The Born Rule and Time-Reversal Symmetry of Quantum Equations of Motion

Abstract

It was repeatedly underlined in literature that quantum mechanics cannot be considered a closed theory if the Born Rule is postulated rather than derived from the first principles. In this work the Born Rule is derived from the time-reversal symmetry of quantum equations of motion. The derivation is based on a simple functional equation that takes into account properties of probability, as well as the linearity and time-reversal symmetry of quantum equations of motion. The derivation presented in this work also allows to determine certain limits to applicability of the Born Rule.

Appendix

Let us consider the Schroedinger equation for an arbitrary state vector $|\psi \rangle$:

$$\begin{aligned} \hat{E}|\psi \rangle =\hat{H}|\psi \rangle , \end{aligned}$$

(17)

where $\hat{E}=i\hbar \tfrac{\partial }{\partial t}$—is the energy operator, $\hat{H}$—is time-independent Hamiltonian.

Upon substituting $t\rightarrow -t$ in (17) one obtains

$$\begin{aligned} -\hat{E}|\psi ^{-}\rangle =\hat{H}|\psi ^{-}\rangle , \end{aligned}$$

(18)

where $|\psi ^{-}\rangle =|\psi (\mathbf {r},-t)\rangle$. Equation (18) does not coincide with the Schroedinger equation (17) due to the “minus” sign on its left side. In order to return to the correct equation of motion while retaining time-reversal, it is necessary to apply the operation of Hermitian conjugation to Eq. (18):

$$\begin{aligned} \langle \psi ^{-}|\hat{E}=\langle \psi ^{-}|\hat{H}, \end{aligned}$$

(19)

where it is taken into account that the Hamiltonian is a Hermitian operator $\hat{H}^{+}=\hat{H}$ while the energy operator is anti-Hermitian $\hat{E}^{+}=-\hat{E}$ because $i\hbar \frac{\partial }{\partial t}$ changes its sign under complex conjugation.

Thus, if we demand that the reversed solution should satisfy the Schroedinger equation then time reversal of a quantum state should result in simultaneous replacement of each ket $|\psi \rangle$ with corresponding bra $\langle \psi ^{-}|$. This operation applied to plane wave (1) results in complex conjugation of amplitude $S\rightarrow S^{*}$, which is taken into account in Eq. (3).