Quantum arrival times for confined autonomous Hamiltonian systems

Abstract

The question of when a given particle prepared in some initial quantum state arrives at a given spatial point remains one of the most challenging problems in quantum mechanics. It has been believed that the time of arrival problem can not be accommodated in the standard single Hilbert space quantum mechanics. This belief traces itself back from the encyclopedic article of Pauli arguing that no self-adjoint time operator canonically conjugate to a semi-bounded Hamiltonian can be constructed (Pauli's Theorem); thus he concluded that the hope of incorporating any dynamical concept of time in quantum mechanics has to be fundamentally abandoned. Most researchers have basically accepted the argument but not the conclusion. This has led to numerous diverse treatments of the problem. While the diversity is broad, proposed solutions have one common element—circumventing Pauli's Theorem. However, we have recently shown that Pauli's arguments can not be correct by demonstrating the consistency of a bounded self-adjoint time operator canonically conjugate to a Hamiltonian with a non-empty point spectrum. Motivated by this development, we treat the problem for spatially confined autonomous systems in one dimension. The reason for treating confined systems is simple: Time of flight experiments are confined.
We advance by attaching the Hilbert space ℋ = L²[−L,L] to our system, where 2L is the available spatial space. For a given Hamiltonian, we proceed to construct the time of arrival operator T acting on ℋ. We require the following of T: (1) canonically conjugate to its Hamiltonian in a not necessarily dense proper subspace of the Hilbert space, (2) reduces to the classical time of arrival in the limit of commuting operators, and (3) self-adjoint. We accomplish our construction by, for a given Hamiltonian H, finding a formal operator T satisfying the formal canonical condition [T,H] = iℏI and which reduces to the classical expression for time of arrival. This requires writing H and solving for T in terms of the (formal) operators q and p. Once T is found we restore rigor by defining T as a self-adjoint operator in ℋ. We then proceed in determining the domain over which H and T are canonically conjugate. Crucial in our procedure is the representation of T as an integral operator on ℋ, the kernel of which is a generalized kernel. In the following we consider the construction of the time of arrival operator for a huge class of potentials. We demonstrate our method for the free particle, harmonic oscillator, and linear potential.