Pre (QC & QE) Pace University START THE. pace.edu. Fly By (C) RIF /9/2003 Pre (QC & QE) Fly By V.6.

Transcription

1 THE START Pre (QC & QE) Fly By Pace University pace.edu 1

2 Fly By Meta Intro* 1. A fly by is a quick look at a large number of topics. 2. No intention to teach, only to pass on a vague idea of what s going on. [A weak form of cultural literacy ]. 3. This is an experiment. I need your feedback (form at the end). 2

3 Complex Numbers & Conjugation TOC -1 [109 Slides + 5 Titles]* (53 * Slides) TOC - 1 [109 Slides + 5 Titles]* I. Space, Vector, Direction [6] II. Complex Numbers & Conjugation (+,-,x,/, ) [20] Z, Z III. Matrix-Vector 1 (k v,+,-,,x, v,,mv, v, v, v w ) [24] IV. Matrix-Vector 2 (Eigenvalue-vector, Basis, Projector) [8] V. Fourier Expansion ( Dimensional Vectors) [10] Fourier Transforms VI. Hermitian Matrix (Spectral Expansion) [4] σ y VII. 2x2 Hermitian Example (Pauli ) [8] 3

4 TOC 2 Complex Numbers & Conjugation TOC- 2 VIII. Unitary Diagonalization (Unitary Matrix) [9] Simultaneous diagonalization of commuting matrices IX. Quantum Mechanics [8] X. Appendix I Differential Operators [2] XI. Appendix II Derivative, Integral & Coordinate Systems [3] XII. Appendix III Probabilities [3] XIII.Appendix IV Polarized Photons [3] XIV. Appendix V - Feed Back Form [1] 4

5 I Space Vector Operator Complex Numbers & Conjugation I Space, Vector, Operator 5

6 Definition of A Space Definition of a Space 1.A SET of thingies called points 2.A SET of of properties of the points Very much like an abstract Data Type 6

7 Definition of Properties Definition of Some Properties 1.A fixed number of coordinate axes (points are n-tuples of coordinates) 2.Addition, Subtraction, Multiplication (coordinate by coordinate +, -, x) Very much like numbers or 3-D vectors 7

8 Usual Space 2 & 3 D Picture Usual Space 2 & 3 D Picture DOT product yields a SCALAR = = a number Right hand thumb along Z axis: fingers curl from x to y z-axis j k i Point: (x, y, z) +-: (x, y, z) + - (x, y, z ) = (x+x, y+y, z+z ) y-axis x-axis Product: (x, y, z) (x, y, z ) DOT = (xx +yy + zz ) 8

9 Definition of Directions Definition of Direction (Vector) 1.Two points determine a direction: (the line between them, with a designated start point and end point) 2.Move the line to the origin and use point Addition, Subtraction, Scalar Multiplication 9

10 Definition of Operator* Definition of Operator* Maps vectors to vectors Example: rotations 3-D rotations have an axis of rotation in the space that stays unchanged. 2-D rotations have NO axis of rotation in the space that stays unchanged. Every vector gets changed. 10

11 II Complex Numbers & Conjugation Complex Numbers & Conjugation II Complex Numbers & Conjugation 11

12 Definition of a Complex Number Z* Definition of a Complex Number Z* Z = a+ ib a and b are real numbers and ( 1) i = 12

13 Definition of the LENGTH of a Complex Number Z Definition of the LENGTH of a Complex Number Z Z = a 2 + b 2 The length is the distance from the origin to the point Z The length is also called the modulus of Z 13

14 Picture of Z and its Conjugate Picture of Z and its Conjugate Imaginary Axis i i b = R sin( θ ) Z b Z is the point in the plane R θ i-axis a = R cos( θ ) a Z 1 = R Real Axis X-AxisR X-AxisR b Z is the point in the plane i i Z 14

15 Alternate Definition of a Complex Number Z Alternate Definition of a Complex Number Z [ cos( ) sin( ) ] Z Z i Z e i θ = + = θ θ Z is the distance from the origin to the point Z θ is the angle that the vector to Z makes with the real axis POLAR FORM z = z e i θ 15

16 Definition of a Conjugate Complex Number Z * Definition of a Conjugate of a Complex Number Z* Z = a ib a and b are real numbers and ( 1) i = 16

17 Alternate Definition of a Complex Conjugate Number Z * *Alternate Definition of a Complex Conjugate Number Z = Z cos( θ) i Z sin( θ) = Z e i θ Z is the distance from the origin to the point Z θ is the angle that the vector to Z makes with the real axis 17

18 Definition of Addition Definition of Addition Z = a+ ib W = c+ id ZW= (a+ib)+(c+id)= (a+c) + i(b+d) Element by element addition. 18

19 Picture of Addition (&-) Picture of Addition (&-) i b=r sin(θ) Imaginary Axis b R θ i-axis X-Axis a Z W THE SUM VECTOR Z+W Subtraction is the opposite Take the difference of the horizontal (real) and vertical (imaginary) components. b+d d φ c Real Axis a+c Vector Addition: horizontal components add (a+c) and vertical components add (b+d). 19

20 Picture of Subtraction (&+) Picture of Subtraction (&+) Imaginary Axis i b+d b=r sin(θ) b d -Z Z W W=(Z+W)-Z THE SUM VECTOR Z+W R θ i-axis X-Axis φ a c Real Axis a+c 20

21 Definition of Multiplication Definition of Multiplication Z = a+ ib W = c+ id Real Part ZW= (a+ib)(c+id)= Imaginary Part (ac-bd) + i(ad+bc) i 2 =-1 21

22 Alternate Definition of the LENGTH of a Complex Number Z * Alternate Definition of the LENGTH of a Complex Number Z* 2 Z = ZZ Z = ZZ 22

23 Definition of Division Definition of Division Z = a+ ib W = c+ id Z a + ib a + ib c id = = = W c+ id c+ id c id ( a ib )( c id ) ( ac bd ) i ( ad bc ) + + = c + d c + d Z ZW ZW = = 2 W WW W 23

24 Reinterpretation of Polar Form Reinterpretation of Polar Form Using definitions from infinite series expansions: Z = Z cos( θ ) + i Z sin( θ ) = i θ e = cos( θ ) + isin( θ ) Z = Z e i θ 24

25 Alternate Definition of Multiplication Alternate Definition of Multiplication Z = a+ ib = e i θ W = c+ id = e i φ ( )( ) iθ iφ ZW = a + id c + id = Z e W e = ( ) iθ iφ i θ+ φ ZW = ZWe e = ZWe 25

26 Geometry of Multiplication Geometry of Multiplication ( ) i ZW = Z W e θ + φ Multiply the length of Z by the length of W and rotate the Z vector by adding the angle φ. 26

27 Alternate Definition of Division Alternate Definition of Division Z = a+ ib = e i θ W = c+ id = e i φ Z ( + ) ( ) a id Z e = = = + i φ W c id W e i θ iθ iφ Z = Ze e = Z W W W e i ( θ φ ) 27

28 Geometry of Division Geometry of Division Z = Z e W W i ( θ φ ) Divide the length of Z by the length of W and rotate the Z vector by subtracting the angle φ. 28

29 Square root of i Square Root of i u = i uu = i π 2 i iθ i2 θ 2 u = u e u e = i = e θ π π π =, u = 1 u = cos isin u = + i 2 2 The 45 degree unit vector multiplied by itself rotates to the vertical. 29

30 A Pure Rotation * A Pure Rotation* Any complex number W, is rotated by the angle θ when multiplied by e iθ iψ iθ i ( + ) W = W e e W = W e ψ θ This is a complex number with the same length as W but with an angle increased by θ. I.e., a pure rotation. 30

31 III Matrix-Vector 1 Complex Numbers & Conjugation III Matrix-Vector 1 31

32 Space & Dual Space Space & Dual Space n-tuple Transpose Row n-tuple v 1 T = = 2 (, ) v v v v v 1 2 v i are scalars (numbers) The set of scalar n-tuples are called a space. The set of transposed scalar n-tuples are called its dual space. 32

33 Vectors & Matrices * Vectors & Matrices* Column v 1 T = = 2 Transpose Row ( ) v v v v v 1 2 v i, w i, m ij are scalars (numbers) m m M = m m rows 2 columns rows index column index 33

34 Vectors & Matrices <Braket> * Vectors & Matrices <Bracket>* T v = 1 = v v a bra v 2 ( ) v = v v = v a ket

35 Scalar Multiplication * Scalar Multiplication* a is a scalar (number) v av av = a 1 1 = = va v av 2 2 T = ( ) = (, ) = T av a v v av av v a av = a v av = a v T 35

36 Vector Addition * Vector Addition* v w av + bw = a 1 b 1 + = v w 2 2 av bw av + bw = av bw av + bw element by element addition 36

37 Vector Space (1/2) Vector Space (1/2) & Dual Space The set of Scalar n-tuples are called a vector space The set of transposed n-tuples are called its dual space. Iff (if & only if) the are closed under scalar multiplication and vector addition. Closed means you get back a vector when you do scalar multiplication and vector addition. 37

38 Usual Space 2 & 3 D Picture Usual Space 2 & 3 D Picture Right hand thumb along Z axis: fingers curl from x to y z-axis j k i y-axis x-axis 38

39 Vector Space (2/2) Vector Space (2/2) & Dual Space ( ) v = 1 = v i + v j v T v ( ) ( ) v = v v = v i + v j We identify the dual space in 2 and 3-D with vectors of the same space. 39

40 (1 of 5) Inner Multiplication (1 of 5) Inner Multiplication== Scalar Product == Inner Product == Dot Product (Sum of Products) Sum over element by element multiplications w T v w = ( v v ) 1 v w 1 2 = w 2 No T ( ) vw + vw = ascalar

41 (2 of 5) Inner Multiplication (2 of 5) Inner Multiplication Sum over element by element multiplications ( )( ) v T w = v i + v j w i + w j = v w = No T ( vw i i+ vw j j) = ( vw + vw ) ( ) = vw = vw cos θ 41

42 (3 of 5) Orthogonal (3 of 5) Orthogonal Two vectors v and w are orthogonal iff (v, w)=0 T cos ( ) 0 v w = v w θ = = v w If v and w are not of 0 length, then cos(θ)=0. θ=90 The vectors are perpendicular to each other. 42

43 (4 of 5) Inner Multiplication (4 of 5) Inner Multiplication Unit orthonormal coordinate vectors: ( i j) ( i k) ( j k) = = = 0 ( i i) ( j j) ( k k) = = = 1 i j = i k = j k = 0 ii = j j = kk = 1 43

44 (5 of 5) Hermitian I.P. * (5 of 5) Hermitian I.P.* w T v w = ( v v ) 1 v w v w 1 2 = = w 2 ( ) v w + v w = a complex scalar In QM only Hermitian Inner Products. vw Means Hermitian Inner Product. 44

45 Vector Length * Vector Length* v T v v = ( v v ) 1 v v 1 2 = v = ( ) vv vv v squared " length " of vector v 45

46 Picture Inner Product Picture of Inner Product (Not obvious but so) v w = v w cos( θ ) v Y-Axis w v cos(θ) θ X-Axis 46

47 Picture of Addition (&-) Picture of Addition (&-) v y-axis Subtraction is the opposite Take the difference of the horizontal (x) and vertical (y) components. v 2 = v sin(ψ) y Axis v 2 +w 2 v + w == v + w THE SUM VECTOR v+w v 2 w 2 ψ φ v 1 w 1 x Axis X-Axis v 1 +w 1 Vector Addition: horizontal components add (v 1 +w 1 ) and vertical components add (v 2 +w 2 ). θ v w 47

48 Outer Multiplication * *Outer Multiplication== Outer Product == Tensor Product == Product Element by element multiplications w Switched T wv = 1 ( v v ) = w v = w v w wv wv wv wv = a D Matrix 48

49 (1 of 2) Cross Product (2-D only) (1 of 2) Cross Product (2-D Only) Determinant Multiplication ( ) ( ) u v = u u v v = ( ) ( ) sin ( θ ) ui + u j vi + v j = uv k i j k det ( ) u u i j uv v u k 1 2 = v v

50 (2 of 2) Cross Product (2-D only) (2 of 2) Cross Product (2-D only) Unit coordinate vectors: Forward Permutation i-j-k ( ), ( ), ( ) i j = k j k = i k i = j ( i i) ( j j) ( k k) = = = 0 ( ), ( ), ( ) j i = k k j = i i k = j Reverse Permutation k-j-i 50

51 Matrix-Vector Multiplication 1 * Matrix-Vector Multiplication 1* ( ) ( ) ( m11 m12 ) ( m m ) m m v = = = Mv v m21 m 22 v ( ) ( ) m11 m12 v m11 v1 + m12 v2 = = m m v m v + m v w Rows come from left factor, columns from the right factor 2 rows, 1 column result [(2x2)(2x1)=(2x1) Mv m m v + ( mv ) 11 1 mv 12 2 ( m v + m v ) = m21 m = 22 v

52 Matrix-Vector Multiplication 2 Matrix-Vector Multiplication 2 ( ) ( ) ( m11 m12 ) ( ) m m v = = = Mv v m21 m 22 v2 m21 m ( + ) ( m v m v ) m11 m12 v mv 11 1 mv 12 2 = = m m v M transforms v into w. M is called an operator. w M( av + bu) = w = amv + bmu M is called a linear operator 52

53 Matrix-Matrix Multiplication Matrix-Matrix Multiplication ( ) ( h h ) h h m m h m m22 h 2 HM = = m m ( ) ( ) ( + ) ( + ) h1 m 1 h1 m 2 h11 m11 + h12 m21 h11 m12 + h12 m22 = h m h m h m h m h m h m rows, 2 column result of 4 scalars HM = h m ij ik kj k 53

54 Summation Convention Summation Convention HM = h m ij ik kj k HM = hm = hm ij ik kj ik kj k A repeated index (here k) is to be summed over all of its values. 54

55 IV Matrix Vector 2 Complex Numbers & Conjugation IV Matrix Vector 2 55

56 Eigenvectors & Eigenvalues * Eigenvectors & Eigenvalues* ( mv mv 12 2 ) ( m v + m v ) Mv = = w When the target vector is a scalar multiple of the source vector, the source vector is called an eigenvector. Mv = M v = w = w = λv = λ v Mv = λ v λ is a stretch factor, called an eigenvalue. 56

57 Normalized Eigenvectors * Normalized Eigenvectors* Mv = λ v Vectors of length 1 are called normalized. 1 1 Mu = Mv = λ v = λ u v v v vv = v & vv = = 1 2 v v v 2 uu = 1 = u 2 57

58 Independent Vectors Independent Vectors k= N k = 1 α k v k = 0 N vectors are DEPENDENT if there are non 0 weights α k making the sum = 0. Then one can be taken to the right hand side defining it in terms of the others. It is dependent on the others. N vectors are INDEPENDENT if there are no such non 0 weights α k (i.e., all are 0). 58

59 Basis Vectors * Basis Vectors* k= N k = 1 α k v = w k If any w in a space can be written as a weighted sum of a fixed set of vectors they are a basis of the space. There are N Basis vectors in an N dimensional space. We usually take them as normalized. The α k are called the coordinates of w in the basis {v i }. The (α k v k ) are the components of w. T (,,,,, ) w = α α α α α α j N 1 N 59

60 Components as Projections Components as Projections α 2 =wsin(θ) w v2 v2 -Axis θ v1 α 1 =wcos(θ) v 1 -Axis 60

61 Projectors * Projectors* k= N k= L αk k = k= L = αl L k = 1 P v P w v k= N k= N v v α v = α v v v = α v L L k k k L L k L L k= 1 k= 1 0, k L v L v k = The v L are orthonormal 1, k = L. The {v i } are orthogonal and normalized. The (α k v k ) are the components of w. 61

62 Commuting Matrices * Commuting Matrices* HM = MH Hv = λ v MHv = λmv MH v = λ M v = H M v Mv = w Hw = λ w w = α v & M v = λ w ( λα ) Mv = v Commuting matrices share eigenvectors. (With maybe different eigenvalues) 62

63 Stuff Complex Numbers & Conjugation V Fourier Expansion 63 V Fourier

64 Fourier Expansion 1 Fourier Expansion 1 (I=[-π, π]) f(x)= a ncos(n x)+b nsin(n x) n=0 Dirichlet Conditions : The Fourier series exists if f(x) is periodic & finite f(x) is absolutely integrable * in each finite interval I I f( x) dx < f(x) doesn t wiggle too much in each finite interval I f(x) has only a finite set of jump discontinuities in I A well behaved function is an infinite sum of basis functions in an infinite dimensional space. 64

65 Basis Vectors and I.P. Basis Vectors & Inner Product Basis vectors sin( mx ) & cos( mx ) Inner Product π π gxhxdx ( ) ( ) = gh 65

66 Fourier Expansion 2 Fourier Expansion 2 (I=[-π, π]) π π π π π π f(x)= a ncos(n x)+b nsin(n x) n=0 Orthogonality Relations sin( mx )sin( nx ) dx cos( mx )cos( nx ) dx π n = m; 0 = 0 n m; 0 π n = m; 0 = 0 n m; 0 { } sin( mx )cos( nx ) dx = 0 all n & m 66

67 Fourier Expansion 3 Fourier Expansion 3 f(x)= a ncos(n x)+b nsin(n x) n=0 1 π 1 n = m; 0 sin( mx )sin( nx ) dx = π π 0 n m; 0 1 π 1 n = m; 0 cos( mx )cos( nx ) dx = π π 0 n m; 0 1 π π π Normalization { } sin( mx )cos( nx ) dx = 0 all n & m 67

68 Fourier Expansion 4 Fourier Expansion 4 f(x)= a ncos(n x)+b nsin(n x) n=0 Coefficients 1 π 1 π π π π π f ( x )sin( nx ) dx = b f ( x )cos( nx ) dx = a n n 68

69 Fourier Expansion 5 Fourier Expansion 5 f(x)= a ncos(n x)+b nsin(n x) n=0 Coefficients 1 π 1 π a + ib π π n n π π inx f( x) e dx = [ ] f( x) cos ( nx) + isin( nx) dx = 69

70 Fourier Transform 1 * Fourier Transform (I=[-, ])* π π = = i(2 p) x i(2 x) p f( x) e dx g( p); f( x) g( p) e dp Dirichlet Conditions : The Fourier Transform exists if f(x) & g(p) bounded & piece-wise continuous f(x) & g(p) are square integrable 2 2 f( x) dx < ; g( p) dp < I I f( x) 0 as x & gp ( ) 0asp A well behaved function is a transform of another well behaved function 70

71 Function Inner Product * Hermitian Inner Product* gxhxdx ( ) ( ) = gh 71

72 Fourier Transform 2 * Fourier Transform 2* Definition of a variables variance ( x) = & ( p) = ( 2) ( 2 ( ) ( ) ) ( ) ( ) 2 2 x f x f x dx p g p g p dp f( x) f( x) dx g( p) g( p) dp Uncertainty Relation of Conjugate Variables (from a transform pair) 1 ( x)( p) 4 π 72

73 VI Hermitian Matrix * Complex Numbers & Conjugation VI Hermitian Matrix* T H = H = def H * 73

74 Properties of H * Properties of H* A Hermitian Matrix H Has Real Eigenvalues A Hermitian Matrix H has a basis of eigenvectors 74

75 The Spectral Expansion of H * The Spectral Expansion of H* Hv = λ v k k k H = λ v v = λ P k k k k k k k v v i j 0, i j = 1, i = j 75

76 Projectors on Eigenspaces * Projectors on Eigenspaces* P = v v L L L w = v i α i i P w = v v w = v v v = L L L L L i i i α i α α v v v = v i L L i L L 76

77 VII 2x2 Hermitian Example Complex Numbers & Conjugation VII 2x2 Hermitian Example 77

78 Hermitian Matrix H * Example:* A 2x2 Hermitian Matrix H [Pauli s σ 2 ] H 0 i = = i 0 σ 2 78

79 The Matrix H & Eigenvalues of H * The Matrix H* 0 i 0 i ( ) T 0 i H =, H =, and H = H * = = H. i 0 i 0 i 0 Real Eigenvalues of H Hv = λ v i = i i ( H) λλ 1 2 ( i)( i) ( ) [ 1, 2 ] i 2x2 det = product of evals & trace = sum of evals. det = = = 1 tr H = + = = λ + λ λ = λ λ1 = 1, λ2 = 1 This will be spin in the y direction. 79

80 Eigenvectors of H * Eigenvectors of H* Hv = v 0 i α iβ α Hv = = v = = α = i β β = iα i 0 β iα β [ ]& [ ] The eigenvector is determined up to a scale factor, so we can assume for now that α = 1. α 1 = = = = = β i V 1 : [ α iβ] &[ β iα] [ ifα 1, β i] Hv = + v. 0 i α iβ α Hv = =+ v = = α = i β β = iα i 0 β iα β α 1 V 2 : [ α = iβ] &[ β = iα] [ ifα = 1, β = i] = β i [ ]& [ ] 80

81 Eigenvector Normalization * Eigenvector Normalization* Now we scale them to length 1. V 1 : normalized [using the hermitian inner product] ( 1 i) = ( 1 i) = 1+ 1= 2= v1 v1 = 2 i i 1 V 1 : = = 2 i i 2 V 2 : normalized [using the hermitian inner product] ( 1 i) = ( 1 i) = 1+ 1= 2= v2 v2 = 2 i i 1 V 2 : = = 2 i i 2 81

82 Orthogonality of the Eigenvectors of H * Orthogonality of the Eigenvectors of H* (normalized or not): ( v ) ( ) 1 v2 v2 v i 2 1 i i 2 2 i = = = = = 82 0

83 Eigenvector Picture for H Eigenvector Picture for H i i 2 V 2 v2 i-axis X-Axis 1 v 1 i 2 -i 1 2 V 1 83

84 The Spectral Expansion of H ** The Spectral Expansion of H** H H = λ v v k k k k i 2 1 i 2 1 i = = ( 1) () 1 i 0 i + = 2 2 i i 1 i i + = i 1 i 1 i QED. 84

85 VIII Unitary Diagonalization Complex Numbers & Conjugation VIII Unitary Diagonalization 85

86 Example:* Title Unitary Matrix U * A 2x2 Unitary Matrix U [Diagonalizes Pauli s σ 2 ] [Columns are the Eigenvectors of H] U = i i

87 The Unitary Matrix U-(1 of 3) The Unitary Matrix U-(1 of 3) i 1 i T T 2 2 U =, U =, and U = = U *. i i 1 i 1 i i T 1 0 UU = UU * = = I. i i 1 i 0 1 =

88 The Unitary Matrix U-(2 of 3) The Unitary Matrix U-(2 of 3) 1 i T 1 0 UU = U * U = = I. 1 i i i 0 1 = i T 0 i 2 2 U HU = U * HU = 1 i i 0 = i i

89 The Unitary Matrix U-(3 of 3) * The Unitary Matrix U-(3 of 3)* 1 i λ 0 1 = = 1 i i i λ I.e., the unitary matrix U made up from the normalized eigenvectors of H, diagonalizes H giving the matrix of the eigenvalues of H.. U rotates the space until the space basis vectors align with the eigenvectors of H. H acting on this rotated space maps the coordinate directions to themselves. 89

90 Commuting Matrices * Commuting Matrices* 0 i 1 1 H = & M = i HM MH 0 i 1 1 i i = = i i i i i i = = 1 1 i 0 i i 90

91 Commuting Matrices Share * Eigenvectors Commuting Matrices* Mv 1 = = (1 + i ) 1 1 i i Mv 2 = = (1 i ) 1 1 i i 2 2 Commuting matrices share eigenvectors. 91

92 Commuting Matrices Eigenvalues * Commuting Matrices Eigenvalues* 1 2 M H Mv 1 = (1 + i ) (1 + i ) = λ 1 λ 1 = 1 i = (1 ) (1 ) = λ λ = Mv i i i M H

93 Diagonalizing Commuting Matrices * Diagonalizing Commuting Matrices* 1 i T U MU = U * MU = = 1 i 1 1 i i i 1+ i 1 i 1+ i 1+ i 1 i i = = 1 i 1 i 1+ i 1+ i 1 i 1 i 1 i M 1+ i i λ = M 0 λ 2 Eigenvalues not real so M is not an Observable. 93

94 IX QM Complex Numbers & Conjugation IX - QM 94

95 QM 1 * Complex Numbers & Conjugation QM 1* Complex Numbers & Conjugation 1. Observables are hermitian matrices H (operators) With a complete (basis) set of eigenvectors 2. On a hermitian inner-product space 4. A pure state is an eigenvector 5. A measurement is a projection onto a pure state vector 6. A possible measured value is the eigenvalue of H 7. The probability of finding that value is the square of the coordinate of the normalized form of the full state vector. 8. A full measurement is set of commuting matrices and their eigenvalues at that common eigenstate 95

96 QM 2 * Complex Numbers & Conjugation QM 2* Complex Numbers & Conjugation 9. A superposition state is a general vector A sum (mixture) of pure (measurable) states 10. Observables come in Fourier Transform (Conjugate) Pairs (that do not commute) Ex: Position x and momentum p Ex: Spin z component and Spin y component 11. Transform Uncertainty Relation == Heisenberg s for Conjugate Variables (amount of non commutation) 12. Non-conjugate variables commute (so are simultaneously measurable) Ex: Spin z component and S (total spin) 13. Time evolution is done by Unitary Operators e. g. i t e θ 96

97 QM 3 * Complex Numbers & Conjugation QM 3* The energy of a moving particle is mv p E = mv 2 = 2 m = 2m De Broglie: A moving particle is represented by a plane wave (or its Fourier Transform) ( Et ) πi ( px ) πi ( Et ) πi ( px ) πi h h h h ψ( xt, ) = e e & ψ( xt, ) = e Φ( pe ) dp Time Space Part Part Separation of Variables 97

98 QM 4 * Complex Numbers & Conjugation QM 4* Taking the time derivative 2πi 2πi ih ih ( Et ) ( px ) h h ψ ( xt, ) = e Φ( pe ) dp 2π t 2π t 2πi 2πi ih 2 π i ( Et ) ( px ) h h = E e ( p) e dp 2 π h Φ = Eψ ( x, t) This is an eigenvalue equation where energy is the eigenvalue 98

99 QM 5 Complex Numbers & Conjugation QM - 5 Taking FTs & Space Derivatives 2πi 2πi h h ( Et ) ( px ) h h ψ ( x, t) = e Φ( p) e dp i x i x 2πi 2πi h ( Et ) 2 i ( px ) h π h = e p ( p) e dp i Φ h 2πi 2πi ( Et ) ( px ) h h = 2 π e pφ( p) e dp h 2πi 2πi ( Et ) ( px ) h h ψ( x, t) = 2 π e pφ( p) e dp 2 i x x 2πi 2πi ( Et ) 2 ( px ) h π i 2 h = 2 π e p Φ( p) e dp h 2 2πi 2πi 4 π ( Et ) ( px ) h 2 h = e p Φ( p) e dp h 2 i x h 2πi 2πi 2 ( Et ) ( Et ) h h 2 1 h 1 2 ( ) h 2 2πi 2πi 4 π ( Et ) ( px ) h 2 h ψ ( x, t) = e p Φ( p) e dp So, taking Fourier Transforms of e E = e 2πi 2πi ( Et ) ( px ) h h e EΦ ( p) e dp = p 2 m ih Eψ( x, t) = ψ( x, t) = 2 π t 2πi 2πi ( Et ) 1 ( px ) h 2 h e p Φ ( p) e dp = 2 m 2m i4π x 2 2 ψ ( xt, ) i ψ ( x, t) 2 2 m x = 99

100 QM 6 * Complex Numbers & Conjugation QM 6* 2 2 p ih h E ( x, t) ( x, t) ( ) ( 2 ) 2 2m 2 t ψ = = π 2m i 4π x ψ 2 i i ψ( x, t) + ψ( x, t) = 0 2 t 2 m x Complex Numbers & Conjugation 13. The eigenvalue equation for Energy in x space is the Schroedinger Equation. The solutions of the Schroedinger Equation are the eigenvectors of the space (plane waves of the moving particle). 100

101 QM 7 * Complex Numbers & Conjugation QM 7* Complex Numbers & Conjugation Notice that the time evolution of the particle s position is executed by: 2πi 2πi ( Et ) ( Et ) h h U = e so that U = e and 2 π i T ( Et ) h 1 U = e = U since 2πi 2πi 2πi 2πi ( Et ) ( Et ) ( Et ) ( Et ) h h h h e e = e ( 0 ) e = 1. I. e., U is Unitary. 101

102 X Appendix I Differential Operators Complex Numbers & Conjugation X Appendix I Differential Operators 102

103 Differential Operators Ex: Differential Operators Ex: π i i x & x. = = 2 D D Let w x x h 2 π i ( px ) = = ( px ) w ( px ) w ( ) ( ) h Dx e Dx e pw e ( ) ( ) ( ) ( ) ( ) 2 ( px ) w ( pxw ) 2 ( pxw ) x = x = D e D pw e pw e 2 D x is adifferential operator witheigenvectors ( px ) w ( e ) and Eigenvalues ( pw ) However, it is not Hermitian. 2 2 p = = 2 ipx ipx ipx ( px ) w T M = e = e so M = e, M = e T M M. It is however Unitary. 103

104 XI Appendix II Derivative, Integral, & Coordinate Systems Complex Numbers & Conjugation X Appendix II Derivative, Integral, & Coordinate Systems 104

105 Derivative & Integral Derivative & Integral Dx f( xyz,, ) = lim ( + ) f ( xyz,, ) f( x x, yz, ) f'( xyz,, ) x x x D x Dx( Dx) f '( x, y, z) f '( ( x + x), y, z) f ''( x, y, z) 2 x x b N 1 ( b a ) ( b a ) f( x, y, z) dx lim f a + n, y, z N a n = 0 N N 105

106 Coordinate System Coordinate Systems Vcos(θ z )=V z V (x, y, z) Z-Axis θz i Vcos(θ y )=V y k j Unit Vectors Vcos(θ x )=V x θy θx Y-Axis X-Axis Pxy[V(x, y, z)] = w(x, y) V( x, y, z) = V + V + V = V cos( θ ) + V cos( θ ) + V cos( θ ) = x y z x y z x y z x y z V cos ( θ ) + cos ( θ ) + cos ( θ ) cos ( θ ) + cos ( θ ) + cos ( θ ) = 1 Vi + V j+ Vk = V( xyz,, ) x y z V( x, y, z) cos( θ ) i+ V( x, y, z) cos( θ ) j+ V( x, y, z) cos( θ ) k = V( x, y, z) x y x 106

107 XII Appendix III Probabilities* Complex Numbers & Conjugation XII Appendix III Probabilities * A set of positive real numbers and p where 0 p 1 0 i < K i i i= K i = 0 1 p i = 1 107

108 Squares of coordinates are probabilities* Squares of State Coordinates Are Probabilities* The squares of the coordinates of a normalized state vector are the probabilities of finding the system in that projected (pure) state. 108

109 Examples* Examples* Ex: The probabilities associated with the tossing of a fair coin are ½ for heads and ½ for tails. Their sum is 1. Ex: The probabilities of passing a vertically polarized photon in either of the two 45 degree filter directions are ½ for and ½ for. Their sum is = + =

110 XIII Appendix IV Polar Photons* XII Appendix IV* Polarized Photons In A Rotated Basis [Have = Probability] = + =

111 Polarized Photons In A Rotated Basis* Polarized Photons Two Axes Rotated 45 Degrees Relative to Each Other The Unit Axis Vectors of Each Project onto the Other as Vectors of Equal Length: Vertical cos π 1 = 4 2 Represented 1 Rotated Vertical In A Rotated cos π 1 = Basis * Have 50% 1 cos π = 4 2 θ = θ = 4 π 4 π θ = θ = 1 cos π = π 4 π 1 cos π = Horizontal (I. E., Equal) cos π 1 = 4 2 cos π 1 = Probabilities Rotated Horizontal 111

112 Filtered Photons Have P=1* Filtered Photons Pass With P = 1 The Photon Polarization Is Parallel To The Filter Axis Vertical cos ( ) cos ( ) P 0 = π =± 1 2 =± 1 = 1 Filtered 1 Rotated Vertical Photons Have P=1* cos π cos π 1 = = 4 2 θ = θ = π 4 4 π θ = θ = cos π π 4 π 4 1 = cos π = Horizontal 1 cos π = cos π = Rotated Horizontal 112

113 IV Appendix V Feedback Form* I Have Studied or I Was Familiar With: YES NO Vaguely The Cartesian model of 3-space Coordinate Systems Vectors Complex Numbers & Conjugation Basic Probability Derivatives Integrals Matrices Eigenvectors & Eigenvalues Hermitian Matrices (HM) Unitary Diagonalization of (HM) The Spectral Expansion Theorem for (HM) Fourier Series Fourier Transforms Basic Quantum Mechanics Spin Quantum Encryption Quantum Information Projectors Did the fly by have any positive benefit for you?

114 The End Pre (QC & QE) Fly By THE END 114