1.1: Fundamentals

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The physical system is a spinless particle of rest mass rn interacting with a t¿ü-get fixed at the origin of the coordinate system. The fundamental dynamical variables of the system are the Cartesian coordìnates and momenta x7, x2, x3., P1 ,P2., p3 (2.1) whioh satisfy the fundamental quantum condrtions [x,,xÀ] : o lr;,rol : o lxi,ehf : m¡x (2.2) (2.3) (2.4) wherej,k-1,2,3. The s tate of the particle I lþ(t) > ar úme ú is I ,þ(t) >: u(t) l,þ > (2.s) where I ty' > is the stato at time zero and t/(l) is the evolution operator, that is, tl(t): "-;nt¡n (2.6) H: HolV (2.1) whe-re- Ho : JF "2 + *:44 (2.8) wlrere P2 : F .F ana v : v (xI, x2, x3, P7, p2, p3) : v (f , F) (2.e) TIre potential v given by (2.9) specif,es the inreraction of the particre with the fixed ttget. V is a local potential if v:v(r) (2.10) V is a central potential if v : v(R) (2.1U / -"=-------= where Ã : t/X X. Comments 1 . Relativistic kinematics (2.8) indicates rhat the speed of the particle is restrjcted only the the principle of special lelativity. Reiativistic krnematics will be used throughout this material unless stated otherwise. 2. Nonrelativistic kinematics When nonrelativistic kinematics are used (2.8) is replaced by 3. Restrictions on the interaction potential We assume that the physical system is invariant unde¡ rotations, space inversion and time reversal. As discussed in QLB: Relativistic Quantum Mechanj¿s, it follows that Y must satisfy where P is the space-inversion operator, T is the time-reversal operator and fi/(d) is the rotation operator for a rotation by d about the j-axis. That is, PJ(0): e-iue/h (2.t6) p2 H¡: - ? '2m, (2.12) f( (qv RIi (q : v PVPT - V TVTT:V (2.13) (2.t4) (2.ts) where L:XxP (2.17) is the angular momentum operator. It follows from (2.13) that lv, ril : o (2.18) that is, angular momentum is conserved and it follows from (2.14) that parity is consered. It follows from (2.13) to (2.15) that v$,F) :v(*R,,Fø) :r( È,-F) :v(Ì,-F) eß) i¡r,: ttçe¡iøtçq Fp,: ø1e¡Føt1s¡ (2.20) (2.2t) (2.19) holds if V is a central potentiat (2.11)