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The probability that an initial quantum state $|\psi_i\rangle$ becomes the final quantum state $|\psi_f\rangle$ is given by

\begin{eqnarray} P(i \rightarrow f) &=& |\langle\psi_f|\psi_i\rangle|^2 \tag{1}\\ &=& \langle\psi_f|\psi_i\rangle^*\langle\psi_f|\psi_i\rangle \\ &=& \langle\psi_i|\psi_f\rangle\langle\psi_f|\psi_i\rangle. \end{eqnarray}

Equation (1) seems to show that the probability for the transition ($i\rightarrow f$) can be interpreted as the system both moving forward in time ($i\rightarrow f$) with amplitude $\langle\psi_f|\psi_i\rangle$ and backward in time ($f\rightarrow i$) with amplitude $\langle\psi_i|\psi_f\rangle$ simultaneously.

Does this reasoning help to explain the Born rule? (Is it like the Transactional Interpretation of QM?)

I guess we must experience the macroscopic direction of time ($i\rightarrow f$) in accord with increasing entropy in an expanding universe whereas microscopically QM works both forwards and backwards in time.

Addition

This is an improved version of the argument including time-evolution operators.

The probability that an initial quantum state $|\psi_i\rangle$ evolves to become the final quantum state $|\psi_f\rangle$ is given by

\begin{eqnarray} P_{i \rightarrow f} &=& |\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle|^2 \tag{2}\\ &=& \langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle^*\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &=& \langle\psi_i|U^\dagger_{i \rightarrow f}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &=& \langle\psi_i|U_{f \rightarrow i}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \end{eqnarray} where $U_{i \rightarrow f}$ is the forward-time evolution operator and $U_{f \rightarrow i}=U^\dagger_{i \rightarrow f}$ is the corresponding backward-time evolution operator.

Equation (2) seems to show that the probability $P_{i\rightarrow j}$ can be interpreted as the system first evolving forwards in time and then evolving backwards in time.

Perhaps this is an example of Murray Gell-Mann's Totalitarian Principle that "Everything not forbidden is compulsory"? At the quantum level, below observable probabilities, there is nothing to stop time flowing both forwards and backwards.

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    $\begingroup$ I think all you're seeing here is that the Born rule has to be consistent with the time-reversal symmetry possessed by quantum mechanics in general. Your way of stating the Born rule also seems wrong to me. The Born rule only makes sense as a way of talking about measurements. It's not a rule for time evolution. Time evolution occurs via the Schrodinger equation. $\endgroup$ – Ben Crowell Sep 25 at 14:18
  • $\begingroup$ @BenCrowell "The Born rule only makes sense as a way of talking about measurements. It's not a rule for time evolution." – this statement is what proponents of different interpretations disagree about. $\endgroup$ – Prof. Legolasov Sep 27 at 11:42
  • $\begingroup$ You should merge the material from your new question into this one; that one was closed as a duplicate of this one (as per general Stack Exchange policy), so it's no longer possible to post answers to it. $\endgroup$ – PM 2Ring Sep 27 at 11:57
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The Born rule is adequately expressed by the first line of the equation (1). It effectively says that the probability of transition from i to f is proportional to the overlap between the two functions- you don't need invoke the additional reasoning in order to explain it.

The transactional interpretation of QM is quite separate from the Born rule. I will post an explanation of it if I can find the time.

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The Born rule and the continuity equation are just the Noether charge conservation law. The Nöther charge of the Schrödinger equation is simply $e|\psi|^2$. The probability density is proportional to the charge density by $e$.

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