Lecture20.pptx

# Lecture20.pptx - 2/27/11 PHYS360QuantumMechanics...

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2/27/11 1 1 PHYS 360 Quantum Mechanics Mon Feb 28, 2011 Lecture 20: Why is it you can only measure ONE eigenvalue at a Ime, no maJer how messy the wave funcIon is? (This is nothing like classical physics!) HW 6 due on Monday Mar 7: Ch 3, #3, 10, 11, 18, 30. 2 Wave functions live in Hilbert space. The set of square‐integrable funcIons “on a speci±ed interval,” for example, that solve a speci±ed equaIon, is called a Hilbert Space. This describes the eigenfuncIons of the Ime‐independent Schrödinger EquaIon: a = a x ˆ i + a y ˆ j + a z ˆ k ( ) Ψ ( x ,0) = c n ψ n ( x ) n = 1 Vector Basis Vector “real” space Hilbert space 3 f x ( ) = c n f n x ( ) n = 1 c n = f n f a = a x ˆ i + a y ˆ j + a z ˆ k ( ) Remember that any point in space can be expressed in terms of the three basis vectors… This property is “completeness.” The members of a Hilbert Space form a complete set if any other funcIon can be expressed as a linear combinaIon of them: …and the coeﬃcients can be determined from: 4 f ˆ Qf = ˆ Qf f for all f x ( ) c n f n x ( ) n Observables are represented by hermitian operators. Q = Q * f ˆ pg = f * i dg dx dx = i f * g −∞ −∞ + i df dx −∞ * gdx = ˆ pf g Check this with the momentum operator: (…use integraIon by parts) 5 General vector: Determinate vector: a = a x ˆ i + a y ˆ j + a z ˆ k ( ) a = a x ˆ i Ψ ( x = c n n ( x ) n = 1 General wave funcIon: Determinate wave funcIon: Ψ ( x = c n n ( x ) 6

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## Lecture20.pptx - 2/27/11 PHYS360QuantumMechanics...

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