12 Quantum states (input)

It bears rep­e­ti­tion: quantum mechanics assigns prob­a­bil­i­ties to pos­sible mea­sure­ment out­comes on the basis of actual mea­sure­ment out­comes. As we just saw, the prob­a­bil­i­ties of all pos­sible out­comes are encoded in a den­sity oper­ator, and the trace rule (or, in an impor­tant spe­cial case, the Born rule) tells us how to extract them. The ques­tions to be addressed next are: how is the den­sity oper­ator deter­mined by actual out­comes, and how does it depend on the time of the mea­sure­ment to the the pos­sible out­comes of which it serves to assign probabilities.

Sup­pose that the den­sity oper­ator W1 is appro­priate for assigning prob­a­bil­i­ties to the pos­sible out­comes of any mea­sure­ment that may be made at the time t1. And sup­pose that a mea­sure­ment M is made at t1, and that the pro­jector P rep­re­sents the out­come. Which den­sity oper­ator W2 is appro­priate for assigning prob­a­bil­i­ties to the pos­sible out­comes of whichever mea­sure­ment is made next, at a later time t2? (As is cus­tomary in dis­cus­sions of this kind, we focus on repeat­able mea­sure­ments. If a phys­ical system is sub­jected con­sec­u­tively to two iden­tical mea­sure­ments, and if the second mea­sure­ment invari­able yields the same out­come as the first, we call these mea­sure­ments “repeatable.”)

W2 is uniquely deter­mined by the fol­lowing requirements:

  • It is con­structed out of W1 and P.
  • It assigns prob­a­bility 1 to the out­come P of M.
  • It assigns prob­a­bility 0 to every other out­come of M.

The World According to Quantum MechanicsIf M is a max­imal test (that is, if all of its out­comes are rep­re­sented by 1-​​dimensional pro­jec­tors), then W2 equals P. Lo and behold, if we update the den­sity oper­ator to take into account the out­come of a max­imal test, it turns into the very pro­jector rep­re­senting this out­come. It is par­tic­u­larly note­worthy that in this case W2 is inde­pen­dent of W1.

Let us now relax the require­ment of repeata­bility and only demand that mea­sure­ments be ver­i­fi­able. A mea­sure­ment M1, per­formed at the time t1, is ver­i­fi­able if it is pos­sible to con­firm its out­come by a mea­sure­ment M2 per­formed at the later time t2. For this con­di­tion to be sat­is­fied, there must be a one-​​to-​​one cor­re­spon­dence between the pos­sible out­comes of M1 and those of M2, such that the actual out­come of M1 can be inferred from that of M2.

If the two mea­sure­ments are max­imal tests, so that there are two sys­tems of mutu­ally orthog­onal 1-​​dimensional sub­spaces, the first rep­re­senting the pos­sible out­comes of M1, the second rep­re­senting the pos­sible out­comes of M2, this amounts to requiring the exis­tence of an oper­ator U (of the kind physi­cists call “uni­tary”), which trans­forms any vector in any sub­space belonging to the first system into a vector in the cor­re­sponding sub­space belonging to the second system.

This oper­ator deter­mines how quantum states “evolve.” In other (and less seri­ously mis­leading) words, if v1 is the state vector appro­priate for cal­cu­lating the prob­a­bil­i­ties of the pos­sible out­comes of any mea­sure­ment that may be made at the time t1, then

v2 = U(t2,t1)v1

is the state vector appro­priate for cal­cu­lating the prob­a­bil­i­ties of the pos­sible out­comes of any mea­sure­ment that may be made at t2. If the time between the two mea­sure­ments is infin­i­tes­imal (see below), U can be cast into the form 1 − (i/​ℏ) H dt, where dt is the now infin­i­tes­imal interval t2 − t1, 1 is the iden­tity oper­ator (it returns unchanged what­ever you insert), and H is some self-​​adjoint oper­ator known as the “Hamilton oper­ator” or, simply, the “Hamil­tonian.” The minus sign is a con­ven­tion, the factor i ensures that U does not change the norm of a vector (plug in a unit vector and out pops a unit vector), and the divi­sion by ℏ ensures that the Hamil­tonian has the dimen­sion of energy. Thus

v2 = v1 − (i/​ℏ) H dt v1.

If dt is infin­i­tes­imal, then so is the dif­fer­ence v2v1, for which we shall write dv. Drop­ping the now super­fluous index from v1, we can mas­sage this equa­tion to take the form

iℏ dv/​dt = Hv.

When­ever we write an equa­tion that con­tains infin­i­tes­imal (“infi­nitely small”) quan­ti­ties, what we tac­itly intend is to let these quan­ti­ties go to zero, one (in this case dt) inde­pen­dently, the others depen­dently. In this limit, ratios of infin­i­tes­imal quan­ti­ties like dv/​dt become well-​​defined ordi­nary quantities.

If v depends not only on t but also on x, y, and z, it is cus­tomary use the symbol ψ(t,x,y,z), and it becomes nec­es­sary to replace the deriv­a­tive dv/​dt (which informs us about the rate of change of v) by the par­tial deriv­a­tive ∂tψ (which informs us about the rate of change of ψ for any given set of values x,y,z). Thus

iℏ ∂tψ = Hψ.

Com­par­ison with Eq. (2.3.7) shows that this the Schrödinger equa­tion with

H = −(ℏ2/​2m) [(∂x)2 + (∂y)2 + (∂z)2] + V.

When we fol­lowed the his­tor­ical route from Planck to Born, the dis­covery that we were dealing with a prob­a­bility cal­culus came last, after the dis­covery of the Schrödinger equa­tion. When we started afresh, we were looking for a prob­a­bility cal­culus capable of han­dling non­trivial prob­a­bil­i­ties and thus suit­able for dealing with the objec­tive fuzzi­ness in nature, and we found (just now) that the Schrödinger equa­tion is an inte­gral part of this cal­culus — nothing less, but also nothing more.

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