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Lecture22 - FriMar4,2011 Lecture22 'tstuck...

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3/4/11 1 1 PHYS 360 Quantum Mechanics Fri Mar 4, 2011 Lecture 22: Surely Quantum Mechanics isn't stuck in just one dimension, is it? HW 6 due on Monday Mar 8: Ch 3, #6, 10, 11, 18, 30. 2 Chapter 3 was all on formalism. What did we learn? 3 Wave functions live in Hilbert space. The set of square‐integrable funcSons “on a specified interval,” for example, that solve a specified equaSon, is called a Hilbert Space. This describes the eigenfuncSons of the Sme‐independent Schrödinger EquaSon: 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 f x ( ) such that f x ( ) a b 2 dx < 4 f g f x ( ) a b * g x ( ) dx Now we define the “inner” product of two func%ons : f m f n = δ mn Orthonormality of any two funcSons can be expressed as: 5 f ˆ Qf = ˆ Qf f for all f x ( ) Observables are represented by hermitian operators. Determinate states are eigenfunctions of ˆ Q. Note: ˆ Qf n ( x ) = q n f n ( x ) Ψ ( x , t ) = c n f n n ( x ) 6 Discrete states: Theorem 1: For eigenfuncSons of a hermiSan operator, the eigenvalues are real . Theorem 2: EigenfuncSons belonging to disSnct eigenvalues are orthogonal . Axiom 3: The eigenfuncSons of an observable operator (that is, a hermiSan operator) are complete . Any funcSon (in Hilbert space) can be expressed as a linear combinaSon of them. f x ( ) = c n f n x ( ) n = 1 c n = f n f
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3/4/11 2 7 The Generalized StaSsScal InterpretaSon of Quantum Mechanics Ψ ( x ,0) = c n ψ n ( x ) n = 1 Assume the iniSal QM wave funcSon is given by: If you measure the energy of this system, you will either get E 1 (with probability |c 1 | 2 ), or you will get E 2 (with probability |c 2 | 2 ), or you will get E 3 (with probability |c 3 | 2 ), and so on.
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