The Time of Arrival Operator for the free particle case in L2(−∞,∞)

Abstract

This paper studies the status of the free particle Time of Arrival Operator T̂ in the Hilbert Space H = L²(−∞, ∞). Specifically, we construct a domain for T̂ˆ,D(T̂). as a linear span of the vectors ψ⁽¹⁾_m,n(x) and ψ⁽²⁾_m,n(x). The vectors were constructed by studying suitable conditions so that they would indeed fall in the D(T̂).  Afterwards, we show that D(T̂) is dense in L²(−∞, ∞). We conclude then, that T̂ is a densely defined operator in L²(−∞, ∞).