Space-time-symmetric extension of quantum mechanics: Interpretation and arrival-time predictions

An alternative quantization rule, in which time becomes a self-adjoint operator and position is a parameter, was proposed by Dias and Parisio [Phys. Rev. A 95, 032133 (2017)]. In this approach, the authors derived a space-time-symmetric (STS) extension of quantum mechanics (QM) where a new quantum state (intrinsic to the particle) $\left|\varphi \right(x)\rangle$ is defined at each point in space. The quantum state $\left|\varphi \right(x)\rangle$ obeys a space-conditional (SC) Schrödinger equation and its projection on $|t\rangle , \langle t\left|\varphi \right(x)\rangle$, represents the probability amplitude of the particle's arrival time at $x$. In this work we provide an interpretation of the SC Schrödinger equation and the eigenstates of observables in the STS extension. Analogous to the usual QM, we propose that by knowing the initial state $\left|\varphi \right(x_0)\rangle$, which predicts any measurement on the particle performed by a detector localized at $x_0$, the SC Schrödinger equation provides $|\varphi \left(x\right)\rangle =\widehat{\mathbb{U}}(x,x_0)|\varphi \left(x_0\right)\rangle$, enabling us to predict measurements when the detector is at $x\lessgtr x_0$. We also verify that for space-dependent potentials, momentum eigenstates in the STS extension $|P_b\left(x\right)\rangle$ depend on position just as energy eigenstates in the usual QM depend on time for time-dependent potentials. In this context, whereas a particle in the momentum eigenstate in the standard QM, $|\psi \left(t\right)\rangle ={|P\rangle |}_t$, at time $t$, has momentum $P$ (and indefinite position), the same particle in the state $|\varphi \left(x\right)\rangle =|P_b\left(x\right)\rangle$ arrives at position $x$ with momentum $P_b\left(x\right)$ (and indefinite arrival time). By investigating the fact that $\left|\psi \right(t)\rangle$ and $\left|\varphi \right(x)\rangle$ describe experimental data of the same observables collected at $t$ and $x$, respectively, we conclude that they provide complementary information about the same particle. Finally, we solve the SC Schrödinger equation for an arbitrary space-dependent potential. We apply this solution to a potential barrier and compare it with a generalized Kijowski distribution, showing that they can predict distinct traversal times.

Figure 1

Illustrations of the (a) time and (b) space translations prescribed by the Schrödinger and the SC Schrödinger equations, respectively. Knowing $\psi \left(x\right|t_0)$ and $\varphi \left(t\right|x_0)$, the Schrödinger and SC Schrödinger equations provide $\psi \left(x\right|t)$ and $\varphi \left(t\right|x)$, respectively.

Figure 2

Probability distributions for the arrival time of the transmitted particles at $x=50$. The initial wave packet $\psi (x,0)$ has $P_0=2, \delta =10, x_0=-50$, and $m=1$. The width of the barrier is $L=10$. The solid and dashed lines illustrate the prediction of $\mathcal{P}\left(t\right|x)$ and ${\mathrm{\Pi }}_K^N\left(t\right|x)$, respectively.