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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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 Spring '07
 M
 mechanics, Momentum

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