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Unformatted text preview: Physics 6210/Spring 2007/Lecture 9 Lecture 9 Relevant sections in text: 1.6, 1.7 Momentum wave functions We have already indicated that one can use any continuous observable to define a class of wave functions. We have used the position observable to this end. Now let us consider using the momentum observable. We define, as usual,* P | p i = p | p i , p R I = Z - dp | p ih p | , ( p ) = h p | i , P ( p ) = p ( p ) , and so forth. The interpretation of the momentum wave function is that | ( p ) | 2 dp is the probability to find the momentum in the range [ p, p + dp ]. In other words, P rob ( P [ a, b ]) = Z b a dp | ( p ) | 2 . Note that a translation of a momentum wave function is a simple phase transformation (exercise): T a ( p ) = h p | e- i h aP | i = e- ipa ( p ) . Physically, this means that a translation has no effect on the momentum probability dis- tribution (exercise). Exercise: use the momentum wave function representation to check the unitarity of T a . A very useful and important relationship exists between the position and momentum (generalized) eigenvectors. To get at it, we study the scalar product h x | p i , which can be viewed as the position wave function representing a momentum eigenvector. It can also be viewed as the complex conjugate of the momentum wave function representing a position eigenvector. This complex function of x must satisfy (for each p ) h x- | p i = h x | T | p i = e- i h p h x | p i ....
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