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Schrodinger versus Heisenberg

When Schrodinger formulated his famous quantum wave equation in 1926, he was using the standard temporal architecture of the Galileo-Newtonian spacetime paradigm in which observers are exophysical and quantum states are described in the SP This picture is an intuitive description of quantum states of SUOs, in which a given state vector Фг at time t obeys the Schrodinger-Dirac equation (24.1).

Observables in QM are operators in H that correspond reasonably closely (but not precisely) to classically observable properties of SUOs. In general, a SP operator At may have an explicit time dependence, in which case we have the rule

where the partial time derivative involves only the explicit dependence in the operator, if any. Dynamical degrees of freedom such as the p' and the q that correspond to momentum and position coordinates in phase space do not have any explicit t-dependence. This reflects the architecture of the SP: the laboratory and all the apparatus in it is fixed.

One of the features of Hilbert space, discussed in the Appendix, is that given any two vectors Ф, Ф, there is an ‘inner product’ denoted (Ф, Ф) that maps the pair of vectors into the complex numbers. One of the deep mysteries surrounding QM is the Born rule [Born, 1926], the proposition that the square modulus |(Ф, Ф)|2 gives the conditional probability ? (Ф | Ф) that, given the SUO was prepared in state Ф, it would be ‘found’ in state Ф.77 Of course, things are never as simple as that, but essentially, that is it: one of the QM mysteries in a nutshell (there are more).

Feynman rephrased this rule in terms of the inner product (Ф, Ф) itself: in his way of describing things he would say that ‘(Ф, Ф) is the amplitude Л(Ф | Ф) for Ф to go to Ф, and the probability for this to happen is the square modulus of this amplitude, i.e., ? (Ф | Ф) = | Л(Ф | Ф)| 2

No one understands the origin of the Born rule, although some theorists claim to be able to derive it. What is bizarre is that this rule is all about time: first we have to prepare Ф, and only then can we see if we have got Ф. Contextuality is inherent in this description, because probabilities in QM are always conditional probabilities. Likewise, amplitudes are always conditional amplitudes. This contextuality reinforces our assertion that QM is not about absolute truths but the correct way of dealing with contextual truths in physics.

Amazingly, the Born rule works even when there is a time interval between state preparation of Ф and outcome detection of Ф. In this case, though, in the intervening time between preparation and detection, the observer must make absolutely no attempt to interfere or intervene with the evolution of Ф. Provided that there is no intervention between initial and final times then the amplitude Л(Ф | Ф(0) for the state Фго prepared at time t0 to be found in state Фг1 at time t1 > t0 is given by the rule

that is, the inner product at time t1. Essentially, this is the amplitude for success in an attempt at time t1 to see if, at that time, the prepared state is actually Ф. [1]

To calculate the right-hand side of (24.3) we need to integrate the Schrodinger- Dirac equation (24.1). The result can be represented in the form

where Ut1,t0 is the temporal evolution operator evolving quantum states from initial time t0 to final time tj. A fundamental feature of Schrodinger mechanics is that if the Hamiltonian operator is self-adjoint[2] then the evolution operator is unitary, meaning that for any state, (Фг1, Ф(1) = (Фг0, Фг0), corresponding to conservation of total probability. Assuming unitary evolution, then Л(Ф | Фго) = (Ф, Utj,0 Фг0), a representation that shows clearly the temporal architecture of the SP.

If the Hamiltonian is explicitly independent of time, then we can show that

In principle the observer is able to reverse the temporal evolution of a state under such circumstances.

  • [1] We ignore any complicating normalization factors here.
  • [2] Corresponding to a real Hamiltonian in CM.
 
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