Qitd113 - qitd113 Hilbert Space Quantum Mechanics Robert B Griffiths Version of 17 January 2012 Contents 1 Introduction 1 1.1 Hilbert space 1 1.2

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Unformatted text preview: qitd113 Hilbert Space Quantum Mechanics Robert B. Griffiths Version of 17 January 2012 Contents 1 Introduction 1 1.1 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Physical interpretation of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Incompatible properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 General d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Operators 4 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Dyads and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Dagger or adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Normal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Hermitian operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.8 Positive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.9 Unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Qubits 9 3.1 Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Intuitive picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Composite systems and tensor products 10 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Product and entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Operators on tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.4 Example of two qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.5 Multiple systems. Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 References: CQT = Consistent Quantum Theory by Griffiths (Cambridge, 2002), Ch. 2; Ch. 3; Ch. 4 except for Sec. 4.3; Ch. 6. QCQI = Quantum Computation and Quantum Information by Nielsen and Chuang (Cambridge, 2000). Secs. 1.1, 1.2; 2.1.1 through 2.1.7; 2.2.1 1 Intr oduc tion 1.1 Hilbert space In quantum mechanics the state of a physical system is represented by a vector in a Hilbert space : a complex vector space with an inner product....
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This note was uploaded on 02/15/2012 for the course PHYS 3101 taught by Professor Staff during the Spring '08 term at Pittsburgh.

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Qitd113 - qitd113 Hilbert Space Quantum Mechanics Robert B Griffiths Version of 17 January 2012 Contents 1 Introduction 1 1.1 Hilbert space 1 1.2

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