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Supplement 13-A Conservation of Total Momentum In our discussion of angular momentum in Chapter 8 we found that the assumption of invariance of the Hamiltonian under rotations led to the appearance of a new constant of motion, the angular momentum. In this supplement we show that the assumption of invariance under spatial displacement leads to the existence of a constant of the motion, the momentum. The requirement that the system be unchanged under the transformation xi l xi a (13A-1) does not change the kinetic energy, since a is independent of time. The potential energy will change, unless it has the form V(x1, x2, x3, . . . , xN) V(x1 x2, x1 x3 . . . x2 x3, . . . , xN1 xN) (13A-2) In classical mechanics, the absence of external forces leads to momentum conservation. This follows from the equations of motion, mi d2xi V(x1 x2, x1 x3, . . . , xN1 xN) 2 x i dt (13A-3) a consequence of which is that d dt m i i x V(x dxi dt i x2, . . . , xN1 xN) 1 i (13A-4) 0 The reason for the vanishing of the right side of the preceding equation is that for every argument in V, there are equal and opposite contributions that come from /xi acting i on it. For example, with u x1 x2, v x1 x3, w x2 x3, V V V 0 x x x V(u, v, w) Vu Vv Vu w v w 1 2 3 Hence P m i i dxi dt (13A-5) is a constant of the motion. In quantum mechanics the same conclusion holds. We shall demonstrate it by using the invariance of the Hamiltonian under the transformation (13A-1). The invariance implies that both HuE(x1, x2, . . . , xN) EuE(x1, x2, . . . , xN) (13A-6) and HuE(x1 a, x2 a, . . . , xN a) EuE(x1 a, x2 a, . . . , XN a) (13A-7) W-49 W-50 Supplement 13-A Conservation of Total Momentum hold. Let us take a infinitesimal, so that terms of 0(a2) can be neglected. Then u(x1 a, . . . , xN a) u(x1, . . . , xN) a a u(x1, . . . , xN) x1 u(x1, . . . , xN) … x2 u(x1, . . . , xN) a (13A-8) x u(x , . . . , x ) i i 1 N and hence x u (x , . . . , x ) aE x u (x , . . . , x ) a Eu (x , . . . , x ) x a Hu (x , . . . , x ) x N N aH i1 i E 1 N i i1 E 1 N E 1 N E 1 N N i1 i (13A-9) N i1 i If we now define P N N p i i1 xi i1 i (13A-10) so that P is the total momentum operator, we see that we have demonstrated that (HP PH) uE(x1, . . . , xN) 0 (13A-11) Since the energy eigenstates for N-particles form a complete set of states, in the sense that any function of x1, x2, . . . , xN can be expanded in terms of all the uE(x1, . . . , xN), the preceding equation can be translated into [H, P](x1, . . . , xN) 0 (13A-12) for all (x1, . . . , xN)—that is, into the operator relation [H, P] 0 (13A-13) This, however, implies that P, the total momentum of the system, is a constant of the motion. This is a very deep consequence of what is really a statement about the nature of space. The statement that there is no origin—that is, that the laws of physics are invariant under displacement by a fixed distance—leads to a conservation law. In relativistic quantum mechanics there are no potentials of the form that we consider here; nevertheless the invariance principle, as stated earlier, still leads to a conserved total momentum.