From Special Relativity to Feynman Diagrams.pdf

# Corresponds to the representation of the vector in

This preview shows pages 251–254. Sign up to view the full content.

corresponds to the representation of the vector in terms of its components along the chosen basis: v ≡ { v i } .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern