The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint

Abstract

We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems -the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone´s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the timedependent Schrödinger equation.

Free fall in KvN mechanics and Einstein's principle of equivalence

“…The unitary operators Û (t) that connect state vectors at time 0 to state vectors at time t, |ψ(t) = Û (t)|ψ(0) , (D.17) constitute a unitary representation of the one-parameter group of time translations. According to Stone's theorem, Û (t) = e − i Ĥ , where Ĥ is Hermitian (a Physicist's proof of Stone's theorem can be found in [85], and some refinements in [86]). Then (D.17 Note that in order to obtain the operator V ′ (x) we need to take the derivative of the potential V (x) by treating x as a c-number and then converting x → x in the result [84].…”

Conceptual inconsistencies in finite-dimensional quantum and classical mechanics

“…( 5) and ( 6): Stone's theorem (see, e.g., Ref. [23,26]) guarantees the uniqueness of the Hermitian operator Ĥ in Eq. ( 3).…”

Escape behavior of a quantum particle in a loop coupled to a lead

“…1 such that H ψ ∈ H for all ψ ∈ D(H ) and {H,D(H )} is self-adjoint. The precise definition of a self-adjoint operator is given in Appendix A and we refer to the paper [20] for some physical motivations. This section is not crucial to understand the remainder of this paper, so the uninterested reader may look at the scattering matrix (7) and then go directly to the solution (19).…”

“…We see that we need to solve the Schrödinger equation (21) inside the loop (x ∈ [0,L]) in order to compute the nonescape probability (20). To determine ψ(x,t) we shall write the initial state ψ 0 (x) as a superposition of generalized eigenstates of H , satisfying (16)- (18), over the spectrum of H , which is (22) where A(k), B(k), C(k), and D(k) are complex numbers.…”