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# onedimsum - Physics 225/315 March 5 2008 One Dimensional QM...

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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/λ = ¯ hk . p = 2 mE and k = 2 π/λ = 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 xpψ ( x ) is not the same as pxψ ( 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 Ψ( x, t ) = i ¯ h Ψ( x, t ) ∂t becomes ˆ - ¯ h 2 2 m 2 ∂x 2 + V ( x ) ! Ψ( x, t ) = i ¯ h Ψ( x, t ) ∂t Separate the variables so Ψ( x, t ) = ψ ( x ) φ ( t ). Then the time equation has the solution φ ( t ) = e - iEt/ ¯ h where E , the separation constant, is the eigenvalue of the Hamiltonian, the energy. The x

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onedimsum - Physics 225/315 March 5 2008 One Dimensional QM...

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