3.4 The Or­tho­dox Sta­tis­ti­cal In­ter­pre­ta­tion

In ad­di­tion to the op­er­a­tors de­fined in the pre­vi­ous sec­tion, quan­tum me­chan­ics re­quires rules on how to use them. This sec­tion gives those rules, along with a crit­i­cal dis­cus­sion what they re­ally mean.

3.4.1 Only eigen­val­ues

Ac­cord­ing to quan­tum me­chan­ics, the only “mea­sur­able val­ues” of po­si­tion, mo­men­tum, en­ergy, etcetera, are the eigen­val­ues of the cor­re­spond­ing op­er­a­tor. For ex­am­ple, if the to­tal en­ergy of a par­ti­cle is mea­sured, the only num­bers that can come out are the eigen­val­ues of the to­tal en­ergy Hamil­ton­ian.

There is re­ally no con­tro­versy that only the eigen­val­ues come out; this has been ver­i­fied over­whelm­ingly in ex­per­i­ments, of­ten to as­ton­ish­ingly many dig­its ac­cu­racy. It is the rea­son for the line spec­tra that al­low the el­e­ments to be rec­og­nized, ei­ther on earth or halfway across the ob­serv­able uni­verse, for lasers, for the black­body ra­di­a­tion spec­trum, for the value of the speed of sound, for the ac­cu­racy of atomic clocks, for the prop­er­ties of chem­i­cal bonds, for the fact that a Stern-Ger­lach ap­pa­ra­tus does not fan out a beam of atoms but splits it into dis­crete rays, and count­less other ba­sic prop­er­ties of na­ture.

But the ques­tion why and how only the eigen­val­ues come out is much more tricky. In gen­eral the wave func­tion that de­scribes physics is a com­bi­na­tion of eigen­func­tions, not a sin­gle eigen­func­tion. (Even if the wave func­tion was an eigen­func­tion of one op­er­a­tor, it would not be one of an­other op­er­a­tor.) If the wave func­tion is a com­bi­na­tion of eigen­func­tions, then why is the mea­sured value not a com­bi­na­tion, (maybe some av­er­age), of eigen­val­ues, but a sin­gle eigen­value? And what hap­pens to the eigen­val­ues in the com­bi­na­tion that do not come out? It is a ques­tion that has plagued quan­tum me­chan­ics since the be­gin­ning.

The most gen­er­ally given an­swer in the physics com­mu­nity is the or­tho­dox in­ter­pre­ta­tion. It is com­monly re­ferred to as the “Copen­hagen In­ter­pre­ta­tion”, though that in­ter­pre­ta­tion, as pro­moted by Niels Bohr, was ac­tu­ally much more cir­cum­spect than what is usu­ally pre­sented.

Ac­cord­ing to the or­tho­dox in­ter­pre­ta­tion, mea­sure­ment causes the wave func­tion $\Psi$ to col­lapse into one of the eigen­func­tions of the quan­tity be­ing mea­sured.

Stay­ing with en­ergy mea­sure­ments as the ex­am­ple, any to­tal en­ergy mea­sure­ment will cause the wave func­tion to col­lapse into one of the eigen­func­tions $\psi_n$ of the to­tal en­ergy Hamil­ton­ian. The en­ergy that is mea­sured is the cor­re­spond­ing eigen­value:

$$\left. \begin{array}{l} \Psi = c_1 \psi_1 + c_2 \psi_2 + ... ...x{Energy} = E_n \end{array} \right. \mbox{for \emph{some} }n$$

This story, of course, is non­sense. It makes a dis­tinc­tion be­tween na­ture (the par­ti­cle, say) and the “mea­sure­ment de­vice” sup­pos­edly pro­duc­ing an ex­act value. But the mea­sure­ment de­vice is a part of na­ture too, and there­fore also un­cer­tain. What mea­sures the mea­sure­ment de­vice?

Worse, there is no de­f­i­n­i­tion at all of what “mea­sure­ment” is or is not, so any­thing physi­cists, and philoso­phers, want to put there goes. Need­less to say, the­o­ries have pro­lif­er­ated, many to­tally de­void of com­mon sense. The more rea­son­able “in­ter­pre­ta­tions of the in­ter­pre­ta­tion” tend to iden­tify mea­sure­ments as in­ter­ac­tions with macro­scopic sys­tems. Still, there is no in­di­ca­tion how and when a sys­tem would be suf­fi­ciently macro­scopic, and how that would pro­duce a col­lapse or at least some­thing ap­prox­i­mat­ing it.

If that is not bad enough, quan­tum me­chan­ics al­ready has a law, called the Schrö­din­ger equa­tion (chap­ter 7.1), that says how the wave func­tion evolves. This equa­tion con­tra­dicts the col­lapse, (chap­ter 8.5.)

The col­lapse in the or­tho­dox in­ter­pre­ta­tion is what the clas­si­cal the­ater world would have called Deus ex Machina. It is a god that ap­pears out of thin air to make things right. A god that has the power to dis­tort the nor­mal laws of na­ture at will. Mere hu­mans may not ques­tion the god. In fact, physi­cists tend to ac­tu­ally get up­set if you do.

How­ever, it is a fact that af­ter a real-life mea­sure­ment has been made, fur­ther fol­low-up mea­sure­ments have sta­tis­tics that are con­sis­tent with a col­lapsed wave func­tion, (which can be com­puted.) The or­tho­dox in­ter­pre­ta­tion does de­scribe what hap­pens prac­ti­cally in ac­tual lab­o­ra­tory set­tings well. It just of­fers no prac­ti­cal help in cir­cum­stances that are not so clear cut, be­ing phrased in terms that are es­sen­tially mean­ing­less.

Key Points

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Even if a sys­tem is ini­tially in a com­bi­na­tion of the eigen­func­tions of a phys­i­cal quan­tity, a mea­sure­ment of that quan­tity pushes the mea­sured sys­tem into a sin­gle eigen­func­tion.

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The mea­sured value is the cor­re­spond­ing eigen­value.

3.4.2 Sta­tis­ti­cal se­lec­tion

There is an­other hot potato be­sides the col­lapse it­self; it is the se­lec­tion of the eigen­func­tion to col­lapse to. If the wave func­tion be­fore a mea­sure­ment is a com­bi­na­tion of many dif­fer­ent eigen­func­tions, then what eigen­func­tion will the mea­sure­ment pro­duce? Will it be $\psi_1$? $\psi_2$? $\psi_{10}$?

The an­swer of the or­tho­dox in­ter­pre­ta­tion is that na­ture con­tains a mys­te­ri­ous ran­dom num­ber gen­er­a­tor. If the wave func­tion $\Psi$ be­fore the mea­sure­ment equals, in terms of the eigen­func­tions,

$$\Psi = c_1 \psi_1 + c_2 \psi_2 + c_3 \psi_3 + \ldots$$

then this ran­dom num­ber gen­er­a­tor will, in Ein­stein's words, throw the dice and se­lect one of the eigen­func­tions based on the re­sult. It will col­lapse the wave func­tion to eigen­func­tion $\psi_1$ in on av­er­age a frac­tion $\vert c_1\vert^2$ of the cases, it will col­lapse the wave func­tion to $\psi_2$ in a frac­tion $\vert c_2\vert^2$ of the cases, etc.

The or­tho­dox in­ter­pre­ta­tion says that the square mag­ni­tudes of the co­ef­fi­cients of the eigen­func­tions give the prob­a­bil­i­ties of the cor­re­spond­ing eigen­val­ues.

This too de­scribes very well what hap­pens prac­ti­cally in lab­o­ra­tory ex­per­i­ments, but of­fers again no in­sight into why and when. And the no­tion that na­ture would some­how come with, maybe not a phys­i­cal ran­dom num­ber gen­er­a­tor, but cer­tainly an end­less se­quence of truly ran­dom num­bers seemed very hard to be­lieve even for an early pi­o­neer of quan­tum me­chan­ics like Ein­stein. Many have pro­posed that the eigen­func­tion se­lec­tions are not truly ran­dom, but re­flect un­ob­served “hid­den vari­ables” that merely seem ran­dom to us hu­mans. Yet, af­ter al­most a cen­tury, none of these the­o­ries have found con­vinc­ing ev­i­dence or gen­eral ac­cep­tance. Physi­cists still tend to in­sist quite force­fully on a lit­eral ran­dom num­ber gen­er­a­tor. Some­how, when be­lief is based on faith, rather than solid facts, tol­er­ance of al­ter­na­tive views is much less, even among sci­en­tists.

While the usual phi­los­o­phy about the or­tho­dox in­ter­pre­ta­tion can be taken with a big grain of salt, the bot­tom line to re­mem­ber is:

Ran­dom col­lapse of the wave func­tion, with chances gov­erned by the square mag­ni­tudes of the co­ef­fi­cients, is in­deed the cor­rect way for us hu­mans to de­scribe what hap­pens in our ob­ser­va­tions.

As ex­plained in chap­ter 8.6, this is de­spite the fact that the wave func­tion does not col­lapse: the col­lapse is an ar­ti­fact pro­duced by lim­i­ta­tions in our ca­pa­bil­ity to see the en­tire pic­ture. We hu­mans have no choice but to work within our lim­i­ta­tions, and within these, the rules of the or­tho­dox in­ter­pre­ta­tion do ap­ply.

Schrö­din­ger gave a fa­mous, rather cruel, ex­am­ple of a cat in a box to show how weird the pre­dic­tions of quan­tum me­chan­ics re­ally are. It is dis­cussed in chap­ter 8.1.

Key Points

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If a sys­tem is ini­tially in a com­bi­na­tion of the eigen­func­tions of a phys­i­cal quan­tity, a mea­sure­ment of that quan­tity picks one of the eigen­val­ues at ran­dom.

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The chances of a given eigen­value be­ing picked are pro­por­tional to the square mag­ni­tude of the co­ef­fi­cient of the cor­re­spond­ing eigen­func­tion in the com­bi­na­tion.