arXiv subjects

Ralph Adrian E. Farrales

Publications and source records attributed to Ralph Adrian E. Farrales.

3 recordsLinked to original sources

arXiv2025

Canonical pairs in finite-dimensional Hilbert space

A pair of Hermitian operators is canonical if they satisfy the canonical commutation relation. It has been believed that no such canonical pair exists in finite-dimensional Hilbert space. Here, we obtain canonical pairs by noting that the canonical commutation relation holds in a proper subspace of the Hilbert space. For a given Hilbert space, we study the many possible canonical pairs and look into the uncertainty relation they satisfy. We apply our results by constructing time operators in finite-dimensional quantum mechanics.

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arXiv2024

Characteristic time operators as quantum clocks

We consider the characteristic time operator $\mathsf{T}$ introduced in [E. A. Galapon, Proc. R. Soc. Lond. A, 458:2671 (2002)] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian $\mathsf{H}$ with some growth condition, $\mathsf{T}$ satisfies the canonical relation $[\mathsf{T},\mathsf{H}]|ψ\rangle=i\hbar|ψ\rangle$ for $|ψ\rangle$ in a dense subspace of the Hilbert space. While $\mathsf{T}$ is not covariant, we show that it still satisfies the canonical relation in a set of times of total measure zero called the time invariant set $\mathscr{T}$. In the neighborhood of each time $t$ in $\mathscr{T}$, $\mathsf{T}$ is still canonically conjugate to $\mathsf{H}$ and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of $\mathscr{T}$, and is proportional to the Pauli matrix $σ_y$. Thus, one can construct a quantum clock that tells the time in the neighborhood of $\mathscr{T}$ by measuring a compatible observable.

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arXiv2022

Conjugates to One Particle Hamiltonians in 1-Dimension in Differential Form

A time operator is a Hermitian operator that is canonically conjugate to a given Hamiltonian. For a particle in 1-dimension, a Hamiltonian conjugate operator in position representation can be obtained by solving a hyperbolic second-order partial differential equation, known as the time kernel equation, with some boundary conditions. One possible solution is the time of arrival operator. Here, we are interested in finding other Hamiltonian conjugates by further studying the boundary conditions. A modified form of the time kernel equation is also considered which gives an even bigger solution space.

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