From Special Relativity to Feynman Diagrams.pdf

This implies that the mapping u between a t and a t a

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This implies that the mapping U between | a , t 0 and | a , t | a , t = U ( t , t 0 ) | a , t 0 (9.73) must be a linear operator. Requiring also the conservation of the norm of a state during its time-evolution (conservation of probability), we must have a , t | a , t = a , t 0 | a , t 0 a , t 0 | U U | a , t 0 = a , t 0 | a , t 0 , implying U U = ˆ I ,
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284 9 Quantum Mechanics Formalism that is, the time-evolution operator U has to be unitary . Moreover if U ( t , t 0 ) maps | a , t 0 into | a , t , U ( t , t 0 ) 1 = U ( t , t 0 ) maps | a , t into | a , t 0 , so that U ( t , t 0 ) = U ( t 0 , t ) . We finally require U to satisfy the condition U ( t 0 , t 0 ) = ˆ I . In order to determine the time-evolution of | a , t we compute the change of | a , t under an infinitesimal change in the parameter t . We have d | a , t dt t = t 0 = lim t t 0 | a , t − | a , t 0 t t 0 = lim t t 0 U ˆ I t t 0 | a , t 0 . (9.74) Let us denote the limit of the operator inside the curly brackets by i ˆ H ; we can the write, at a generic time t , the differential equation ˆ H | a , t = i d dt | a , t . (9.75) The operator ˆ H is the infinitesimal generator of time-evolution and, in analogy with classical mechanics, is identified with the quantum Hamiltonian . If we substitute ( 9.73 ) in ( 9.74 ) we obtain an equation for the evolution operator: i dU ( t , t 0 ) dt = ˆ HU ( t , t 0 ) i dU ( t , t 0 ) dt = − U ( t , t 0 ) ˆ H , (9.76) where we have used the hermiticity property of ˆ H : ˆ H = ˆ H . If the Hamiltonian is time-independent, as it is the case for a free particle, we can easily write the formal solution to the above equation with the initial condition U ( t 0 , t 0 ) = ˆ I : U ( t , t 0 ) = U ( t t 0 ) = e i ˆ H ( t t 0 ) . (9.77) The equation for the wave function ψ( x , t ) = x | a , t , is obtained by scalar multiplication of both sides of ( 9.75 ) by the bra x | . Taking into account ( 9.19 ) we obtain ˆ H ψ( x , t ) = i t ψ( x , t ). (9.78) that is the Schrödinger equation , where ˆ H is now the Hamiltonian operator realized as a differential operator on wave functions. For a free particle ˆ H = p | 2 2 m = − 2 2 m 2 , and ( 9.78 ) reads:
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9.3 Unitary Operators 285 2 2 m 2 ψ( x , t ) = i t ψ( x , t ), (9.79) where 2 · = 3 i = 1 2 i . Note that in this formulation the dynamic vari- ables described by hermitian operators are not evolving in time, that is they are time-independent, while states, or equivalently wave functions, are time-dependent. Thinking of time-evolution as of a particular kind of transformation, we have previ- ously referred to such description as the Schroedinger picture . In the Heisenberg picture on the other hand, transformations (including time- evolution) act on operators while states stay inert. In this representation therefore states are time independent while operators ˆ O ( t ) representing observables evolve in time. To see how, let us specialize ( 9.38 ) to the time-evolution and apply it to an observable ˆ O ( t ) : ˆ O ( t ) = U ( t t 0 ) ˆ O ( t 0 ) U ( t t 0 ) = e i ˆ H ( t 0 t ) ˆ O ( t 0 ) e i ˆ H ( t t 0 ) , (9.80) where we have used the property U ( t t 0 ) = U ( t 0 t ) . Clearly at t = t 0 , being U ( t 0 , t 0 ) = ˆ I , any Heisenberg dynamic variable, as well as the state of the system, is the same as the corresponding one in the Schroedinger picture. To find the equation
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