Postulates of Quantum Theory

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.

$\boxed{Postulate \ 1}$: Any system can be described by a wave function $\psi(t, x)$, where $t$ is a parameter representing the time and $x$ represents the coordinates of the system. Function $\psi(t, x)$ must be continuous, single valued and square integrable.R1(57)

Note 1: As a consequence of Postulate 4, we will see that $P(t,x) = \psi^*(t,x) \psi(t,x) \it dx$ represents the probability of finding the system between $x$ and $x + dx$ at time $t$.

$\boxed{Postulate \ 2}$: 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 $\hat{O}$ is a mathematical entity that transforms a function $f(x)$ into another function $g(x)$ as follows,R4(96)

$$\hat{O} \it f(x)=g(x), \qquad \normalfont$$

where $f$ and $g$ are functions of $x$.

Definition 2: R1(190) An operator $\hat{O}$ that represents an observable $O$ is obtained by first writing the classical expression of such observable in Cartesian coordinates (e.g., $O=O(x,p)$) and then substituting the coordinate $x$ in such expression by the coordinate operator $\hat{x}$ as well as the momentum $p$ by the momentum operator $\hat{p}=-i\hbar \partial/\partial{x}$.

Definition 3: An operator $\hat{O}$ is linear if and only if (iff),

$$\hat{O}( a f(x) + b g(x) ) = a \hat{O} f(x) + b \hat{O} g(x),$$

where a and b are constants.

Definition 4: An operator $\hat{O}$ is hermitian iff,         R1(164)

$$\int{dx\phi_n^*(x) \hat{O} \psi_m(x)} = \left [\int{dx\psi_m^*(x) \hat{O} \phi_n(x)} \right ]^*,$$

where the asterisk represents the complex conjugate of the expression embraced by brackets.

Definition 5: A function $\phi_n(x)$ is an eigenfunction of $\hat{O}$ iff,

$$\hat{O} \phi_n(x) =O_n \phi_n(x),$$

where $O_n$ is a number called eigenvalue.

Property 1: The eigenvalues of a hermitian operator are real.R1(166)(167)

Proof: Using Definition 4, we obtain

$$\int{dx\phi_n^*(x) \hat{O} \phi_n(x)} - \left[\int{dx\phi_n^*(x) \hat{O} \phi_n(x)}\right]^* = 0,$$

therefore,

$$\left[ O_n - O_n^* \right] \int{dx \phi_n(x)^* \phi_n(x)} = 0.$$

Since $\phi_n(x)$ are square integrable functions, then,

$$O_n = O_n^*.$$

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 $\hat{O}\phi_n = O_n\phi_n$, and $\hat{O}\phi_m = O_m\phi_m$, with $O_n \neq O_m$, then $\int{dx\phi_n^*\phi_m} = 0$.

Proof:

$$\int{dx\phi_m^*\hat{O}\phi_n} - \left[\int{dx\phi_n^*\hat{O}\phi_m} \right]^* = 0,$$

and

$$\left[O_n - O_m\right]\int{dx\phi_m^*\phi_n} = 0.$$

Since $O_n \neq O_m$, then $\int{dx\phi_m^*\phi_n} = 0$.

$\boxed{Postulate \ 3}$:The only possible experimental results of a measurement of an observable are the eigenvalues of the operator that corresponds to such observable.

$\boxed{Postulate \ 4}$: The average value of many measurements of an observable $O$, when the system is described by function $\psi(x)$, is equal to the expectation value $\bar{O}$, which is defined as follows,

$$\qquad \bar{O}=\frac{\int{dx \psi(x)^* \hat{O} \psi(x)}}{\int{dx \psi(x)^* \psi(x)}}. \qquad \normalfont$$

$\boxed{Expansion \ Postulate}$: R1(191), R5(15)), R4(97) here

The eigenfunctions of a linear and hermitian operator form a complete basis set. Therefore, any function $\psi(x)$ that is continuous, single valued, and square integrable can be expanded as a linear combination of eigenfunctions $\phi_n(x)$ of a linear and hermitian operator $\hat{A}$ as follows,

$$\psi (x) = \sum_j{C_j \phi_j (x) },$$

where $C_j$ are numbers (e.g., complex numbers) called expansion coefficients.

Exercise 1: Show that $\bar{A} = \sum_j {C_j C_j^* a_j}$, when $\psi (x) = \sum_j{C_j \phi_j (x) },$

$$\hat{A} \phi_j (x) = a_j \phi_j (x),$$   and$$\qquad \int{dx {\phi_j (x)}^* \phi_k (x)} = \delta_{jk}.$$

Note that (according to Postulate 3) eigenvalues $a_j$ are the only possible experimental results of measurements of $\hat{A}$, and that (according to Postulate 4) the expectation value $\bar{A}$ is the average value of many measurements of $\hat{A}$ when the system is described by the expansion $\psi (x) = \sum_j{C_j \phi_j (x) }$. Therefore, the product $C_j C_j^*$ can be interpreted as the probability weight associated with eigenvalue $a_j$ (i.e., the probability that the outcome of an observation of $\hat{A}$ will be $a_j$).

Hilbert-Space

According to the Expansion Postulate (together with Postulate 1), the state of a system described by the function $\Psi(x)$ can be expanded as a linear combination of eigenfunctions $\phi_j(x)$ of a linear and hermitian operator (e.g., $\Psi(x) = C_1 \phi_1(x) + C_2 \phi_2(x) + \ldots$). 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 $\Psi(x)$ in such space of functions corresponds to a vector-function,

where $C_1$ and $C_2$ are the projections of $\Psi(x)$ along $\phi_1(x)$ and $\phi_2(x)$, respectively. All other components are omitted from the representation because they are orthogonal to the ``plane'' defined by $\phi_1(x)$ and $\phi_2(x)$.

Continuous Representation

Certain operators have a continuous spectrum of eigenvalues. For example, the coordinate operator is one such operator since it satisfies the equation $\hat{x}$ $\delta (x_0 -x)$ = $x_0$ $\delta (x_0 -x)$, where the eigenvalues $x_0$ define a continuum. Delta functions $\delta (x_0 -x)$ define a continuum representation and, therefore, an expansion of $\psi(x)$ in such representation becomes,

$$\psi(x) = \int{d x_0 C_{x_0} \delta (x_0 - x)},$$

where $C_{x_0}=\psi(x_0)$, since

$$\int{dx \delta(x -\beta)\psi (x)}=\int{dx\int{d\alpha C_\alpha\delta (x-\beta) \delta (\alpha-x)}}=\psi(\beta).$$

According to postulates 3 and 4 (see Exercise 1), the probability of observing the system with coordinate eigenvalues between $x_0$ and $x_0 + d x_0$ is $P(x_0) = C_{x_0} C_{x_0}^* d x_0 = \psi(x_0) \psi(x_0)^* d x_0$ (see Note 1).

In general, when the basis functions $\phi (\alpha, x)$ are not necessarily delta functions but nonetheless define a continuum representation,

$$\psi(x) = \int{d \alpha C_\alpha \phi (\alpha, x)},$$

with $C_{\alpha} = \int{dx \phi(\alpha, x)^* \psi(x)}$.

Note 2: According to the Expansion Postulate, a function $\psi(x)$ is uniquely and completely defined by the coefficients $C_j$, associated with its expansion in a complete set of eigenfunctions $\phi_j(x)$. However, the coefficients of such expansion would be different if the same basis functions $\phi_j$ depended on different coordinates (e.g., $\phi_j(x')$ with $x' \neq x$). In order to eliminate such ambiguity in the description it is necessary to introduce the concept of vector-ket space.R4(108)

Vector-Ket Space $\varepsilon$

The vector-ket space is introduced to represent states in a convenient space of vectors $\vert\phi_j>$, instead of working in the space of functions $\phi_j(x)$. 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 $\psi(x)$ is the component of vector $\vert\psi >$ associated with index $x$ (vide infra) . Therefore, for any function $\psi (x) = \sum_j{C_j \phi_j (x) },$ we can define a ket-vector $\vert\psi >$ such that,

$$\vert\psi > = \sum_j{C_j \vert \phi_j>}.$$

The representation of $\mid \psi >$ in space $\varepsilon$ is,

Note that the expansion coefficients $C_j$ depend only on the kets $\mid \psi_j >$ 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.

Linear functionals

A functional $\chi$ is a mathematical operation that transforms a function $\psi(x)$ into a number. This concept is extended to the vector-ket space $\varepsilon$, as an operation that transforms a vector-ket into a number as follows,

$\chi(\psi(x)) = n$, or $\chi(\vert\psi>) = n,$

where $n$ is a number. A linear functional satisfies the following equation,

$\chi(a \psi(x) + b f(x))= a \chi(\psi(x)) + b \chi(f(x))$,

where $a$ and $b$ are constants.

Example: The scalar product,R4(110)

$$n= \int{dx \psi^*(x) \phi(x)},$$

is an example of a linear functional, since such an operation transforms a function $\phi(x)$ into a number $n$. In order to introduce the scalar product of kets, we need to introduce the bra-space.

Bra Space $\varepsilon^*$

For every ket $\vert\psi >$ we define a linear functional $<\psi\vert$, called bra-vector, as follows:

$<\psi\vert(\vert\phi>) = \int{dx \psi^*(x) \phi(x)}.$

Note that functional $<\psi\vert$ is linear because the scalar product is a linear functional. Therefore,

$$<\psi\vert(a\vert\phi> + b\vert f>) = a<\psi\vert(\vert\phi>) + b<\psi\vert(\vert f>).$$

Note: For convenience, we will omit parenthesis so that the notation $<\psi\vert(\vert\phi>)$ will be equivalent to $<\psi\vert\vert\phi>$. 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,

$$<\psi\vert\vert\phi>=<\psi\vert\phi>.$$

The space of bra-vectors is called dual space $\varepsilon^*$ simply because given a ket $\vert\psi> = \sum_j C_j \vert\phi_j>$, the corresponding bra-vector is $<\psi\vert = \sum_j C_j^*<\phi_j\vert$. In analogy to the ket-space, a bra-vector $<\psi\vert$ is represented in space $\varepsilon^*$ according to the following diagram:

where $C_j^*$ is the projection of $< \psi \mid$ along $< \phi_j \mid$.

Projection Operator and Closure Relation

Given a ket $\mid \psi >$ in a certain basis set $\vert\phi_j>$,

$$\vert\psi> = \sum_j C_j\vert\phi_j>,$$ (1)

where $<\phi_k\vert\phi_j>=\delta_{kj}$,

$$C_j=<\phi_j\vert\psi>.$$ (2)

Substituting Eq. (2) into Eq.(1), we obtain

$$\vert\psi> = \sum_j \vert \phi_j> <\phi_j \vert\psi>.$$ (3)

From Eq.(3), it is obvious that

$$\sum_j \vert \phi_j><\phi_j\vert = \hat{1}, \qquad Closure \ Relation \normalfont$$

where $\hat{1}$ is the identity operator that transforms any ket, or function, into itself.

Note that $\hat{P}_j = \vert\phi_j><\phi_j\vert$ is an operator that transforms any vector $\vert\psi >$ into a vector pointing in the direction of $\vert\phi_j>$ with magnitude $<\phi_j\vert\psi>$. The operator $\hat{P}_j$ is called the Projection Operator. It projects $\vert\phi_j>$ according to,

$$\hat{P}_j \vert \psi > = <\phi_j \vert \psi> \vert \phi_j>.$$

Note that $\hat{P}_j^2 = \hat{P}_j$, where $\hat{P}_j^2 = \hat{P}_j \hat{P}_j$. This is true simply because $<\phi_j\vert\phi_j>=1$.

$\boxed{Postulate \ 5}$: The evolution of $\psi(x, t)$ in time is described by the time dependent Schrodinger equation:

$$i\hbar \frac{\partial{\psi(x,t)} }{ \partial{t}} = \hat{H} \psi (x, t),$$

where $\hat{H}=-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial{x}^2} + \hat{V}(x)$, is the operator associated with the total energy of the system, $E=\frac{p^2}{2m} + V(x)$.

Continuity Equation

Exercise 2: Prove that

$$\frac{\partial{(\psi^*(x,t) \psi (x,t))}}{\partial{t}} + \frac{\partial}{\partial{x}}j(x,t) = 0,$$

where

$$j(x,t)=\frac{\hbar}{2mi}\left(\psi^*(x,t) \frac{\partial{\psi (x,t)}}{\partial{x}}- \psi(x,t) \frac{\partial{\psi^*(x,t)}}{\partial{x}}\right).$$

In general, for higher dimensional problems, the change in time of probability density, $\rho({\bf x},t)=\psi^*({\bf x},t) \psi({\bf x},t)$, is equal to minus the divergence of the probability flux ${\bf j}$,

$$\frac{\partial \rho({\bf x}, t)}{\partial t} = - \nabla \cdot {\bf j}.$$

This is the so-called Continuity Equation.

Note: Remember that given a vector field ${\bf j}$, e.g., ${\bf j}(x, y, z)=j_1(x, y, z)\hat{i} +j_2(x,y,z) \hat{j} +j_3(x,y,z) \hat{k}$, the divergence of ${\bf j}$ is defined as the dot product of the ``del'' operator $\nabla=(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})$ and vector ${\bf j}$ as follows:

$$\nabla \cdot {\bf j} = \frac{\partial j_1}{\partial x} + \frac{\partial j_2}{\partial y} + \frac{\partial j_3}{\partial z}.$$