In quantum theory we refer to the probability of finding a particle between positions $x$ and $x+dx$ at the instant $t$, although we have no capacity of predicting exactly when the detection occurs. In this work, we first present an extended nonrelativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between $x$ and $x+dx$ during [$t,\phantom{\rule{0.16em}{0ex}}t+dt$]. Then, we find a Schr\"odinger-like equation for a ``mirror'' wave function $\ensuremath{\phi}(t,x)$ associated with the probability of measuring the system between $t$ and $t+dt$, given that detection occurs at $x$. In this framework, it is shown that energy measurements of a stationary state display a nonzero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to ad hoc assumptions, is a natural, built-in part of the formalism presented here.
In quantum theory we refer to the probability of finding a particle between positions x and x+dx at the instant t, although we have no capacity of predicting exactly when the detection occurs. In this work, we first present an extended nonrelativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between x and x+dx during [t,t+dt]. Then, we find a Schr"odinger-like equation for a ``mirror'' wave function \ensuremathϕ(t,x) associated with the probability of measuring the system between t and t+dt, given that detection occurs at x. In this framework, it is shown that energy measurements of a stationary state display a nonzero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to ad hoc assumptions, is a natural, built-in part of the formalism presented here.
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) $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ is defined at each point in space. The quantum state $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ obeys a space-conditional (SC) Schr\"odinger equation and its projection on $|t\ensuremath{\rangle}, \ensuremath{\langle}t|\ensuremath{\phi}(x)\ensuremath{\rangle}$, represents the probability amplitude of the particle's arrival time at x. In this work we provide an interpretation of the SC Schr\"odinger equation and the eigenstates of observables in the STS extension. Analogous to the usual QM, we propose that by knowing the initial state $|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, which predicts any measurement on the particle performed by a detector localized at x0, the SC Schr\"odinger equation provides $|\ensuremath{\phi}(x)\ensuremath{\rangle}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbb{U}}(x,{x}_{0})|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, enabling us to predict measurements when the detector is at $x\ensuremath{\lessgtr}{x}_{0}$. We also verify that for space-dependent potentials, momentum eigenstates in the STS extension $|{P}_{b}(x)\ensuremath{\rangle}$ 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, $|\ensuremath{\psi}(t)\ensuremath{\rangle}={|P\ensuremath{\rangle}|}_{t}$, at time t, has momentum P (and indefinite position), the same particle in the state $|\ensuremath{\phi}(x)\ensuremath{\rangle}=|{P}_{b}(x)\ensuremath{\rangle}$ arrives at position x with momentum Pb(x) (and indefinite arrival time). By investigating the fact that $|\ensuremath{\psi}(t)\ensuremath{\rangle}$ and $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ 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 Schr\"odinger 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.
We generalize a space-time-symmetric (STS) extension of non-relativistic quantum mechanics (QM) to describe a particle moving in three spatial dimensions. In addition to the conventional time-conditional (Schrödinger) wave function ψ(x,y,z∣t), we introduce space-conditional wave functions such as ϕ(t,y,z∣x), where x plays the role of the evolution parameter. The function ϕ(t,y,z∣x) represents the probability amplitude for the particle to arrive on the plane x=constant at time t and transverse position (y,z). Within this framework, the coordinate xμ∈{t,x,y,z} can be conveniently chosen as the evolution parameter, depending on the experimental context under consideration. This leads to a unified formalism governed by a generalized Schrödinger-type equation, P^μ∣ϕμ(xμ)⟩=−iℏημνdxνd∣ϕμ(xμ)⟩. It reproduces standard QM when xμ=t, with ∣ϕ0(x0)⟩=∣ψ(t)⟩, and recovers the STS extension when xμ=xi∈{x,y,z}. For a free particle, we show that ϕ(t,y,z∣x)=⟨t,y,z∣ϕ(x)⟩ naturally reproduces the same dependence on the momentum wave function as the axiomatic Kijowski distribution. Possible experimental tests of these predictions are discussed. Finally, we demonstrate that the different states ∣ϕμ(xμ)⟩ can emerge by conditioning (i.e., projecting) a timeless and spaceless physical state onto the eigenstate ∣xμ⟩, leading to constraint equations of the form P^μ∣Φμ⟩=0. This formulation generalizes the spirit of the Wheeler-DeWitt-type equation: instead of privileging time as the sole evolution parameter, it treats all coordinates on equal footing.
In quantum theory it is usual one refers to the probability of finding a particle between the positions x and x + dx at time t, although the theory does not provide when detection occurs.In the primordies of the theory, Pauli presented an argument refuting the idea of introducing an operator describing the probability of the instant in which the detection occurs.Later, Page and Wootters, motivated by the fact that one is not able to observe energy eigenstates superposition, presented a non-relativistic quantum formalism in an extended space in which space and time play equivalent roles, being consistent with Pauli's inquiries.This leads to a description given by the conditional probability of finding a particle between x and x + dx in respect to the time [t, t + dt] given by another particle, called the clock particle.This dissertation consists in a review of the Page and Wootters proposal to address the time problem in quantum mechanics as well as a detailed analysis of this using recent tools introduced in the area of quantum information for the treatment of symmetries in physics.
Time continues to be an intriguing physical property in the modern era. On the one hand, we have the Classical and Relativistic notion of time, where space and time have the same hierarchy, which is essential in describing events in spacetime. On the other hand, in Quantum Mechanics, time appears as a classical parameter, meaning that it does not have an uncertainty relation with its canonical conjugate. In this work, we use a recent proposed spacetime-symmetric formalism~\href{https://doi.org/10.1103/PhysRevA.95.032133}{[Phys.~Rev.~A {\bf 95}, 032133 (2017)]} that tries to solve the unbalance in nonrelativistic Quantum Mechanics by extending the usual Hilbert space. The time parameter t and the position operator X^ in one subspace, and the position parameter x and time operator T in the other subspace. Time as an operator is better suitable for describing tunnelling processes. We then solve the novel 1/2-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunnelling time through a rectangular barrier. We compare to previous works, obtaining pure imaginary times for energies below the barrier and a fast-decaying imaginary part for energies above the barrier, indicating the anti-hermiticity of the time operator for tunnelling times. We also show that the expected time of arrival in the tunnelling problem has the form of an energy average of the classical times of arrival plus a quantum contribution.
The question of how long a particle takes to pass through a potential barrier is still a controversial topic in quantum mechanics. One of the main theoretical problems in obtaining estimates for measurable times is the fact that several previously defined time operators, which remained within the borders of standard quantum mechanics, present some kind of pathology. Recently, a time operator acting on an additional Hilbert space has been shown to support both Hermiticity and canonical relation with an energy observable. The theory is built in a framework that treats space and time as symmetrically as possible, in the nonrelativistic regime. In this work, we use this formalism to derive a closed analytical expression for the traversal time of a quantum particle impinging on a rectangular potential barrier. To do this, first we discuss the meaning of the expectation values in the additional Hilbert space and, then, we apply it to traversal times. We test our formalism in the specific experimental scenario of a realization by Ranfagni et al. [A. Ranfagni et al., Appl. Phys. Lett. 58, 774 (1991)]. The proposed approach displays a better performance in comparison with the B\"uttiker-Landauer and the phase-time approximations.
Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities.