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Unformatted text preview: Physics 225/315 March 5 2008 One Dimensional QM Recall T = p 2 / 2 m in classical physics, The de Broglie relation is p = h/ = hk . p = 2 mE and k = 2 / = q 2 mE/ h 2 The momentum operator in the x representation is h i d dx . The kinetic energy part of the Hamiltonian is T = p 2 / 2 m =- h 2 2 m d 2 dx 2 In general H = T + V ( x ) where the potential energy depends on x but we assume is inde- pendent of time. Notice that the momentum operator, p , does not commute with x ; that is xp ( x ) is not the same as px ( x ) since p involves a derivative. In fact the commutator is [ x, p ] = i h . As in the previous case of the spin operators this failure to commute is related to the uncertainty principle relating x and p . The Schrodinger equation H ( x, t ) = i h ( x, t ) t becomes - h 2 2 m 2 x 2 + V ( x ) ! ( x, t ) = i h ( x, t ) t Separate the variables so ( x, t ) = ( x ) ( t )....
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