1. Mathematical Tools of Quantum Mechanics
Chapter 2
Mathematical Tools of Quantum Mechanics

2. Hilbert space Let’s recall for Cartesian 3D space:
A vector is a set of 3 numbers, called components – it can be expanded in terms of three unit vectors (basis)
The basis spans the vector space
Inner (dot, scalar) product of 2 vectors is defined as:
Length (norm) of a vector

3. Hilbert space

4. Hilbert space Hilbert space:
Its elements are functions (vectors of Hilbert space)
The space is linear: if φ and ψ belong to the space then φ + ψ, as well as aφ (a – constant) also belong to the space
David Hilbert
(1862 – 1943)

5. Hilbert space Hilbert space:
Inner (dot, scalar) product of 2 vectors is defined as:
Length (norm) of a vector is related to the inner product as:
David Hilbert
(1862 – 1943)

6. Hilbert space Hilbert space:
The space is complete, i.e. it contains all its limit points (we will see later)
Example of a Hilbert space: L2, set of square-integrable functions defined on the whole interval
David Hilbert
(1862 – 1943)

7. Wave function space Recall:
Thus we should retain only such functions ψ that are well-defined everywhere, continuous, and infinitely differentiable
Let us call such set of functions F
F is a subspace of L2
For two complex numbers λ1 and λ2 it can be shown that if

8. Scalar product In F the scalar product is defined as:
Properties of the scalar product:
φ and ψ are orthogonal if
Norm is defined as

9. Karl Hermann Amandus Schwarz
Scalar product
Schwarz inequality
Karl Hermann Amandus Schwarz
(1843 – 1921)

10. Linear operators Linear operator A is defined as:
Examples of linear operators:
Parity operator:
(Multiplication by) coordinate operator:
Differentiation operator:

11. Linear operators Product of operators: In general: Commutator:
Example:

12. Orthonormal bases A countable set of functions
is called orthonormal if:
It constitutes a basis if every function in F can be expanded in one and only one way:
Recall for 3D vectors:

13. Orthonormal bases For two functions a scalar product is:
Recall for 3D vectors:

14. 2.A.2
Orthonormal bases
This means that
Closure relation

15. Orthonormal bases
δ-function:

16. Orthonormal bases A set of functions labelled by a continuous index α
is called orthonormal if:
It constitutes a basis if every function in F can be expanded in one and only one way:

17. 2.A.3
Orthonormal bases
For two functions
a scalar product is:

18. 2.A.3
Orthonormal bases
This means that
Closure relation

19. 2.A.3
Orthonormal bases
Useful relationship:

20. Examples of orthonormal bases
Let us apply Fourier transform to function ψ(x):
Using functions of plane waves
we can write:

21. Examples of orthonormal bases
For two functions
a scalar product is:

22. Examples of orthonormal bases
This means that
Closure relation

23. Examples of orthonormal bases
Let us consider a set of functions:
The set is orthonormal:
Functions in F can be expanded:

24. Examples of orthonormal bases
For two functions
a scalar product is:

25. Examples of orthonormal bases
This means that
Closure relation

26. State vectors and state space
2.B.1
State vectors and state space
The same function ψ can be represented by a multiplicity of different sets of components, corresponding to the choice of a basis
These sets characterize the state of the system as well as the wave function itself
Moreover, the ψ function appears on the same footing as other sets of components

27. State vectors and state space
2.B.1
State vectors and state space
Each state of the system is thus characterized by a state vector, belonging to state space of the system Er
As F is a subspace of L2, Er is a subspace of the Hilbert space

28. Paul Adrien Maurice Dirac
2.B.2
Dirac notation
Bracket = “bra” x “ket”
< > = < | > = “< |” x “| >”
Paul Adrien Maurice Dirac
(1902 – 1984)

29. Paul Adrien Maurice Dirac
2.B.2
Dirac notation
We will be working in the Er space
Any vector element of this space we will call a ket vector
Notation:
We associate kets with wave functions:
F and Er are isomporphic
r is an index labelling components
Paul Adrien Maurice Dirac
(1902 – 1984)

30. Paul Adrien Maurice Dirac
2.B.2
Dirac notation
With each pair ok kets we associate their scalar product – a complex number
We define a linear functional (not the same as a linear operator!) on kets as a linear operation associating a complex number with a ket:
Such functionals form a vector space
We will call it a dual space Er*
Paul Adrien Maurice Dirac
(1902 – 1984)

31. Paul Adrien Maurice Dirac
2.B.2
Dirac notation
Any element of the dual space we will call a bra vector
Ket | φ > enables us to define a linear functional that associates (linearly) with each ket | ψ > a complex number equal to the scalar product:
For every ket in Er there is a bra in Er*
Paul Adrien Maurice Dirac
(1902 – 1984)

32. Paul Adrien Maurice Dirac
2.B.2
Dirac notation
Some properties:
Paul Adrien Maurice Dirac
(1902 – 1984)

33. Linear operators Linear operator A is defined as:
2.B.3
Linear operators
Linear operator A is defined as:
Product of operators:
In general:
Commutator:
Matrix element of operator A:

34. Linear operators Example: What is ?
2.B.3
Linear operators
Example:
What is ?
It is an operator – it converts one ket into another

35. Linear operators Example: Let us assume that Projector operator
2.B.3
Linear operators
Example:
Let us assume that
Projector operator
It projects one ket onto another

36. Linear operators Example: Let us assume that
2.B.3
Linear operators
Example:
Let us assume that
These kets span space Eq, a subspace of E
Subspace projector operator
It projects a ket onto a subspace of kets

37. Linear operators Recall matrix element of a linear operator A:
2.B.4
Linear operators
Recall matrix element of a linear operator A:
Since a scalar product depends linearly on the ket, the matrix element depends linearly on the ket
Thus for a given bra and a given operator we can associate a number that will depend linearly on the ket
So there is a new linear functional on the kets in space E, i.e., a bra in space of E*, which we will denote
Therefore

38. Linear operators Operator A associates with a given bra a new bra
Let’s show that this correspondence is linear

39. Linear operators For each ket there is a bra associated with it
Hermitian conjugate (adjoint) operator:
This operator is linear (can be shown)
Charles Hermite
(1822 – 1901)

40. 2.B.4
Linear operators
Some properties:
Charles Hermite
(1822 – 1901)

41. Hermitian conjugation
2.B.4
Hermitian conjugation
To obtain Hermitian conjugation of an expression:
Replace constants with their complex conjugates
Replace operators with their Hermitian conjugates
Replace kets with bras
Replace bras with kets
Reverse order of factors
Charles Hermite
(1822 – 1901)

42. Hermitian operators For a Hermitian operator:
2.B.4
Hermitian operators
For a Hermitian operator:
Hermitian operators play a fundamental role in quantum mechanics (we’ll see later)
E.g., projector operator is Hermitian:
If:
Charles Hermite
(1822 – 1901)

43. Representations in state space
In a certain basis, vectors and operators are represented by numbers (components and matrix elements)
Thus vector calculus becomes matrix calculus
A choice of a specific representation is dictated by the simplicity of calculations
We will rewrite expressions obtained above for orthonormal bases using Dirac notation

44. Orthonormal bases A countable set of kets is called orthonormal if:
It constitutes a basis if every vector in E can be expanded in one and only one way:

45. 2.C.2
Orthonormal bases
Closure relation
1 – identity operator

46. 2.C.3
Orthonormal bases
For two kets
a scalar product is:

47. Orthonormal bases A set of kets labelled by a continuous index α
is called orthonormal if:
It constitutes a basis if every vector in E can be expanded in one and only one way:

48. 2.C.2
Orthonormal bases
Closure relation
1 – identity operator

49. 2.C.3
Orthonormal bases
For two kets
a scalar product is:

50. Representation of kets and bras
2.C.3
Representation of kets and bras
In a certain basis, a ket is represented by its components
These components could be arranged as a column-vector:

51. Representation of kets and bras
2.C.3
Representation of kets and bras
In a certain basis, a bra is also represented by its components
These components could be arranged as a row-vector:

52. Representation of operators
2.C.4
Representation of operators
In a certain basis, an operator is represented by matrix components:

53. Representation of operators
2.C.4
Representation of operators

54. Representation of operators
2.C.4
Representation of operators

55. Representation of operators
2.C.4
Representation of operators

56. Representation of operators
2.C.4
Representation of operators

57. Representation of operators
2.C.4
Representation of operators
For Hermitian operators:
Diagonal elements of Hermitian operators are always real

58. Change of representations

59. Change of representations

60. 2.D.1
Eigenvalue equations
A ket is called an eigenvector of a linear operator if:
This is called an eigenvalue equation for an operator
This equation has solutions only when λ takes certain values - eigenvalues
If:
then:

61. 2.D.1
Eigenvalue equations
The eigenvalue is called nondegenerate (simple) if the corresponding eigenvector is unique to within a constant
The eigenvalue is called degenerate if there are at least two linearly independent kets corresponding to this eigenvalue
The number of linearly independent eigenvectors corresponding to a certain eigenvalue is called a degree of degeneracy

62. 2.D.1
Eigenvalue equations
If for a certain eigenvalue λ the degree of degeneracy is g:
then every eigenvector of the form
is an eigenvector of the operator A corresponding to the eigenvalue λ for any ci:
The set of linearly independent eigenvectors corresponding to a certain eigenvalue comprises a g-dimensional vector space called an eigensubspace

63. 2.D.1
Eigenvalue equations
Let us assume that the basis is finite-dimensional, with dimensionality N
This is a system of N linear homogenous equations for N coefficients cj
Condition for a non-trivial solution:

64. 2.D.1
Eigenvalue equations
This equation is called the characteristic equation
This is an Nth order equation in and it has N roots – the eigenvalues of the operator
Condition for a non-trivial solution:

65. Eigenvalue equations Let us select λ0 as one of the eigenvalues
2.D.1
Eigenvalue equations
Let us select λ0 as one of the eigenvalues
If λ0 is a simple root of the characteristic equation, then we have a system of N – 1 independent equations for coefficients cj
From linear algebra: the solution of this system (for one of the coefficients fixed) is

66. Eigenvalue equations Let us select λ0 as one of the eigenvalues
2.D.1
Eigenvalue equations
Let us select λ0 as one of the eigenvalues
If λ0 is a multiple (degenrate) root of the characteristic equation, then we have less than N – 1 independent equations for coefficients cj
E.g., if we have N – 1 independent equations then (from linear algebra) the solution of this system is

67. Eigenproblems for Hermitian operators
2.D.2
Eigenproblems for Hermitian operators
For:
Therefore λ is a real number
Also:
If:
Then:
But:

68. 2.D.2
Observables
Consider a Hermitian operator A whose eigenvalues form a discrete spectrum
The degree of degeneracy of a given eigenvalue an will be labelled as gn
In the eigensubspace En we consider gn linearly independent kets: If
Then

69. Observables Inside each eigensubspace Therefore:
If all these eigenkets form a basis in the state space, then operator A is called an observable

70. Observables For an eigensubspace projector
2.D.2
Observables
For an eigensubspace projector
These relations could be generalized for the case of continuous bases
E.g., a projector is an observable

71. Observables If Then If a is non-degenerate then
so this ket is also an eigenvector of B
If a is degenerate then
Thereby, if A and B commute, each eigensubspace of A is globally invariant (stable) under the action of B

72. 2.D.3
Observables If
Then
If two operators commute, there is an orthonormal basis with eigenvectors common to both operators

73. 2.D.3
Observables
A set of observables, commuting by pairs, is called a complete set of commuting observables (CSCO) if there exists a unique orthonormal basis of common eigenvectors
If all the eigenvalues of a certain operator are non-degenerate, this operator constitutes CSCO by itself
If one ore more eigenvalues of a certain operator are degenerate, there is no unique orthonormal basis of eigenvectors
Then at least one more operator commuting with the first one is used to construct a unique orthonormal basis of common eigenvectors, an thus a CSCO

74. Examples of representations
Let us consider a set of functions:
The set is orthonormal:
Kets can be expanded:

75. Examples of representations
Closure relation

76. Examples of representations
For two kets
a scalar product is:

77. Examples of representations
Let us consider a set of functions:
The set is orthonormal:
Kets can be expanded:

78. Examples of representations
Closure relation

79. Examples of representations
For two kets
a scalar product is:

80. Change of representations
Recall:
Choosing
we obtain:

81. R and P operators For we obtain: where Similarly “Vector” operator R:

82. 2.E.2
R and P operators
“Vector” operator P:
Then:

83. 2.E.2
R and P operators
Analogously:
Then:

84. 2.E.2
R and P operators
Calculating a commutator:
Similarly:

85. R and P operators Calculating a matrix element: Similarly:
Position and momentum operators are Hermitian

86. R and P operators Calculating a matrix element: Thus: Similarly:
Since |r > and |p > constitute complete bases, therefore operators R and P are observables
Sets of operators {X,Y,Z} as well as {Px,Py,Pz} comprise a CSCO each, however, separate operators don’t, since they are degenerate (in other directions)

87. Tensor products of state spaces
Spaces of square-integrable functions in 1D, 2D, and 3D are not the same (e.g., Er and Ex are different)
How are those spaces related?
In general, if there are two or more mutually isolated subsystems of a certain system, each of which has its own space, what is the space of the entire system?
Such questions are resolved via introduction of tensor products of spaces

88. Tensor products of state spaces
Let there be two spaces E1 and E2 with dimensions N1 and N2
Tensor product of E1 and E2 is a vector field E with the following properties:
Notation:
If vectors belonging to E1 and E2 are
Then vectors belonging to E are
Tensor product is linear:

89. Tensor products of state spaces
Let there be two spaces E1 and E2 with dimensions N1 and N2
Tensor product of E1 and E2 is a vector field E with the following properties:
Tensor product is distributive:
Tensor product of bases is a basis

90. Tensor products of state spaces
If:
Then:
Components of a tensor product of two vectors are products of the components
Not all the vectors in E can be represented as tensor products of vectors from E1 and E2:

91. Tensor products of state spaces
Scalar product:
For orthonormal bases:
Tensor product of operators:
Projector:

92. Tensor products of state spaces
If:
Then:
And:
Eigenvectors of A(1) + B(2) are tensor products of eigenvectors of A(1) and eigenvectors of B(2)
If there is one CSCO in E1 and another CSCO in E2 their tensor product is a CSCO in E

93. Tensor products of state spaces
If the problem is strictly 1D (e.g. x-dependent), then the state space is Ex
In the x-representation the basis kets:
Similarly we can consider Ey and Ez:
Introducing
we get

94. Tensor products of state spaces
Exyz is the state space of a 3D particle
A ket in 3D can be represented:
In general, ψ cannot be factorized
Operator X in Ex is a CSCO by itself, but in Exyz its eigenvalues would be infinitely degenerate, because Ey and Ez in are infinitely-dimensional
On the other hand, the {X,Y,Z} set is a CSCO

95. States of a two-particle system
For two particles the state space is:
and the basis is:
A ket can be represented:
If:
Then:
In this case there is no correlation between the particles