1. Chapter 3
Formalism

2. Hilbert space Let’s recall for Cartesian 3D space:
3.1
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. 3.1
Hilbert space

4. Hilbert space Hilbert space:
3.1
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:
3.1
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:
3.1
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: L 2, set of square-integrable functions defined on the whole interval
David Hilbert
(1862 – 1943)

7. Wave function space Recall:
3.1
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 L 2
For two complex numbers λ1 and λ2 it can be shown that if

8. Scalar product In F the scalar product is defined as:
3.1
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
3.1
Scalar product
Schwarz inequality
Karl Hermann Amandus Schwarz
(1843 – 1921)

10. 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:

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

12. Orthonormal bases
This means that
Closure relation

13. 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:

14. Orthonormal bases
For two functions
a scalar product is:

15. Orthonormal bases
This means that
Closure relation

16. Example of an orthonormal basis
Let us consider a set of functions:
The set is orthonormal:
Functions in F can be expanded:

17. Example of an orthonormal basis
For two functions
a scalar product is:

18. Example of an orthonormal basis
This means that
Closure relation

19. 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

20. 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 L 2, Er is a subspace of the Hilbert space

21. Paul Adrien Maurice Dirac
3.6
Dirac notation
Bracket = “bra” x “ket”
< > = < | > = “< |” x “| >”
Paul Adrien Maurice Dirac
(1902 – 1984)

22. Paul Adrien Maurice Dirac
3.6
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)

23. Paul Adrien Maurice Dirac
3.6
Dirac notation
With each pair of 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)

24. Paul Adrien Maurice Dirac
3.6
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)

25. Paul Adrien Maurice Dirac
3.6
Dirac notation
Some properties:
Paul Adrien Maurice Dirac
(1902 – 1984)

26. Linear operators Linear operator A is defined as:
Product of operators:
In general:
Commutator:
Matrix element of operator A:

27. Linear operators Example: What is ?
It is an operator – it converts one ket into another

28. Linear operators Example: Let us assume that Projector operator
It projects one ket onto another

29. 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

30. 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

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

32. 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)

33. Linear operators
Some properties:
Charles Hermite
(1822 – 1901)

34. 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)

35. Hermitian operators For a Hermitian operator:
3.2
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)

36. 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

37. 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:

38. Orthonormal bases
Closure relation
1 – identity operator

39. Orthonormal bases
For two kets
a scalar product is:

40. 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:

41. Orthonormal bases
Closure relation
1 – identity operator

42. Orthonormal bases
For two kets
a scalar product is:

43. 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:

44. 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:

45. Representation of operators
In a certain basis, an operator is represented by matrix components:

46. Representation of operators

47. Representation of operators

48. Representation of operators

49. Representation of operators

50. Representation of operators
For Hermitian operators:
Diagonal elements of Hermitian operators are always real

51. Change of representations
How do representations change when we go from one basis to another?
Let’s denote
Some properties:

52. Change of representations

53. Change of representations

54. 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:

55. 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

56. 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

57. 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:

58. 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:

59. 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

60. 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 – 2 independent equations then (from linear algebra) the solution of this system is

61. Eigenproblems for Hermitian operators
3.2
Eigenproblems for Hermitian operators
For:
Therefore λ is a real number
Also:
If:
Then:
But:

62. 3.2
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

63. Observables Inside each eigensubspace Therefore:
3.2
Observables
Inside each eigensubspace
Therefore:
If all these eigenkets form a basis in the state space, then operator A is called an observable

64. Observables For an eigensubspace projector
3.2
Observables
For an eigensubspace projector
These relations could be generalized for the case of continuous bases
E.g., a projector is an observable

65. Observables If Then If a is non-degenerate then
3.2
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

66. 3.2
Observables If
Then
If two operators commute, there is an orthonormal basis with eigenvectors common to both operators

67. 3.4
Questions QM answers
1) How is the state of a system described mathematically? (In CM – via generalized coordinates and momenta)
2) For a given state, how can one predict results of measurements of various physical quantities? (In CM – unambiguously, via the calculated trajectory in a phase space)
3) For a given state of the system known at time t0, how can one find a state of this system at an arbitrary time t? (In CM – using Hamilton’s equations)
Answers to these questions are given by the postulates of QM

68. State of a system
1st postulate: At certain time t0 a state of this system is defined by a ket belonging to the state space E

69. Physical quantities
2nd postulate: Every measurable physical quantity is described by an observable operator acting in E

70. Measurement
3rd postulate: Measurements of a physical quantity result only in (real) eigenvalues of a corresponding observable

71. Measurement
3rd postulate: Measurements of a physical quantity result only in (real) eigenvalues of a corresponding observable
It is not obvious a priori whether the spectrum of the measured quantity is continuous or discrete (e.g., a system consisting of a proton and an electron)

72. Spectral decomposition
3.4
Spectral decomposition If
Then the state of the system
4th postulate: The probability of measuring an eigenvalue an of an observable A in a certain state of the system is:

73. Spectral decomposition
3.4
Spectral decomposition

74. Spectral decomposition
3.4
Spectral decomposition
The mean value of an observable:

75. Spectral decomposition
3.4
Spectral decomposition If
Then the state of the system
4th postulate: The probability of measuring an eigenvalue of an observable A between α and α+dα in a certain state of the system is:
ρ – probability density

76. Spectral decomposition
3.4
Spectral decomposition
The mean value of an observable:

77. 3.2
RMS deviation
How can one quantify the dispersion of the measurements around the mean value?
Averaging a deviation from the average is not adequate:
Instead, the RMS deviation is used:

78. 3.2
RMS deviation
How can one quantify the dispersion of the measurements around the mean value?
Averaging a deviation from the average is not adequate:
Instead, the RMS deviation is used:

79. Reduction via measurement
When the measurement is performed only one possible result is obtained
Then the state of the system after the measurement of an eigenvalue is:
We can write this as:

80. Reduction via measurement
5th postulate: If measurement of a physical quantity in a given state of the system yields a certain eigenvalue, the state of the system immediately after the measurement is the normalized projection of the initial state onto a state associated with that eigenvalue
The state of the system after the measurement is the eigenvector corresponding to that eigenvlaue

81. Reduction via measurement
We shall consider only ideal measurements
This means that the perturbations the measurement devices produce are only due to the quantum-mechanical aspect of measurement
We will consider the studied system and the measurement device together as a whole

82. Time evolution of the system
6th postulate: The time evolution of the state vector of the system is determined by the Schrödinger equation:
H – is the Hamiltonian operator, observable associated with the total energy of the system
Sir William Rowan
Hamilton
(1805 – 1865)

83. Time evolution of the system
3.5
Time evolution of the system
How does the mean value of an observable evolve?
Recall the CM result:

84. Compatibility of observables
3.5
Compatibility of observables
If two (observable) operators commute, there exists a basis common to both operators
There is at least one state that will simultaneously yield specific eigenvalues for these two operators, thereby these two observable can be measured simultaneously
Such operators are called compatible with each other
If, on the other hand, the operators do not commute, a state cannot in general be an eigenvector of both observables, thus these operators are called incompatible

85. Compatibility of observables
3.5
Compatibility of observables
When two observables are compatible, the measurement of the second does not produce any loss of the information obtained from the measurement of the first
When two observables are incompatible, the measurement of the second does produces a loss of the information obtained from the measurement of the first

86. The uncertainty principle
3.5
The uncertainty principle
Recall Schwarz inequality:
In Dirac’s notation:
Since
Then:

87. The uncertainty principle
3.5
The uncertainty principle
Let us calculate:
Similarly:
On the other hand:

88. The uncertainty principle
3.5
The uncertainty principle
Synopsizing:
Hence:
This is the generalized uncertainty principle
Recall:
Then:

89. The uncertainty principle
3.5
The uncertainty principle
Synopsizing:
Hence: If
And operator Q doesn’t depend on time explicitly
Then:

90. The uncertainty principle
3.5
The uncertainty principle
Recall:
Hence:
Then:

91. The uncertainty principle
3.5
The uncertainty principle
Introducing Δt as the time it takes the expectation value of Q to change by one standard deviation:
Then: