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.