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PhyChem3_vector_matrix_mechanics.pptx
- 1.
- 2. Review of MathematicalConcepts 3D Vector – can be represented by specifying its components 𝑎𝑖 with respect to a set of three mutually perpendicular unit vectors 𝑒𝑖 as; 𝑎 = 𝑒1𝑎1 + 𝑒2𝑎2 + 𝑒3𝑎3 = 𝑖=1 3 𝑒𝑖𝑎𝑖 the basis is not unique; 𝑎 = 𝜀1𝑎1 + 𝜀2𝑎2 + 𝜀3𝑎3 = 𝑖=1 3 𝜀𝑖𝑎𝑖
- 3. 𝑒 = 1 00 0 1 0 0 0 1 𝜀 = 0.354 0.612 0.707 −0.927 0.127 0.354 0.127 −0.78 0.612 Same vector under two basis 𝑎1 𝑎2 𝑎3 = −0.5 0.5 0.75 𝑎1 𝑎2 𝑎3 = −0.55 −0.83 0.28 0.28 Red Blue Green -0.83 -0.55
- 4. What is abasis 𝑒 = 1 0 0 0 1 0 0 0 1 𝑒1 = 1 0 0 𝑒2 = 0 1 0 𝑒3 = 0 0 1 𝜀 = 0.354 0.612 0.707 −0.927 0.127 0.354 0.127 −0.78 0.612 𝜀1 = 0.354 −0.927 0.127 𝜀2 = 0.612 0.12 −0.78 𝜀3 = 0.707 0.354 0.612
- 5. 𝑎 = 𝑒1𝑎1+ 𝑒2𝑎2 + 𝑒3𝑎3 = 𝑖=1 3 𝑒𝑖𝑎𝑖 𝑎 = 1 0 0 −0.5 + 0 1 0 0.5 + 1 0 0 0.75 = −0.5 0.5 0.75 𝑎 = 0.354 −0.927 0.127 −0.55 + 0.612 0.12 −0.78 −0.83 + 0.707 0.354 0.612 0.28 𝑎 = −0.5 0.5 0.75 𝑎 = 𝑒𝑖|𝑎𝑖 0.354 0.612 0.707 −0.927 0.127 0.354 0.127 −0.78 0.612 𝑇 −0.55 −0.83 0.28
- 6. Properties of basis 𝑒𝑖|𝑒 𝑗 = 1 𝑖𝑓 𝑖 = 𝑗 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 0.354 −0.927 0.127 0.612 0.12 −0.78 = 0 𝑒 = 1 0 0 0 1 0 0 0 1 𝜀 = 0.354 0.612 0.707 −0.927 0.127 0.354 0.127 −0.78 0.612 0.354 −0.927 0.127 0.354 −0.927 0.127 = 1
- 7. 𝑎 = 𝑒1𝑎1+ 𝑒2𝑎2 + 𝑒3𝑎3 = 𝑖=1 3 𝑒𝑖𝑎𝑖 𝑒𝑗𝑎 = 𝑎𝑗 𝑒 𝑖𝑒 𝑗 = 1 𝑖𝑓 𝑖 = 𝑗 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑒 2𝑎 = 𝑒 2𝑒1𝑎1 + 𝑒 2𝑒2𝑎2 + 𝑒 2𝑒3𝑎3 𝑒 2𝑎 = 0 ∗ 𝑎1 + 1 ∗ 𝑎2 + 0 ∗ 𝑎3 𝑒 2𝑎 = 𝑎2 Since Properties of basis
- 8. Coordinate systems Cartesian andPolar coordinate
- 9. Scalar Product ofTwo Vectors 𝑎 ∗ 𝑏 = 𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3 = 𝑖=1 3 𝑎𝑖𝑏𝑖 𝑎 ∗ 𝑏 = 𝑖 𝑗 𝑒 𝑖𝑒 𝑗𝑎𝑖𝑏𝑖 𝑒 𝑖|𝑒 𝑗 = 𝛿𝑖𝑗 = 1 𝑖𝑓 𝑖 = 𝑗 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Orthonormal -mutually perpendicular and have unit length 𝑖 𝑒 𝑖𝑒 𝑖 = 1
- 10. What is anOperator? Operator – a quantity that when acting on a vector 𝑎 converts it into a vector 𝑏 𝒪𝑎 = 𝑏 𝒪 𝑥𝑎 + 𝑦𝑏 = 𝑥𝒪𝑎 + 𝑦𝒪𝑏 Linear Operator Completely determined linear operator – its e 𝑂13 on every possible vector is known 𝒪𝑒𝑖 = 𝑗=1 3 𝑒𝑗 𝑂𝑗𝑖 𝑂 = 𝑂11 𝑂12 𝑂13 𝑂21 𝑂22 𝑂23 𝑂31 𝑂32 𝑂33 Matrix representation of 𝒪 in 𝑒𝑖
- 11. Permutation Operator 𝓟 = 01 0 1 0 0 0 0 1 𝜀1 = 0.354 −0.927 0.127 𝓟𝜀1 = 0 1 0 1 0 0 0 0 1 0.354 −0.927 0.127 = −0.927 0.354 0.127 𝓟𝜀1 = 𝑗=1 3 𝜀𝑗1 𝑃𝑗1 How to perform column permutations? 𝜀2 = 0.612 0.12 −0.78 How to get P?
- 12. 𝜀 = 0.354 0.6120.707 −0.927 0.127 0.354 0.127 −0.78 0.612 𝓟𝜀 = −0.927 0.127 0.354 0.354 0.612 0.707 0.127 −0.78 0.612 𝓟 = 0 1 0 1 0 0 0 0 1 𝑃 = 𝜀 𝓟 𝜀 = 0.354 0.612 0.707 −0.927 0.127 0.354 0.127 −0.78 0.612 𝑇 𝓟𝜀 𝑃 = −0.640 −0.621 −0.621 −0.621 0.764 −0.171 −0.452 −0.171 0.875
- 13. 𝓟𝜀𝑖 = 𝑗=1 3 𝜀𝑗 𝑃𝑗𝑖 𝑃= −0.640 −0.621 −0.621 −0.621 0.764 −0.171 −0.452 −0.171 0.875 𝓟 = 0 1 0 1 0 0 0 0 1 𝜀 = 0.354 0.612 0.707 −0.927 0.127 0.354 0.127 −0.78 0.612 𝓟𝜀2 = 𝑗=1 3 𝜀𝑗 𝑃𝑗2 𝓟𝜀2 = 0.127 0.612 −0.78 = 0.354 −0.927 0.127 −0.621 + 0.612 0.12 −0.78 0.764 + 0.707 0.354 0.612 −0.171
- 14.
- 15. 𝜀 = 0.354 0.6120.707 −0.927 0.127 0.354 0.127 −0.78 0.612 𝑅𝑥𝑦 = 0.707 0.000 0.707 −0.500 0.707 0.500 −0.5 −0.707 0.500 𝑅𝑥𝑦𝜀 = 0.340 −0.119 0.933 −0.769 −0.606 0.203 0.542 −0.786 −0.298
- 17. Matrix multiplication 𝓒 =𝓐𝓑 𝓒𝑒𝑖 = 𝑗=1 3 𝑒𝑗 𝐶𝑗𝑖 𝓐𝓑𝑒𝑖 = 𝑗=1 3 𝑒𝑗 𝐶𝑗𝑖 𝓐 𝑗=1 3 𝑒𝑘 𝐵𝑘𝑖 = 𝑗=1 3 𝑒𝑗 𝐶𝑗𝑖 𝑗=1 3 𝓐𝑒𝑘 𝐵𝑘𝑖 = 𝑗=1 3 𝑒𝑗 𝐶𝑗𝑖 𝑗=1 3 𝑘=1 3 𝑒𝑗𝐴𝑗𝑘𝐵𝑘𝑖 = 𝑗=1 3 𝑒𝑗 𝐶𝑗𝑖 𝐶𝑗𝑖 = 𝑘=1 3 𝐴𝑗𝑘𝐵𝑘𝑖
- 18. Order of Matrixmultiplication 𝓐𝓑 ≠ 𝓑𝓐 Commutator of operators Do not commute 𝐴, 𝐵 = 𝐴𝐵 − 𝐵𝐴 Anti-commutator of operators 𝐴, 𝐵 = 𝐴𝐵 + 𝐵𝐴
- 19. Matrices : Definition 𝐴= 𝐴11 𝐴12 ⋯ 𝐴1𝑀 𝐴21 𝐴22 ⋯ 𝐴2𝑀 ⋮ ⋮ ⋯ ⋮ 𝐴𝑁1 𝐴𝑁2 … 𝐴𝑁𝑀 If N = M, matrix is square matrix 𝑎 = 𝑎1 𝑎2 ⋮ 𝑎𝑁 𝐴𝑎 = 𝑏 𝑏𝑖 = 𝑖=1 𝑀 𝐴𝑖𝑗𝑎𝑗
- 20. Adjoint and Transpose 𝑂= 𝑂11 𝑂12 𝑂13 𝑂21 𝑂22 𝑂23 𝑂31 𝑂32 𝑂33 𝑂𝑇 = 𝑂11 𝑂21 𝑂31 𝑂12 𝑂22 𝑂32 𝑂13 𝑂23 𝑂33 𝐴𝑇 𝑖𝑗 = 𝐴𝑗𝑖 𝐴† 𝑖𝑗 𝑇 = 𝐴∗ 𝑗𝑖 Take complex conjugate of each element and interchange the rows interchange the rows complex conjugate - the sign of the imaginary part of all its complex numbers have been changed
- 21. complex numbers Complex numbers- extends the real numbers with a specific element denoted i, called the imaginary unit. 𝑖 = −1 𝑖2 = −1 If a and b are real numbers, then 𝑎 + 𝑏𝑖 is a complex number 𝑎 + 𝑏𝑖 The Complex Plane Basic operations 𝑧1 = 𝑥1 + 𝑖𝑦1 𝑧2 = 𝑥2 + 𝑖𝑦2 𝑧1 + 𝑧2 = 𝑥1 + 𝑥1 + 𝑖 𝑦1 + 𝑦2 𝑧1 − 𝑧2 = 𝑥1 − 𝑥1 + 𝑖 𝑦1 − 𝑦2 𝑎𝑧1 = 𝑎𝑥1 + 𝑎𝑖𝑦1
- 22. Definition in squareMatrices Diagonal matrix – all off diagonals are zero Trace of matrix – sum of the diagonal elements Unit matrix – Identity matrix, diagonal with 1 in diagonal Inverse; 𝐴−1 𝐴−1 𝐴 = 𝐴𝐴−1 = 𝐼 = 1 Unitary matrix 𝐴−1 = 𝐴† Orthogonal matrix – real unitary matrix
- 23. Hermitian matrix –self adjoint 𝐴 = 𝐴† Symmetric matrix – real Hermitian matrix Determinants 𝑑𝑒𝑡 𝐴 = 𝐴 = 𝐴11 ⋯ 𝐴1𝑁 ⋮ ⋮ ⋮ 𝐴𝑁1 ⋯ 𝐴𝑁𝑁 𝐴 = 𝑖=1 𝑁! −1 𝑝𝑖 𝓟𝑖𝐴11𝐴12 ⋯ 𝐴𝑁𝑁 −1 0 𝐴11𝐴22𝐴33 + −1 1 𝐴12𝐴21𝐴33 + −1 1 𝐴13𝐴22𝐴31 + −1 1 𝐴11𝐴23𝐴32 + −1 2 𝐴13𝐴21𝐴32 + −1 2 𝐴12𝐴23𝐴31 𝐴11 𝐴12 𝐴13 𝐴21 𝐴22 𝐴23 𝐴31 𝐴32 𝐴33 =
- 24. N-dimensional vector spaces 𝑒|𝑖 𝑖 = 1,2, … , 𝑁 |𝑎 = 𝑖=1 𝑁 |𝑖 𝑎𝑖 𝑎 = 𝑎1 𝑎2 ⋮ 𝑎𝑁 ket vectors 𝑎† = 𝑎1 𝑎2 ⋯ 𝑎𝑁 bra vectors Braket notation 𝑎||𝑏 = 𝑎|𝑏 = 𝑖=1 𝑁 𝑎∗ 𝑖𝑏𝑖 = 𝑎1 𝑎2 ⋯ 𝑎𝑁 𝑏1 𝑏2 ⋮ 𝑏𝑁 𝑎|𝑎 = 𝑖=1 𝑁 𝑎∗ 𝑖𝑎𝑖 = 𝑖=1 𝑁 𝑎𝑖 2
- 25. 𝑎 = 𝑎1 𝑎2 ⋮ 𝑎𝑁 |𝑎 = 𝑖=1 𝑁 |𝑖𝑎𝑖 𝑎† = 𝑎1 𝑎2 ⋯ 𝑎𝑁 𝑎| = 𝑖=1 𝑁 𝑎𝑖 ∗ 𝑖| 𝑎|𝑏 = 𝑖=1 𝑁 𝑎𝑖 ∗ 𝑖| 𝑗=1 𝑁 |𝑗 𝑏𝑗 = 𝑖=1 𝑁 𝑗=1 𝑁 𝑎𝑖 ∗ 𝑖|𝑗 𝑏𝑗 𝑖|𝑗 = 𝛿𝑖𝑗
- 26. How to determinecomponents with respect to a basis? |𝑎 = 𝑖=1 𝑁 |𝑖 𝑎𝑖 𝑎| = 𝑖=1 𝑁 𝑎𝑖 ∗ 𝑖| 𝑎|𝑗 = 𝑖=1 𝑁 𝑎𝑖 ∗ 𝑖|𝑗 𝑗|𝑎 = 𝑖=1 𝑁 𝑗|𝑖 𝑎𝑖 𝑎|𝑗 = 𝑎𝑗 ∗ 𝑗|𝑎 = 𝑎𝑗 |𝑎 = 𝑖=1 𝑁 |𝑖 𝑖|𝑎 𝑎| = 𝑖=1 𝑁 𝑎|𝑖 𝑖| 𝑖=1 𝑁 |𝑖 𝑖| = 1 Completeness of the basis
- 27. 0.354 −0.927 0.127 0.354 −0.927 0.127+ 0.612 0.127 −0.78 0.612 0.127 −0.78 + 0.707 0.354 0.612 0.707 0.354 0.612 = 1 0 0 0 1 0 0 0 1 |𝑖 = 0.354 0.612 0.707 −0.927 0.127 0.354 0.127 −0.78 0.612 𝑖=1 𝑁 |𝑖 𝑖| = 1
- 28. 𝒪|𝑎 = |𝑏 𝒪|𝑖= 𝑗 𝑁 |𝑗 𝑂𝑗𝑖 𝑂 is the matrix representation of 𝒪 in the basis |𝑖 𝑘|𝒪|𝑖 = 𝑗 𝑁 𝑘|𝑗 𝑂𝑗𝑖 = 𝑂𝑘𝑖 Using completeness relation 𝒪|𝑖 = 1𝒪|𝑖 = 𝑖=1 𝑁 |𝑖 𝑖| 𝒪|𝑖 𝑗 𝒪|𝑖 = 𝑗 𝒪|𝑖 = 𝑂𝑗𝑖 𝓒 = 𝓐𝓑 𝑖|𝓒|𝑗 = 𝑖|𝓐𝓑|𝑗 𝑖|𝓐1𝓑|𝑗 𝑖|𝓐 𝑘=1 𝑁 |𝑘 𝑘| 𝓑|𝑗 𝑘=1 𝑁 𝑖|𝓐|𝑘 𝑘| 𝓑|𝑗 𝐶𝑖𝑗 = 𝑘=1 𝑁 𝐴𝑖𝑘𝐵𝑘𝑗
- 29. Hermitian operator –self adjoint 𝒪∗† = 𝒪 Change of basis We can express any ket in the basis |α as a linear combination of the kets in basis |𝑖 |α = 1|α |α = 𝑖=1 𝑁 |𝑖 𝑖|α |α = 𝑖=1 𝑁 |𝑖 𝑈𝑖∝ 𝑖|α = 𝑈 Transformation matrix
- 30. 𝑖|α = 𝑈 Propertiesof transformation matrix 𝑈†𝑈 = 1 𝑈𝑈† = 1 𝙊, 𝞨, Matrix representation two operators in basis |𝑖 and ∝ respectively 𝞨 = 𝑈𝙊𝑈† 𝙊 = 𝑈†𝞨U