From Special Relativity to Feynman Diagrams.pdf

Of motion for the operator ˆ o t we differentiate

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of motion for the operator ˆ O ( t ) we differentiate both sides of ( 9.80 ) with respect to t : d dt ˆ O ( t ) = d dt U ( t t 0 ) ˆ O ( t 0 ) U ( t t 0 ) + U ( t t 0 ) ˆ O ( t 0 ) d dt U ( t t 0 ) = i ˆ HU ( t t 0 ) ˆ O ( t 0 ) U ( t t 0 ) U ( t t 0 ) ˆ O ( t 0 ) U ( t t 0 ) ˆ H , where we have used ( 9.76 ). Using ( 9.80 ) again we find d dt ˆ O ( t ) = i [ ˆ H , ˆ O ( t ) ] , (9.81) which is referred to as the quantum Hamilton equations of motion. Let us compare this equation with the Hamilton equations of motion of the clas- sical theory, ( 8.97 ). We see that the time-evolution of a dynamic variable in quantum mechanics can be obtained from the classical formula ( 8.97 ) by replacing the Poisson bracket between the classical observable quantities with the commutator between the corresponding quantum operators, according to the prescription ( 9.40 ). We give another example of this procedure by examining the condition under which a quantum dynamic variable is conserved. In the classical case this happens when the Hamiltonian of the system is invariant under the action of a group of transformations G . Quantum mechanically the transformation of the Hamiltonian operator ˆ H under the transformations U ( g ) of G reads g G : ˆ H = U ( g ) ˆ HU ( g ). The infinitesimal form of the above transformation is given by ( 9.39 ) with ˆ O = ˆ H :
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286 9 Quantum Mechanics Formalism δ ˆ H = ˆ H ˆ H = i δθ r [ ˆ H , ˆ G r ] , θ r , (9.82) where ˆ G r denote the infinitesimal generators of G . As in the classical case, in quan- tum mechanics the group G is a symmetry or an invariance of the theory if under the action of G -transformations, the Hamiltonian is left invariant (here we assume ˆ G g not to explicitly depend on time): δ ˆ H = 0 ⇒ ∀ r : [ ˆ H , ˆ G r ] = 0 . On the other hand from ( 9.81 ), using the invariance condition, we obtain d dt ˆ G r ( t ) = i [ ˆ H , ˆ G r ] = 0 , that is the generators ˆ G r of G are conserved . Equation ( 9.82 ) amounts to saying that, a system is invariant with respect to the transformations in G if and only if the Hamiltonian operator commutes with all the infinitesimal generators of G . Using the exponential representation of a finite time-evolution operator U ( t t 0 ) and of a finite G -transformation U ( g ), this property implies that for any g G and t , t 0 : U ( t t 0 ) U ( g ) = U ( g ) U ( t t 0 ), that is the result of a time-evolution and of a G -transformation (e.g. a change in the RF) does not depend on the order in which the two are effected on the system. Let us now mention an important application of Schur’s Lemma, see Sect.7.2 , to quantum mechanics. Let G be a symmetry group of a quantum mechanical system. We know, from our previous discussion, that the Hamiltonian operator ˆ H commutes with the action U of G on the Hilbert space V ( c ) . Its matrix representation on the states will then have the form ( 7.27 ), where c 1 , . . . , c s ( s may be infinite!) are the energy levels E 1 , . . . , E s of the system, and k 1 , . . . , k s their degeneracies. This means that the k states | E of the system corresponding to a given energy level E , define a subspace of V ( c ) on which an irreducible representation D k of the symmetry group G acts. We can easily show this by writing the Schroedinger equation for a
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