On the Relation between Momentum and Velocity for Elementary Systems

— ~ 2 + ‘2~ in the case of arbitrary s. In the former case the moo — / . “ ‘ mentum-space wave function of the only state which The situation in quantum mechanics is not immeis localized at the origin, at time t = 0, is, according diately obvious because although P has a natural deto ref. [3], finition — it is the generator of spatial translations of = ‘2~)—312F112 ‘3 the state vector [1, 2] — this is not so foru. For an 0 elementary system (the definition [3] of which is and it follows that the expectation value, at t = 0, of broader than that of an elementary particle) of definite the position operator q is mass m it was shown [4] using invariance principles / . only, that eq.(1) holds in quantum mechanics if the (q(0)) = (27r)3J’(,fei” p~i/2cI~(p) ~) velocity is defined as /~r/&,with the time interval 0 ~t so long that the initial and final uncertainties of / dF’ the position are of no consequence. If this is the case, X r~fe1rPP~/2t(P)~~_jd, it is not necessary to define these positions exactly 0/ i.e. the position operator does not have to be specified dF r p fully. It is our purpose here toextend these considera= ~ 4~(F) — _]~(F)~ (4) tions to include infinitesimal time intervals. In this 0 2F~ case, of course, the “velocity” does depend on the definition of the position, that is on the form chosen for where 4(P) is normalized so that the position operator and we’ll show that eq.(1) bedF comes valid if the position operator has the form given fT I 4(P)12 = 1 . (5) inref.[3]. 0 Localized states exist [3] for all systems of nonThe calculation leading to the rhs of eq. (4) has been given previously [3]. Now since the time displacement * Permanent address: Department of Physics, Joseph Henry operator in momentum space is simply multiplication Laboratory, Princeton University, Princeton, NewJersey with ehJ~otit follows that 08540.