From Special Relativity to Feynman Diagrams.pdf

Corresponds to the representation of the vector in

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corresponds to the representation of the vector in terms of its components along the chosen basis: v ≡ { v i } .
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9.2 Wave Functions, Quantum States and Linear Operators 269 We see that for consistency we must set x | x = δ 3 ( x x ). (9.18) The above normalization equation can be interpreted as the definition of the wave function ψ x ( x ) describing the ket | x in the coordinate representation. Such eigen- function is no ordinary function, but belongs to the class of generalized or improper functions. The reader can then easily verify that the identity operator ˆ I can be expressed in this basis as follows: ˆ I = d 3 x | x x | , which generalizes ( 9.10 ) to a basis labeled by a triplet of continuously varying variables [i.e. ( x , y , z ) ]. Restor- ing for the moment the explicit dependence of the quantum state | a , t on time, the (time-dependent) wave function is defined as ψ ( a ) ( x , t ) = x | a , t . (9.19) Note that since d 3 x has dimension L 3 , in order for ( 9.16 ) to be consistent the state | x has to be dimensionful, of dimension L 3 2 . This is in agreement with the nor- malization ( 9.18 ). There is a one-to-one correspondence between states and wave-functions which satisfies the property that a linear combination of states corresponds to the same linear combination of the wave functions representing them (the space of wave function is said to be isomorphic to V ( c ) ). In particular we can write a hermitian scalar product on wave functions which reproduces with the inner product between states: b | a = b | ˆ I | a = V d 3 x b | x x | a = V d 3 x ψ ( b ) ( x ) ψ ( a ) ( x ), (9.20) so that we can write the squared norm a 2 of a state as a 2 = a | a = d 3 x | ψ ( a ) ( x ) | 2 . (9.21) We conclude that states with finite norm (i.e. normalizable) correspond to square integrable wave functions, belonging to the Hilbert space L 2 ( V ) . Let us recall, for completeness, the probabilistic interpretation of a wave function ψ( x , t ), normalized to one, in quantum mechanics: The quantity | ψ( x , t ) | 2 dV mea- sures the probability of finding the particle within an infinitesimal volume dV about x at a time t . To complete the correspondence between abstract states and their wave function representation, we observe that operators acting on states correspond to differential operators acting on the corresponding wave functions: | b ˆ O | a ψ ( b ) ( x ) = ˆ O ( x , ( a ) ( x ), (9.22) where ˆ O ( x , ) is a local differential operator . For example, as we shall review in Sect.9.3.1 , the momentum operator ˆ p is implemented on wave functions by the oper- ator i . Observables quantities are represented by differential operators which
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270 9 Quantum Mechanics Formalism are hermitian with respect to the inner product ( 9.20 ). For the time being, we shall denote abstract operators and their differential representation on wave functions by the same symbol. Eigenstates | λ i of an observable ˆ O are represented by eigen- functions ψ i ( x ) of the corresponding differential operator, solution to a differential equation: ˆ O | λ i = λ i | λ i ˆ O ( x , i ( x ) = λ i ψ i ( x ). (9.23) The eigenstates of ˆ p are then represented by the functions ψ p ( x ) e i p · x .
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