Quantum Theory can be formulated according to a few postulates (i.e., theoretical principles based on experimental observations). The goal of this section is to introduce such principles, together with some mathematical concepts that are necessary for that purpose.R1(190) To keep the notation as simple as possible, expressions are written for a 1-dimensional system. The generalization to many dimensions is usually straightforward.
: Any system can be described by a wave function , where is a parameter representing the time and represents the coordinates of the system. Function must be continuous, single valued and square integrable.R1(57)
Note 1: As a consequence of Postulate 4, we will see that represents the probability of finding the system between and at time .
: Any observable (i.e., any measurable property of the system) can be described by an operator. The operator must be linear and hermitian.
What is an operator ? What is a linear operator ? What is a hermitian operator?
Definition 1: An operator is a mathematical entity that transforms a function into another function as follows,R4(96)
Definition 2: R1(190) An operator that represents an observable is obtained by first writing the classical expression of such observable in Cartesian coordinates (e.g., ) and then substituting the coordinate in such expression by the coordinate operator as well as the momentum by the momentum operator .
Definition 3: An operator is linear if and only if (iff),
Definition 4: An operator is hermitian iff, R1(164)
Definition 5: A function is an eigenfunction of iff,
Property 1: The eigenvalues of a hermitian operator are real.R1(166)(167)
Proof: Using Definition 4, we obtain
Property 2: Different eigenfunctions of a hermitian operator (i.e., eigenfunctions with different eigenvalues) are orthogonal (i.e., the scalar product of two different eigenfunctions is equal to zero). Mathematically, if , and , with , then .
:The only possible experimental results of a measurement of an observable are the eigenvalues of the operator that corresponds to such observable.
: The average value of many measurements of an observable , when the system is described by function , is equal to the expectation value , which is defined as follows,
: R1(191), R5(15)), R4(97) here
The eigenfunctions of a linear and hermitian operator form a complete basis set. Therefore, any function that is continuous, single valued, and square integrable can be expanded as a linear combination of eigenfunctions of a linear and hermitian operator as follows,
Exercise 1: Show that , when
According to the Expansion Postulate (together with Postulate 1), the state of a system described by the function can be expanded as a linear combination of eigenfunctions of a linear and hermitian operator (e.g., ). Usually, the space defined by these eigenfunctions (i.e., functions that are continuous, single valued and square integrable) has an infinite number of dimensions. Such space is called Hilbert-Space in honor to the mathematician Hilbert who did pioneer work in spaces of infinite dimensionality.R4(94)
A representation of in such space of functions corresponds to a vector-function,
where and are the projections of along and , respectively. All other components are omitted from the representation because they are orthogonal to the ``plane'' defined by and .
Certain operators have a continuous spectrum of eigenvalues. For example, the coordinate operator is one such operator since it satisfies the equation = , where the eigenvalues define a continuum. Delta functions define a continuum representation and, therefore, an expansion of in such representation becomes,
In general, when the basis functions are not necessarily delta functions but nonetheless define a continuum representation,
Note 2: According to the Expansion Postulate, a function is uniquely and completely defined by the coefficients , associated with its expansion in a complete set of eigenfunctions . However, the coefficients of such expansion would be different if the same basis functions depended on different coordinates (e.g., with ). In order to eliminate such ambiguity in the description it is necessary to introduce the concept of vector-ket space.R4(108)
The vector-ket space is introduced to represent states in a convenient space of vectors , instead of working in the space of functions . The main difference is that the coordinate dependence does not need to be specified when working in the vector-ket space. According to such representation, function is the component of vector associated with index (vide infra) . Therefore, for any function we can define a ket-vector such that,
Note that the expansion coefficients depend only on the kets and not on any specific vector component. Therefore, the ambiguity mentioned above is removed.
In order to learn how to operate with kets we need to introduce the bra space and the concept of linear functional. After doing so, this section will be concluded with the description of Postulate 5, and the Continuity Equation.
A functional is a mathematical operation that transforms a function into a number. This concept is extended to the vector-ket space , as an operation that transforms a vector-ket into a number as follows,
where is a number. A linear functional satisfies the following equation,
where and are constants.
Example: The scalar product,R4(110)
For every ket we define a linear functional , called bra-vector, as follows:
Note that functional is linear because the scalar product is a linear functional. Therefore,
Note: For convenience, we will omit parenthesis so that the notation will be equivalent to . Furthermore, whenever we find two bars next to each other we can merge them into a single one without changing the meaning of the expression. Therefore,
where is the projection of along .
Projection Operator and Closure Relation
Given a ket in a certain basis set ,
Note that is an operator that transforms any vector into a vector pointing in the direction of with magnitude . The operator is called the Projection Operator. It projects according to,
Note that , where . This is true simply because .
: The evolution of
in time is described by the time
dependent Schrodinger equation:
In general, for higher dimensional problems, the change in time of probability density,
, is equal to minus the divergence of the probability flux ,
Note: Remember that given a vector field , e.g.,
, the divergence of is defined as the dot product of the ``del'' operator
and vector as follows:
Exercise 2: Prove that
In general, for higher dimensional problems, the change in time of probability density, , is equal to minus the divergence of the probability flux ,
Note: Remember that given a vector field , e.g., , the divergence of is defined as the dot product of the ``del'' operator and vector as follows: