.6×10^-7, and the same asymp­totic be­hav­iors.

(No com­ment on the physics part of your ar­gu­ment.)

• For the ball, I calcu­late an over­lap frac­tion of 5.6×10^-7, and the same asymp­totic be­hav­iors.

Hmm, my in­tu­ition was that dis­plac­ing a n-ball di­ag­o­nally is equiv­a­lent to dis­plac­ing it ax­i­ally, and similar to dis­plac­ing a hy­per­cube ax­i­ally. I could very well be wrong but I’d be in­ter­ested to see how you calcu­lated this.

• The in­tu­ition: For a high di­men­sional ball, most of the vol­ume is near the sur­face, and most of the sur­face is near the equa­tor (for any given choice of equa­tor). The ex­trem­ity of “most” and “near” in­creases with num­ber of di­men­sions. The in­ter­sec­tion of two equal-size balls is a ball minus a slice through the equa­tor, and thus miss­ing most of its vol­ume even if it’s a pretty thin slice.

The calcu­la­tion: Let $v\(n,r\$%20=%202\int%20_{y=0}%5E{r}%20v(n-1,\sqrt{r%5E2-y%5E2})%20dy%20=%20(2%20\pi%5E{n/​2}%20r%5En)%20/​%20(n%20\Gamma(n/​2))) which is the vol­ume of a n-di­men­sional ball of ra­dius r.
Then the frac­tion of over­lap be­tween two balls dis­placed by x is $\\tfrac\{2\\int \_\{y=x/2\}^\{r\} v\(n\-1,\\sqrt\{r^2\-y^2\}\$%20dy}{v(n,r)}) (The in­te­grand is a cross-sec­tion of the in­ter­sec­tion (which is a lower-di­men­sional ball), and y pro­ceeds along the axis of dis­place­ment.) Numeric re­sult.

• Thanks for both the math and the in­tu­itive ex­pla­na­tion. Now I’m re­ally cu­ri­ous what the right an­swer is to the physics ques­tion...

• The po­si­tion of the ap­para­tus has to be un­cer­tain enough for you to be able to mea­sure mo­men­tum (i.e. ac­cel­er­a­tion) pre­cisely enough. It works out just fine to pat­terns be­ing smeared, an in­ter­est­ing ex­er­cise to do math­e­mat­i­cally though.

edit: didn’t see con­text, thought you were speak­ing of the reg­u­lar dou­ble slit ex­per­i­ment. It still ap­plies though.

With re­gards to the M1 I don’t quite un­der­stand the ques­tion as the spin is not an ar­row that snaps from ar­bi­trary ori­en­ta­tion to par­allel or anti-par­allel. When it in­ter­acts with field, af­ter the speed of light lag, there’s re­coil.

• Alright, I’ll take a crack at this. I haven’t read the com­ments, so likely (I hope?) there’s a lot of du­pli­cate in­for­ma­tion here.

1: When does the atom al­ign it­self?

I’m not 100% sure what you mean by this, so let me know if I mis­in­ter­preted the ques­tion. Con­sider a sin­gle elec­tron. Think of the wave­func­tion as the product of the spin wave­func­tion (which is rep­re­sentable as some lin­ear com­bi­na­tion of spin up and spin down), and the po­si­tion-space wave­func­tion (which is prob­a­bly a pretty tight gaus­sian wavepacket). This goes prop­a­gat­ing along hap­pily un­til it reaches the SG aper­a­tus. Up un­til now, the spin wave­func­tion didn’t af­fect the time evolu­tion of the po­si­tion wave­func­tion at all: this is no longer true.

Quan­tum me­chan­ics is lin­ear: If you calcu­late what hap­pens to one half of the wave­func­tion, and what hap­pens to the other half, and add them to­gether, you get the whole thing. So, you start with:

$\\left | \\text\{packet moving forward\}\\right>\\times \\left\(A\\left|\\uparrow\\right>\+B\\left|\\downarrow\\right>\\right\$=%0AA\left%20%7C%20\text{packet%20mov­ing%20for­ward}\right%3E%20\left%7C\up­ar­row\right%3E+%0AB\left%20%7C%20\text{packet%20mov­ing%20for­ward}\right%3E\left%7C\dow­nar­row\right%3E)

A and B are some com­plex co­effi­cients whose squares add to one. Now, the first term is pure spin up. As such, it will move up, while the sec­ond term will move down. This means your new wave­func­tion will be:

$A\\left | \\text\{packet moving forward and up\}\\right> \\left|\\uparrow\\right>\+ B\\left | \\text\{packet moving forward and down\}\\right>\\left|\\downarrow\\right>$

Now we get into in­ter­pre­ta­tional differ­ences. I be­lieve in the Copen­hagen in­ter­pre­ta­tion, it goes some­thing like this:

The atom hits the screen: this is a po­si­tion mea­sure­ment. The wave­func­tion col­lapses, and one of the po­si­tion eigen­states is cho­sen. It’s ei­ther go­ing to be an eigen­state of a po­si­tion in the top sec­tion of the screen, or of a po­si­tion in the bot­tom sec­tion of the screen. Now, there are mul­ti­ple eigen­states for the same po­si­tion: spin up or spin down, be­cause pos­i­ton mea­sure­ments don’t care about spin. How­ever, if you wind up with a top sec­tion po­si­tion eignen­state, it must be spin up, as your pre-mea­surent wave­func­tion didn’t have any com­po­nent that was both in the top sec­tion and spin down. Like­wise, if you mea­sure the po­si­tion of the atom as in the bot­tom sec­tion, you must be spin down.

Now on to MWI. We need to de­scribe the state of the screen now, so I’m go­ing to add an­other term to the wave­func­tion. Be­fore the silver atom hits the screen:

$\\left|\\text\{blank screen\}\\right>\\left\(A\\left | \\text\{packet moving forward and up\}\\right> \\left|\\uparrow\\right>\+ B\\left | \\text\{packet moving forward and down\}\\right>\\left|\\downarrow\\right>\\right\$)

Now, similarly to how the po­si­tion of the atom got en­tan­gled with the spin of the par­ti­cle, the state of the screen is go­ing to be­come en­tan­gled with both of them, re­sult­ing in:

$A\\left | \\text\{packet moving forward and up\}\\right> \\left|\\uparrow\\right> \\left|\\text\{screen with top mark\}\\right>\+ B\\left | \\text\{packet moving forward and down\}\\right>\\left|\\downarrow\\right> \\left|\\text\{screen with bottom mark\}\\right>$

Now here comes the “world split­ting” bit. When you look at the screen, or oth­er­wise be­come causally en­tan­gled with the state of the screen in any way (that is to say, when your wave­func­tion de­pends on the state of the screen in some way). Be­fore this hap­pens, you have:

$\\left|\\text\{You\}\\right>\\left\(A\\left | \\text\{packet moving forward and up\}\\right> \\left|\\uparrow\\right> \\left|\\text\{screen with top mark\}\\right>\+ B\\left | \\text\{packet moving forward and down\}\\right>\\left|\\downarrow\\right> \\left|\\text\{screen with bottom mark\}\\right>\\right\$)

And af­ter­wards, you have:

$A\\left | \\text\{packet moving forward and up\}\\right> \\left|\\uparrow\\right> \\left|\\text\{screen with top mark\}\\right> \\left|\\text\{You\}\_A\\right>\+ B\\left | \\text\{packet moving forward and down\}\\right>\\left|\\downarrow\\right> \\left|\\text\{screen with bottom mark\}\\right> \\left|\\text\{You\}\_B\\right>$

Now, re­mem­ber that QM is lin­ear. As such, you can treat each term com­pletely sep­a­rately. The first term looks like a world where the screen is marked near the top, and the atom is purely spin up. The sec­ond term looks like a world where the screen is marked near the top, and the atom is purely spin down. As soon as you be­come en­tan­gled with the state of the screen, the spin no longer seems to be in su­per­po­si­tion at all, but is sim­ply up or down, de­pend­ing on if we’re dis­cussing the ex­pe­riences of You_A or You_B. I would say that the world splits when you in­ter­act with the screen.

I may con­tinue, but this level of de­tail is ex­cru­ci­at­ing and I’m a bit burned out from it atm.

• Your MWI anal­y­sis is close to the mark. One thing that is not quite right is mul­ti­ple states in­side each term. Note that once an atom in­ter­acts with the screen, it no longer has a definite spin or even po­si­tion. It be­comes a part of the blob on screen, en­tan­gled with the atoms around it. Thus the in­ter­ac­tion is bet­ter de­scribed as

blank screen(Awave packet mov­ing for­ward and up­atom spin up + Bwave packet mov­ing for­ward and dow­natom spin down) → Ascreen with top mark + Bscreen with bot­tom mark.

The atom state is buried some­where in­side the mark on screen.

The ob­server’s in­ter­ac­tion with the screen is similarly sim­plified:

You(Ascreen with top mark + Bscreen with bot­tom mark) → AYou-top­screen with top mark+BYou-bot­tom*screen with bot­tom mark.

Note that you still have to ap­ply the Born rule to calcu­late the prior prob­a­bil­ities of You be­com­ing You-top and You-bot­tom.

* any po­ten­tial dou­ble en­ten­dre is com­pletely ac­ci­den­tal.

• Note that once an atom in­ter­acts with the screen, it no longer has a definite spin or even po­si­tion.

I don’t fol­low, on po­si­tion. The atom had a dis­tri­bu­tion over po­si­tion which was a smooth wave while it was in tran­sit, and then it has a dis­tri­bu­tion over po­si­tion which is some com­pli­cated func­tion of po­si­tion along the screen. It hasn’t lost any definite­ness in po­si­tion—and in­deed it is much less spread out in space than be­fore (went from 3 ex­tended di­men­sions to 2). It has only lost sim­plic­ity of rep­re­sen­ta­tion.

• How do you know which atom in­side the blob is the one that hit it a mo­ment ago? After all, they are in­dis­t­in­guish­able. All you have left is the Hamil­to­nian of the blob of in­ter­act­ing silver atoms (and the screen, and the field) and its eigen­states, which might not even cor­re­spond to sin­gle atoms, but in­stead to some lat­tice states.

• What? The note I quoted was in the ap­prox­i­ma­tion of ‘atoms have iden­tity’.

If you’re go­ing that far from the first, then my point about re­duc­tion of di­men­sions still ap­plies.

Plus, these en­ergy eigen­states are each for a well-defined num­ber of atoms.

• Sorry, I don’t fol­low...

• You said, Once an atom in­ter­acts with the screen, it no longer has a definite spin or even po­si­tion.

In no event did any atom be­come LESS con­strained in po­si­tion by hit­ting the screen, and this ap­plies whether or not you take in­di­vi­d­ual iden­tities or not. That’s the first two lines.

The last line points out that you seem to think that the en­ergy eigen­states of the screen and field might cor­re­spond to sin­gle atoms—but the eigen­states for an ex­tended ob­ject will be mul­ti­par­ti­cle states of ex­treme com­plex­ity—and, at least within the en­ergy regime we’re talk­ing about, a fixed num­ber of par­ti­cles.

• Some of these “quan­tum” con­cepts that you men­tion don’t re­quire ac­tual quan­tum physics in the first place. “Quan­tum dice” is just a short-hand for “truly ran­dom de­ci­sion mak­ing method”, and a quan­tum mul­ti­verse is for de­ci­sion-the­o­ret­i­cal pur­poses equiv­a­lent to hav­ing an in­finite en­sem­ble of clas­si­cal uni­verses with small differ­ences be­tween them, plus self-lo­ca­tion un­cer­tainty.

That’s my im­pres­sion any­way.

• “Quan­tum dice” is just a short-hand for “truly ran­dom de­ci­sion mak­ing method”, and a quan­tum mul­ti­verse is for de­ci­sion-the­o­ret­i­cal pur­poses equiv­a­lent to [...]

Un­for­tu­nately, some peo­ple take it overly se­ri­ously, just look at the re­cent posts about in­fini­ties and death in many wor­lds, and about whether a copy of you pre­serves your iden­tity. Some are filled with quan­tum angst.

• If, as ShardPhoenix says, “quan­tum mul­ti­verse is for de­ci­sion-the­o­ret­i­cal pur­poses equiv­a­lent to hav­ing an in­finite en­sem­ble of clas­si­cal uni­verses with small differ­ences be­tween them, plus self-lo­ca­tion un­cer­tainty”, (BTW I would use “similar” in­stead of “equiv­a­lent”), “posts about in­fini­ties and death in many wor­lds, and about whether a copy of you pre­serves your iden­tity” seem perfectly jus­tified, so I find your com­ment puz­zling as a re­sponse to his com­ment.

• Is there rea­son not to take it se­ri­ously, from a quan­tum-me­chan­i­cal per­spec­tive?

I may not agree with their con­clu­sions, but I’m fairly sure the MWI wor­lds ac­tu­ally ex­ist.

• Agreed. The mul­ti­verse idea is older than, and in­de­pen­dent of quan­tum the­ory. Ac­tu­ally, a sin­gle in­finitely large clas­si­cal uni­verse will do, since statis­ti­cally, ev­ery pos­si­bil­ity should play out. Niet­zsche even had a ver­sion of im­mor­tal­ity based on an in­finitely old uni­verse. Though it’s not clear whether he ever meant it liter­ally, he very well could have, be­cause it was con­sis­tent with the sci­en­tific un­der­stand­ing of the time.

That said, I like the idea of sminux’s post. I try to steer clear of quan­tum lan­guage my­self, and think oth­ers should too, if all they mean by “quan­tum” is “ran­dom”.

• For those in­ter­ested in the proper QM calcu­la­tion of what is ob­served in the clas­sic S-G ex­per­i­ment, I warmly recom­mend this old pa­per (PDF), es­pe­cially Ap­pendix 1. It shows why it is pos­si­ble to treat the wave func­tion of a spin-half atom as a su­per­po­si­tion of spin-up and spin-down tra­jec­to­ries, why the spin pre­ces­sion in mag­netic field can be ig­nored, and how to ac­count for the ve­loc­ity vari­a­tion and beam width in calcu­lat­ing the deflec­tion dis­tri­bu­tion.

• I ob­ject to “the fol­low­ing ques­tions (usu­ally called med­i­ta­tions in the idiosyn­cratic lan­guage used on this fo­rum).” The term “med­i­ta­tion” was in­tro­duced rel­a­tively re­cently, not as an al­ter­na­tive to “ques­tion”, but a ques­tion serv­ing a cer­tain role, which these ques­tions do not. I find this both a mi­suse of jar­gon and un­nec­es­sar­ily bizarre in this con­text. I re­quest that you use what you think is or­di­nary lan­guage, in­stead.

• 1: What­ever the origi­nal al­ign­ment of the atom was, the N and S com­po­nents ex­pe­rience a differ­ent lat­eral force, so the am­pli­tude blobs cor­re­spond­ing to those two cases will be shoved apart side­ways. I guess you could say they be­gin al­ign­ing the mo­ment the blobs be­come dis­t­in­guish­able—and that’ll de­pend on a va­ri­ety of other fac­tors like how you’re mea­sur­ing the deflec­tion. This is the case whether you are work­ing in MWI or Copen­hagen.

2: back to un­al­igned, if you can com­pletely re­verse the forces in all cases and equal­ize the path-lengths—though the way the S-G ap­para­tus re­ally works, that’s not go­ing to hap­pen. So you’re go­ing to end up with a com­pli­cated mish­mash of al­tered phases, and thus re­ori­en­ta­tions.

3: The atoms head­ing through the hole in the first screen would end up with two sin­gle-slit in­terfer­ence pat­terns—though since atoms are re­ally heavy I wouldn’t count on this be­ing a very no­tice­able in­terfer­ence pat­tern. It’s two sin­gle-slit pat­terns be­cause the left and right con­tri­bu­tions are not feed­ing into the same quan­tum state (differ­ent spin), so they can’t in­terfere.

4: By mea­sur­ing the deflec­tion, you’ve pro­vided an ear­lier mechanism for dis­t­in­guish­ing the blobs, so it would be at that ear­lier time.

• M2: Ap­par­ently it’s not pos­si­ble to re­cover the origi­nal spin, so I guess you just end up with a ran­dom spin.

Ab­stract: In this re­port, we in­ves­ti­gate the spin dy­nam­ics of a neu­tron beam in a Stern-Ger­lach ex­per­i­ment. In con­trast to the sim­ple con­stant gra­di­ent mag­netic field as­sumed in most liter­a­tures which vi­o­lates Maxwell Equa­tions, we work with a model of mag­netic field which sa­tises Maxwell Equa­tions. The spin dy­nam­ics is in­ves­ti­gated by solv­ing the Schrod­inger equa­tion us­ing Fast Fourier Trans­form method. The spin co­her­ence of the neu­tron beam is found to ex­hibit the humpty-dumpty be­havi­our[1, 2, 3] even though there is no fluc­tu­a­tion to the mag­netic field nor any en­vi­ron­men­tal noise. The main cause of the spin de­co­her­ence is iden­ti­fied as the in­ho­mo­gene­ity of the mag­netic field in the x and y com­po­nents. In ad­di­tion, the non­lin­ear terms in the z com­po­nent of the mag­netic field, zn, n > 1 also con­tribute to the loss of the spin co­her­ence. Although only mag­netic field model con­sid­ered is very spe­cific, the cause of the loss of spin co­her­ence iden­ti­fied us­ing this model is a com­mon fea­tures of any re­al­is­tic mag­netic to be used in a SGA ex­per­i­ment. There­fore the humpty-dumpty na­ture of the spin co­her­ence ex­ist even with­out any fluc­tu­a­tion to the mag­netic field and is in­her­ent to the SGA ex­per­i­ment it­self.

And see also this ear­lier pa­per, Is spin co­her­ence like Humpty-Dumpty? I. Sim­plified treat­ment (free full text), which gives a more un­der­stand­able qual­i­ta­tive ar­gu­ment:

Why should it be so difficult to main­tain spin co­her­ence? The es­sen­tial fea­tures of the fol­low­ing qual­i­ta­tive ar­gu­ment are due to Heisen­berg.

I won’t try to quote the ac­tual ar­gu­ment, which is on page 3 of the PDF, but it seems that the main prob­lem isn’t en­tan­gle­ment be­tween the par­ti­cle and the ap­para­tus.

• 27 Dec 2012 18:30 UTC
2 points

shminux,

I think this test demon­strates how lit­tle I un­der­stand quan­tum me­chan­ics, but the test also serves as an im­pe­tus for me to learn physics at a more rigor­ous level. So thank you for that!

Do you have any text­book recom­men­da­tions for teach­ing my­self quan­tum me­chan­ics? Ideally, I need an in­tro­duc­tory text­book suit­able for some­one already study­ing math­e­mat­ics at an un­der­grad­u­ate level.

(I also wel­come oth­ers to an­swer my ques­tion and de­bate each oth­ers’ recom­men­da­tions. An­swers in this style get ex­tra Paragon points.)

• I learned the ba­sics of QM, as well as a few in­ter­pre­ta­tional bits from Griffiths. I went through three or four other un­der­grad and grad QM texts, as well, but Griffiths is still my fa­vorite by far, even though it does not ad­dress some of the stan­dard grad cur­ricu­lum top­ics, like par­tial waves. YMMV.

Bear in mind that none of the stan­dard QM texts ad­dress the in­ter­pre­ta­tions in any depth, mostly em­brac­ing the in­stru­men­tal “shut up and calcu­late” ap­proach. The most you will get is the Bell the­o­rem and the dis­cus­sion of hid­den vari­ables. I am not sure which texts ad­dress this is­sue head on, but my guess is that it would be in some of the Quan­tum In­for­ma­tion ones. Maybe some­one else can chime in.

• You’re do­ing a world of good with this. Peo­ple take their un­e­d­u­cated guesses far, far too se­ri­ously (QM and moral­ity etc).

• Thank you!

• Thanks! Great post.

• You are wel­come. One of my goals was to show how non-triv­ial Quan­tum Me­chan­ics is, es­pe­cially where “mea­sure­ment” is con­cerned.

• 23 Dec 2012 0:24 UTC
2 points

I was ex­pect­ing more math-look­ing stuff in physics prob­lems. Is it im­plicit?

• Yes. Part of the prob­lem is to un­der­stand what math­e­mat­i­cal for­mal­isms are rele­vant to the ques­tion in the first place.

• When will the solu­tions be posted? If you post any more of these tests, could you men­tion that in the post it­self, so that we know how much time we have to finish them, and also when to re­mind you to post the an­swers in case you for­get?

• This ba­si­cally nailed it, I’ll try to write it up in more de­tail this week.

• 23 Dec 2012 13:55 UTC
1 point

In other words, in the MWI pic­ture, when does the world split into two, one with the atom al­igned and one with the atom anti-al­igned?

This one I’m go­ing to do un-rot13ed be­cause this mis­con­cep­tion both­ers me: The world was split all along ac­cord­ing to the mod­ern un­der­stand­ing of the MWI.

• Well, once you want to solve real prob­lems, talk­ing about “wor­lds” be­comes less helpful than talk­ing about en­tan­gle­ment. Quan­tum me­chan­ics is con­tin­u­ous along a di­men­sion that think­ing us­ing the word “wor­lds” tends to be dis­crete. In fact, the name “many-wor­lds” was pro­posed about 13 years af­ter Everett’s sem­i­nal pa­per.

• i think you still have to talk about wor­lds to make ac­tual pre­dic­tions. You need to spec­ify “a world where ob­server X sees Y” and a prob­a­bil­ity weight. of some sort for that world (or that ob­server, how­ever you want to phrase it).

• Nay. In fact, be­ing able to make pre­dic­tions just from a quan­tum state, no la­bels on “wor­lds,” is an im­por­tant part of mak­ing a quan­tum-me­chan­ics-only in­ter­pre­ta­tion a re­duc­tion­ist suc­cess.

• How can you pre­dict ‘ob­server A sees the elec­tron al­igned with the mag­netic field’ with­out be­ing able to point to the sec­tion of the wave­func­tion that rep­re­sents ‘ob­server A sees the elec­tron al­igned with the mag­netic field’?

All de­co­her­ence can tell me is that my den­sity ma­tri­ces evolve to­wards be­ing di­ag­o­nal. I still need a frame­work to ex­tract ac­tual mean­ing from it.

I’ve never seen a pa­per make a pre­dic­tion with­out some im­plicit defi­ni­tion of world (or some­times equiv­a­lently of ‘ob­server’)- if you have a refer­ence, please point me to one.

• Yes, definitely, one needs to “point to the sec­tion of the wave­func­tion that rep­re­sents ‘ob­server A sees the elec­tron al­igned with the mag­netic field’.” So to the ex­tent do­ing that is an im­plicit defi­ni­tion of “world,” then I agree on that part too.

The rea­son why I replied, er, em­phat­i­cally, to “you need to spec­ify [...]” is be­cause any the­ory treat­ing wor­lds as ba­sic should not be the sim­plest way to un­der­stand what’s go­ing on—that should be plain ol’ quan­tum me­chan­ics if the rel­a­tive state in­ter­pre­ta­tion is to be be­lieved, even though that in­ter­pre­ta­tion is some­times called “many-wor­lds.” So you shouldn’t have to talk about wor­lds.

And some­times it’s a good idea not to talk about wor­lds, like when shminux asks you “when does the atom get po­larized when it’s in a mag­netic field?” One definitely has to use math for that one.

• The only way I know to an­swer shminux’s ques­tion rigor­ously (with the de­co­her­ence time scale) in MWI is to make the as­sump­tion that a di­ag­o­nal den­sity of states of rep­re­sents a clas­si­cal en­sem­ble of wor­lds (with weights cor­re­spond­ing to prob­a­bil­ities)- which means ex­plic­itly talk­ing about wor­lds.

Then you can define a de­co­her­ence time-scale of some sort. To the best of my knowl­edge, with­out some rule-of-thumb for what a world is, all you can say is that “the sys­tem evolves to a a den­sity ma­trix that has 1/​sqrt(2) Aligned on one di­ag­o­nal el­e­ment, and 1/​sqrt(2) anti-al­igned on the other.”

• The only way I know to an­swer shminux’s ques­tion rigor­ously (with the de­co­her­ence time scale) in MWI is to make the as­sump­tion that a di­ag­o­nal den­sity of states of rep­re­sents a clas­si­cal en­sem­ble of wor­lds (with weights cor­re­spond­ing to prob­a­bil­ities)- which means ex­plic­itly talk­ing about wor­lds.

Yeah, that’s a cool re­al­iza­tion: “Hey, this all works if we as­sume that a den­sity ma­trix is what an en­tan­gled state looks like from in­side!” And this hap­pens to be pretty true, though if it’s your defi­ni­tion of “world” then there will be differ­ent “wor­lds” from differ­ent per­spec­tives.

But any­how, to an­swer shminux I wouldn’t use wor­lds, I’d just give the de­cay rate of the top state cou­pled to a pho­ton mode, to the bot­tom state plus a pho­ton. The as­sump­tion there is not about wor­lds, but sim­ply that the am­pli­tude-squared mea­sure should be used to de­scribe the prop­er­ties of the atom.

• And this hap­pens to be pretty true, though if it’s your defi­ni­tion of “world” then there will be differ­ent “wor­lds” from differ­ent per­spec­tives.

I can’t parse what you mean by per­spec­tives here- do you mean differ­ent non-rel­a­tivis­tic ob­servers (no rel­a­tivity, we are us­ing Schroed­inger quan­tum), or do you mean putting a differ­ent ba­sis on the Hilbert space?

But any­how, to an­swer shminux I wouldn’t use wor­lds, I’d just give the de­cay rate of the top state cou­pled to a pho­ton mode, to the bot­tom state plus a pho­ton.

Sh­minux’s ques­tion in­volves an un­po­larized beam en­ter­ing a mag­netic field and al­ign­ing, not po­larized atoms flip­ping state. Th­ese are very differ­ent prob­lems.

In the shminux ques­tion, you have an en­tirely off-di­ag­o­nal den­sity ma­trix that evolves to­ward di­ag­o­nal very quickly when the sys­tem be­comes en­tan­gled with the screen. To ex­tract in­for­ma­tion, you im­plic­itly or ex­plic­itly as­sume that a di­ag­o­nal den­sity of states rep­re­sents an en­sem­ble of clas­si­cal wor­lds.

In the ques­tion you are an­swer­ing, you start with only one di­ag­o­nal el­e­ment in the den­sity ma­trix non-zero, and over time the am­pli­tude of that el­e­ment shrinks while the el­e­ment of the other di­ag­o­nal el­e­ment grows. This is a to­tally differ­ent prob­lem. You still are im­plic­itly think­ing about wor­lds, you’ve just cre­ated a differ­ent prob­lem where there is no en­tan­gle­ment so it dodges the messy ques­tion.

• I can’t parse what you mean by per­spec­tives here- do you mean differ­ent non-rel­a­tivis­tic ob­servers (no rel­a­tivity, we are us­ing Schroed­inger quan­tum), or do you mean putting a differ­ent ba­sis on the Hilbert space?

Hm. So what I mean is that if you have sev­eral par­ti­cles en­tan­gled with each other, and you want to know what that “looks like” to one sub­sys­tem (or, ex­per­i­men­tally, if you’re go­ing to pro­duce a beam of these par­ti­cles and do a bunch of sin­gle-par­ti­cle mea­sure­ments), then you have to trace over the other par­ti­cles in the en­tan­gled state. This gives you a re­duced den­sity ma­trix, which is then in­ter­preted as a mixed state. A mixed state be­tween what? Well, “wor­lds,” of course.

This the defi­ni­tion of “world” I meant when I said “if that’s your defi­ni­tion of world...”

But any­how, why did I say that differ­ent ob­servers will see differ­ent lists of wor­lds? Well, be­cause when you take the par­tial trace, what you trace over de­pends on which per­spec­tive you want (ex­per­i­men­tally, what par­tial mea­sure­ment you’re mak­ing). If you’re an elec­tron, your wor­lds are bor­ing—we traced all the com­pli­cated ex­ter­nals away and now your per­spec­tive just looks like a dis­tri­bu­tion over sin­gle-elec­tron states. If you’re a per­son, your per­spec­tive is much more in­ter­est­ing, your den­sity ma­trix is much big­ger. Or to put it an­other way, you have more “wor­lds” to have a dis­tri­bu­tion over.

You still are im­plic­itly think­ing about wor­lds,

And you’re im­plic­itly think­ing about wa­ter­falls, be­cause ev­ery cog­ni­tive al­gorithm is iso­mor­phic to a thoght about a wa­ter­fall :P

• Whose mod­ern un­der­stand­ing of MWI? Give me a math­e­mat­i­cal for­mal­ism for “world” that al­lows world count to be con­served as in­ter­ac­tions go for­ward (if the wor­lds ‘were split all along’, then some­thing like ‘world count’ is con­stant).

Your un­der­stand­ing also seems to have im­pli­ca­tions for (“the mod­ern un­der­stand­ing of”) MWI and Bell’s the­o­rem- since you are ap­ply­ing if I know enough about the de­grees of free­dom “I don’t care about” I can track which world I am in. This should re­duce to some sort of hid­den vari­able ap­proach, I would think.

• Give me a math­e­mat­i­cal for­mal­ism for “world” that al­lows world count to be con­served as in­ter­ac­tions go for­ward (if the wor­lds ‘were split all along’, then some­thing like ‘world count’ is con­stant).

As Man­fred says, think­ing of wor­lds as dis­crete isn’t helpful. Any­way, the time evolu­tion dic­tated by Schröd­inger’s equa­tion is uni­tary, so if it always ap­plies (i.e. there’s no such thing as ob­jec­tive col­lapse), the mea­sure stays con­stant.

I know enough about the de­grees of free­dom “I don’t care about” I can track which world I am in.

If I un­der­stand de­co­her­ence well enough (prob­a­bly I don’t), the an­swer to that is “in you could, yes, but you can’t, be­cause ther­mo­dy­nam­ics”. The differ­en­tial equa­tions of quan­tum field the­ory are more-or-less time-sym­met­ric (i.e. re­vers­ing time is equiv­a­lent to bor­ing stuff such as con­ju­gat­ing com­plex phases and swap­ping par­ti­cles with anti-par­ti­cles and left with right), so the rea­son stuff ap­pears to be ir­re­versible is the bound­ary con­di­tion that the past had very lit­tle en­tropy. (And I think I’ve seen the ar­gu­ment that given that T-sym­me­try is equiv­a­lent to CP-sym­me­try, the fact that the past had that lit­tle en­tropy may (or may not) have some­thing to do with the fact that there are so many more par­ti­cles than an­tipar­ti­cles.)

• You have to have a mechanism to sep­a­rate world as dis­crete or you have a the­ory that can’t make pre­dic­tions. If you want to talk about the al­igned/​anti-al­igned beam in the Stern-Ger­lach ex­per­i­ment you have to be able to point and say “this rep­re­sents the world where ob­servers mea­sure al­igned, and this bit over here rep­re­sents the world where ob­servers mea­sure anti-al­igned.” If you can’t do that, you have no the­ory.

If I un­der­stand de­co­her­ence well enough (prob­a­bly I don’t), the an­swer to that is “in you could, yes, but you can’t, be­cause ther­mo­dy­nam­ics”.

This has to be wrong, oth­er­wise MWI would pre­dict vi­o­la­tions of Bell in­equal­ities.

I think your ‘the world is already split’ in­ter­pre­ta­tion is ac­tu­ally the fun­da­men­tal mi­s­un­der­stand­ing- I can’t make any sense of it other than as a hid­den vari­able the­ory of the type already ex­per­i­men­tally ruled out by Aspect-like ex­per­i­ments.

Edit: Un­re­lated, but to clar­ify- you can show that (as­sum­ing en­ergy is bounded be­low) a Lorentz in­var­i­ant Hamil­to­nian has a com­bined CPT sym­me­try, which can mean a lot of things, de­pend­ing on di­men­sion. T has to be re­lated to CP, but not nec­es­sar­ily the-same-as, un­less you have a state where CP^2 = 1.

• un­less you have a state where CP^2 = 1

How can it be any­thing else? Even then, T would equal (CP)^-1.

• Gen­er­ally, you pick up a phase fac­tor af­ter CP^2. The story is ex­actly like par­ity (P) if you can em­bed P^2 in a con­tin­u­ous sym­me­try, you can define away the phase fac­tor, but if you can’t you are just stuck with it.

• Wasn’t sure if post­ing my not-very-con­sid­ered an­swers was in the spirit of the ex­er­cise, but since you’re hop­ing for more an­swers, here are mine. (And de­spite tak­ing a QM class that dis­cussed the Stern-Ger­lach ex­per­i­ment, I’d some­how for­got­ten that S-G used an in­ho­mo­ge­neous mag­netic field. So I might as well so­licit some schoolin’ from LW on this topic.)

Q1. If “When ex­actly does the atom al­ign it­self?” means “When does the atom start chang­ing tra­jec­tory in re­sponse to the mag­netic field?”, I’d say it does that as soon as it en­ters the field. If it means “When can we say the atom has a sin­gle, un­am­bigu­ous spin di­rec­tion?”, I’d say that’s not un­til the atom hits the screen — be­fore then it’s in a su­per­po­si­tion of spin up and spin down states.

Q2. As the atoms all travel unim­peded through the first S-G field, through the holes, and then through the re­verse field, I guess they’d re­main in a 5050 su­per­po­si­tion the whole way along. So they’re just as “un­al­igned” or “al­igned” as they were when they started. Pre­sum­ably most of us would say the silver atoms were origi­nally “un­al­igned” when they came fly­ing out of the fur­nace, in which case we’d have to say they’re still “un­al­igned”.

Q3. I’d see two equally bright blobs, like those I’d have seen on the origi­nal screen, only fur­ther apart (be­cause the two groups of atoms move fur­ther apart as they prop­a­gate, since the spin down par­ti­cles have down­ward ve­loc­ity rel­a­tive to the spin up par­ti­cles).

Q4. I’d ob­serve ei­ther an atom con­tin­u­ously ac­cel­er­at­ing up­wards or an atom con­tin­u­ously ac­cel­er­at­ing down­wards. Once the atom en­ters the S-G ap­para­tus and I see the ac­celerom­e­ter nee­dle start mov­ing, it’s no longer in a su­per­po­si­tion; it’s ei­ther spin up or spin down, and I ob­serve it ac­cel­er­ate ac­cord­ingly.

• It’s not clear to me what you’re en­vi­sion­ing for M4. I’m able to trans­late M2 and M3 into ques­tions that I would have seen in my quan­tum classes, but M1 (and by ex­ten­sion M4) re­mind me more of elec­tro­dy­nam­ics. Is that in­ten­tional?

Similarly, the com­pass nee­dle only point­ing up or down doesn’t fit with the stan­dard de­scrip­tion what un­der­lies Stern-Ger­lach, and I can’t tell if dis­cov­er­ing that sub­tlety is an in­tended effect of the med­i­ta­tions or not.

• It’s not clear to me what you’re en­vi­sion­ing for M4.

The ques­tion is straight­for­ward, pre­dict what an ac­celerom­e­ter sig­nal would show. For ex­am­ple, clas­si­cally one ex­pects to see the sig­nal show near-con­stant re­coil level dur­ing the time the small mag­net trav­els through this in­ho­mo­ge­neous field, as­sum­ing it’s al­igned. The situ­a­tion is not nec­es­sar­ily the same in QM. If you ex­pect the mea­sure­ment to hap­pen at the screen, you’d pre­dict no sig­nal un­til the spike at the mo­ment of col­li­sion. Note that M4 is not nec­es­sar­ily the same ex­per­i­ment as M1.

the com­pass nee­dle only point­ing up or down doesn’t fit with the stan­dard de­scrip­tion what un­der­lies Stern-Gerlach

Right, the setup is not ex­actly the same, but it de­ter­mines es­sen­tially the same phys­i­cal prop­erty,

• The ques­tion is straight­for­ward, pre­dict what an ac­celerom­e­ter sig­nal would show.

Read­ing Benja’s an­swers, I think my is­sue was that I thought “ap­para­tus” referred to the ac­celerom­e­ter, not the large mag­net used in S-G. Re­plac­ing that word with “large mag­net” makes the in­tended ques­tion clear to me, and I see how to an­swer it with just QM.

• M1: The wor­lds split over a short pe­riod of time start­ing as soon as the atom en­ters the field. They don’t split very far. In the Copen­hagen in­ter­pre­ta­tion no mea­sure­ment oc­curs (EDIT: un­til they hit the screen) (you can tell be­cause later on we’ll “un­mea­sure” this alleged mea­sure­ment, which would be im­pos­si­ble if the wave­func­tion had col­lapsed.). EDIT: The wor­lds don’t split very far while the atoms are in the field (they only split in one di­rec­tion). When they hit the screen they sud­denly split a lot fur­ther. In Copen­hagen this would con­sti­tute a mea­sure­ment.

M2: The atoms be­come un­al­igned again. EDIT: You can’t nec­es­sar­ily tell this apart from 5050 al­igned un­al­igned. If I could be both­ered I’d calcu­late the den­sity ma­tri­ces. The situ­a­tions are dis­t­in­guish­able iff the den­sity ma­tri­ces are differ­ent. My in­tu­ition says they’re the same.

M3: Two blobs. No in­terfer­ence. EDIT: Be­cause the spins are differ­ent.

M4: The de­tec­tor an­swers soon af­ter the atom en­ters the field. It will of course end up agree­ing with the light formed by the atom hit­ting the screen. EDIT: In par­tic­u­lar the ac­cel­er­a­tion would vary with time ex­actly as if the atom were clas­si­cal.

• The wor­lds split over a short pe­riod of time start­ing as soon as the atom en­ters the field. They don’t spilt (sic.) very far

Nei­ther state­ment is clear to me. How short a time? How do you define the dis­tance be­tween wor­lds?

Re M2: As­sume you are deal­ing with a sin­gle atom, so it’s a pure state, no need for den­sity ma­tri­ces. What would be your an­swer?

• M1: The atom has a wave­func­tion, a func­tion from {up,down} x R^3 to C. We can view this as two spa­cial wave­func­tions (i.e. from R^3 to C) one for spin up, one for spin down. Be­fore en­ter­ing the field the up and down wave func­tions are the same, and lo­cal­ised in space (i.e. all the am­pli­tude is in a small lump around a point). This lump of am­pli­tude moves along un­til it reaches the field. At this point the two wave­func­tions cease to be the same (the spin of the par­ti­cle be­comes en­tan­gled with its po­si­tion). The one as­so­ci­ated with spin up trans­lates up­ward, its ve­loc­ity in­creas­ing just like a par­ti­cle in a clas­si­cal field. Similarly the spin down wave­func­tion moves down­ward. The time that the wor­lds take to split is the time un­til the two lumps (cor­re­spond­ing to up and down) cease to over­lap spa­tially. I don’t view the “wor­lds” as on­tolog­i­cally fun­da­men­tal, only the wave­func­tion, so the pre­vi­ous sen­tence is close to tau­tol­ogy. If we al­low the wave­func­tion to have thin tails off to in­finity then the lumps never truly split, but they do still mostly sep­a­rate.

Since the two lumps haven’t sep­a­rated very far (say at most a few cms, or how­ever big the S-G ap­para­tus is) and the atom hasn’t en­tan­gled with any­thing else, it will be easy to re­move the en­tan­gle­ment be­tween spin and po­si­tion by re­vers­ing the field. This is what I mean by “the wor­lds aren’t very far apart”. To for­mal­ise the no­tion of dis­tance I sup­pose one could take the root of the sum of the squares of the dis­place­ments of ev­ery par­ti­cle in the uni­verse be­tween the two wor­lds. So in this case the dis­tance be­tween wor­lds would be the literal dis­tance be­tween the two lumps of am­pli­tude.

Once they hit the screen we sud­denly have that lots of par­ti­cle’s po­si­tions differ be­tween the two wor­lds, and so the dis­tance be­tween them be­comes very great. This is de­co­her­ence.

M2: In the ex­per­i­ment given, any sin­gle par­ti­cle is es­sen­tially re­turned to ex­actly the same su­per­po­si­tion of states it was in be­fore it en­tered the fields. Ex­actly like nei­ther of the fields was ever there. (Also, I hold that I can still use den­sity ma­tri­ces to deal with my sub­jec­tive un­cer­tainty, even about sin­gle par­ti­cles).

• At this point the two wave­func­tions cease to be the same (the spin of the par­ti­cle be­comes en­tan­gled with its po­si­tion).

So it seems that you define the de­gree of sep­a­ra­tion of wor­lds as the spa­tial over­lap of the spin-up and spin-down com­po­nents of the wave func­tion, prob­a­bly the in­ner product of the two nor­mal­ized terms. I did not fol­low your mus­ings on “dis­place­ments of ev­ery par­ti­cle in the uni­verse be­tween the two wor­lds”, how­ever.

M2: In the ex­per­i­ment given, any sin­gle par­ti­cle is es­sen­tially re­turned to ex­actly the same su­per­po­si­tion of states it was in be­fore it en­tered the fields.

Are you say­ing that the screen with the two holes has no effect what­so­ever? Just won­der­ing.

(Also, I hold that I can still use den­sity ma­tri­ces to deal with my sub­jec­tive un­cer­tainty, even about sin­gle par­ti­cles).

You sure can, but it seems un­nec­es­sary for a pure state. Or maybe I mi­s­un­der­stand what you mean by “sub­jec­tive un­cer­tainty”.

• 23 Dec 2012 18:43 UTC
0 points

D1: Gur ngbz vz­zrqvn­gryl ra­gref va n fhcrecbfvgvba bs fcva-hc naq fcva-qbja, naq rnpu grez bs gur fhcrecbfvgvba trgf qvf­cyn­prq, ohg (cneqba gur un­aqjnil Pbcraun­tral ynath­ntr) Angher qbrfa’g qr­pvqr ju­vpu grez bs gur fhcrecbfvgvba lbh bofreir (va ZJV + qrp­bu­r­erapr, gur fhcrecbfvgvba qbrfa’g qrp­bu­rer) hagvy vg uvgf gur fperra.

D2: Gurer’f ab jnl gb gryy. Gur qrafvgl zngevk (|hc><qverpgvba|), fb gurer’f ab jnl gb gryy na rafr­zoyr ju­rer unys gur ng­bzf unir fcva hc naq unys unir fcva qbja sebz bar ju­rer nyy gur fc­vaf ner qvfgevo­hgrq ng enaqbz vfbge­bcvp­nyyl. Guvf obguref zr orpn­hfr gur oryvrsf “gur ngbz vf rvgure fcva-hc be fcva-qbja jvgu ce­bonovyvgl 50% rnpu” naq “gur ngbz vf rvgure fcva-rnfg be fcva-jrfg jvgu ce­bonovyvgl 50% rnpu” srry vap­bzc­ngvoyr ohg gurl yrnq gb gur rknpg fnzr nagvpvc­n­grq rkcrevraprf.

D3: Ubj jvqr ner gur ubyrf?

D4: Gur gjb ornzf qba’g va­gres­rer fv­tavsvp­nagyl ba gur fperra, fb vg qbrfa’g zng­gre gung lbh qb gung. BGBU vs lbh qvq gung jvgu n gjb-fyvg rkcrevzrag, lbh’q qrfgebl gur va­gres­r­erapr cng­grea.

• 23 Dec 2012 0:05 UTC
0 points

Wouldn’t a mean­ingful test in­clude some sort of math?

• This ex­per­i­ment doesn’t seem to re­quire any non-clas­si­cal be­hav­ior be­yond quan­tized ba­sic prop­er­ties and a bet­ter un­der­stand­ing of elec­tro­mag­netism.

• Right; S-G gets in­ter­est­ing when you chain it.

• I think that in­di­cates in­ter­est­ing non­clas­si­cal fea­tures of the elec­tro­mag­netic effect. There’s a hand­ful of in­ter­est­ing ex­per­i­ments I’d like to see the re­sults of be­fore go­ing any fur­ther.