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348-problem19 - p and p 1 2 ipx ipx x c e c e-Ψ = h h Show...

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Chem. 540 Instructor: Nancy Makri PROBLEM 19 In class we showed that the eigenstates of the momentum operator (in one dimension) have the form ( 29 1 2 / ( ) 2 ipx p x e φ π - = h h . Show that these states are also eigenstates of the Hamiltonian for a free particle, 2 0 ˆ ˆ 2 p H m = . Consider a wavefunction formed by taking a linear combination of two momentum eigen- states corresponding to the momentum values
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Unformatted text preview: p + and p-: / / 1 2 ( ) ipx ipx x c e c e-Ψ = + h h . Show that Ψ is an eigenstate of the free particle Hamiltonian, but is not an eigenstate of mo-mentum. Yet, ˆ p and ˆ H commute. Does this fact contradict the theorem about simultaneous ei-genstates? Explain....
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