Lecture22.pptx

Lecture22.pptx - 3/4/11 PHYS360QuantumMechanics...

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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 func±ons “on a speci²ed interval,” for example, that solve a speci²ed equa±on, is called a Hilbert Space. This describes the eigenfunc±ons of the ±me‐independent Schrödinger Equa±on: 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 de²ne the “inner” product of two func%ons : f m f n = δ mn Orthonormality of any two func±ons 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 eigenfunc±ons of a hermi±an operator, the eigenvalues are real . Theorem 2: Eigenfunc±ons belonging to dis±nct eigenvalues are orthogonal . Axiom 3: The eigenfunc±ons of an observable operator (that is, a hermi±an operator) are complete . Any func±on (in Hilbert space) can be expressed as a linear combina±on 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 Stas±scal Interpretason of Quantum Mechanic± Ψ ( x ,0) = c n ψ n ( x ) n = 1 A±±ume the inisal QM wave funcson i± given by: If you mea±ure the energy of thi± ±y±tem, 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 ±o on.
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This note was uploaded on 04/23/2011 for the course PHYS 360 taught by Professor Durbin,stephen during the Spring '11 term at Purdue.

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Lecture22.pptx - 3/4/11 PHYS360QuantumMechanics...

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