FOUNDATIONS OF QUANTUM MECHANICS

CONTENTS
I CONCEPTUAL PROBLEMS 7
I. 1 Introduction . . . . . . . . . . . . . . . 7
I. 2 Incompleteness and locality . . . . . . . . . . . 11
II THE FORMALISM 17
II. 1 Finite - dimensional Hilbert spaces . . . . . . . . . . 17
II. 2 Operators . . . . . . . . . . . . . . . . 20
II. 3 Eigenvalue problem and spectral theorem . . . . . . . . 24
II. 3. 1 Appendix . . . . . . . . . . . . . . . 26
II. 4 Functions of normal operators . . . . . . . . . . . 27
II. 5 Direct sum and direct product . . . . . . . . . . . 30
II. 5. 1 Direct sum . . . . . . . . . . . . . . 30
II. 5. 2 Direct product . . . . . . . . . . . . . 31
II. 6 Addendum: Infinite - dimensional Hilbert spaces . . . . . . 34
II. 6. 1 The structure of vector spaces . . . . . . . . . . 34
II. 6. 2 Operators . . . . . . . . . . . . . . . 36
II. 6. 2. 1 Unbounded operators . . . . . . . . . . 37
II. 6. 2. 2 Continuous spectra . . . . . . . . . . . 38
II. 6. 2. 3 Spectral theorem . . . . . . . . . . . . 39
II. 6. 3 Dirac . . . . . . . . . . . . . . . . 40
II. 6. 4 Summary . . . . . . . . . . . . . . . 40
III THE POSTULATES 41
III. 1 Von Neumann’s postulates . . . . . . . . . . . . 41
III. 2 Pure and mixed states . . . . . . . . . . . . . 45
III. 3 The interpretation of mixed states . . . . . . . . . . 51
III. 4 Composite systems . . . . . . . . . . . . . 55
III. 4. 1 Summary . . . . . . . . . . . . . . . 63
III. 5 Proper and improper mixtures . . . . . . . . . . . 63
III. 6 Spin 1/2 particles . . . . . . . . . . . . . . 64