From Special Relativity to Feynman Diagrams.pdf

It is apparent from our analysis thus far that our

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It is apparent from our analysis thus far, that our space V ( c ) also contains states with no finite norm, whose wave functions are therefore not in L 2 ( V ) . Simple exam- ples are given by eigenstates of the ˆ x or of the ˆ p operators, represented, respectively, by delta functions and by e i p · x . The norm of the latter is indeed infinite if the space V is infinite: V d 3 x | e i p · x | 2 = V d 3 x = ∞ . 3 Although the physical (probabilis- tic) interpretation of non-normalizable wave functions is more problematic (we can however define relative probabilities as the ratio of the probabilities associated with two finite space intervals), as we saw for the case of the position eigenstates, they are useful to express wave functions which are L 2 ( V ) . 4 Let us emphasize here the different role played in non-relativistic quantum mechanics by the space and time variables x , t . Just as in classical mechanics, the former are dynamical variables while the latter is a parameter. By this we do not mean that the argument x in ψ( x , t ) should be intended as the position of the particle at the time t , since we adopt for the probability distribution in space the analogue of the Eulerian point of view in describing the velocity distribution of a fluid in fluid-dynamics. If we have a system of N non-interacting particles, the corresponding space of quantum states is the tensor product of the spaces describing the quantum states of each particle (see Chap. 4, Sect. 4.2). We can therefore consider as a basis of the N -particle states the vectors: | x 1 | x 2 . . . | x N ≡ | x 1 ⊗ | x 2 . . . ⊗ | x N , 3 As we shall see in Sect.9.3.1 , when dealing with free one-particle states, we can avoid the use of non-renormalizable wave functions, generalized functions Dirac delta-functions etc, by quantizing the physical system in a box. In this case, instead of considering the whole R 3 as the domain of integration, we take a large box of finite volume V , so that the functions which were not L 2 ( −∞; +∞ ) -integrable become now L 2 ( V ) -integrable. In this way we may always restrict our- selves to considering the Hilbert space of functions defined over a finite volume. 4 This is not an uncommon feature. For example in the Fourier integral transform f ( x ) = 1 2 π dpF ( p ) e ipx , if f ( x ) L 2 ( −∞; +∞ ), so does its Fourier transform. However the basis functions 1 2 π e ipx are not in L 2 ( −∞; +∞ ) since | 1 2 π e ipx | 2 = 1 2 π .
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9.2 Wave Functions, Quantum States and Linear Operators 271 also denoted by | x 1 , x 2 , . . . x N , describing the particles located in x 1 , x 2 , . . . , x N . The corresponding representation of a state | a , t is described by the wave function: ψ( x 1 , x 2 , . . . , x N , t ) x 1 | x 2 | . . . x N | a , t . Let us come back now to a single particle system. Similarly to what we have done when defining the coordinate representation, we can choose to describe a state | a in the momentum representation by expanding it in a basis of eigenvectors | p of the momentum operator: | a = d 3 p p | a | p = d 3 p ˜ ψ ( a ) ( p ) | p , (9.24) where ˆ p | p = p | p , and ˜ ψ ( a ) ( p ) is the wave function in the momentum repre- sentation.
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