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# Lect5_XandP - Lecture 5 Coordinate and Momentum...

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Lecture 5: Coordinate and Momentum Representations We will start by considering the quantum description of the motion of a particle in one dimension. In classical mechanics, the state of the particle is given by its position and momentum coordinates, x and p . In quantum mechanics, we will consider position and momentum as observables and therefore represent them by Hermitian operators, X and P , respectively. According to the Seventh Postulate, these two operators obey the commutation relation: [ ] h i P X = , Incompatible Observables If two operators do not commute, then an eigenstate of one cannot be an eigenstate of the other. Proof: Let [ A,B ]= M Assume that | a,b ! is simultaneously an eigenstate of A and B : Operate on | a,b ! with M : If A and B commute ( M =0 ) then | a,b !" 0 A and B are then called ‘compatible’ If A and B do not commute (M!= 0) then there is no solution (i.e. | a,b ! =0 is only solution) A and B are then called ‘incompatible’ b a b b a B b a a b a A , , , , = = ( ) ( ) 0 , ) ( , , , , , , , = ! = ! = ! = ! = b a ab ba b a ab b a ba b a a B b a b A b a BA b a AB b a M

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Coordinate Representation Thus X and P are incompatible, so a particle cannot simultaneously have a well-defined position and momentum
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Lect5_XandP - Lecture 5 Coordinate and Momentum...

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