[NOTES/QM-06002] Postulates of Quantum Mechanics

1. States of physical system
The state of a quantum mechanical system is represented by a vector in a complex vector space with inner product (Hilbert Space).

A vector that represents state of physical system will be called state vector.
The null vector does not represent any state.
Two non-null vectors represent the same state if and only if they are linearly dependent.
Thus $\ket{\psi_1}$ and $\ket{\psi_2}$ represent the same state if there exists a complex number $c$ such that \begin{equation} \ket{\psi_2} = c\ket{\psi_1} \label{EQ01} \end{equation}
The quantum mechanical states represented by vectors in Hilbert space are called pure states.

Density Matrix: There are other possible states which are called mixed states. These are represented by a density matrix $\rho$. The density matrix is an operator having properties that it is hermitian and that its eigenvalues are between $0$ and $1$. A density matrix $\rho$ corresponds to a pure state if and only if $\rho^2 = \rho$.

2. Dynamical Variables
The dynamical variables of a physical system are represented by linear operators in the vector space.

A linear operator representing a dynamical variable must have real eigenvalues and their eigenvectors must form a complete set. These properties are satisfied by self-adjoint operators (hermitian operators). So we demand that the dynamical variables be represented by self-adjoint operators in Hilbert space. An operator representing a dynamical variable will also be called an observable.

3. Measurement postulate and probabilities

If the system is in a state \(\ket(\psi)\), a measurement of dynamical variable \(A\) will give one of its eigenvalues \(\alpha_k\)  with probability equal to \(|\innerproduct{u_k}{\psi}|^2\), where \(\ket{u_k}\) is the eigenvector of \(\widehat{A}\) corresponding to eigenvalue \(\alpha_k\)

As an example, let $\ket{u_1}, \ket{u_2},\cdots, \ket{u_n}$ represent the eigenvectors of an observable $\hat{A}$. If the state vector of a physical system, $\ket{u_i}$, is an eigenvector of an operator $\hat{A}$ representing a dynamical variable $A$, a measurement of the dynamical variable gives value $\alpha$ with probability 1. Here $\alpha$ is the eigenvalue of $\hat{A}$ corresponding the eigenvector $\ket{u_i}$. Conversely, if the measurement of $A$ gives the value $\alpha$ with probability 1, the state of system will be represented by a vector which will be eigenvector of the operator $\hat{A}$ corresponding to the eigenvalue $\alpha$

In general, state vector \(\ket{\psi}\) will not be an eigenvector of the given dynamical variable. In such a case a measurement of the variable $A$ will results in values $\alpha_1, \alpha_2,\cdots, \alpha_n$ with probabilities $c_1\ket{\alpha_2}, c_1\ket{\alpha_2}, \cdots, c_n\ket{\alpha_n}$ where $c_1, c_2,\cdots, c_n$ are the coefficients in the expansion of the {\sf state vector} $\ket{\psi}$ \begin{equation} \ket{\psi}  = \sum_k c_k \ket{u_k} \label{EQ02} \end{equation} in terms of eigenvectors of $\hat{A}$. Here $\ket{\psi}$ and $\ket{u_k}$ are assumed to be normalized.\begin{equation} \innerproduct{\psi}{\psi}= 1; \qquad \innerproduct{u_k}{u_k} = 1, k=1,2,\ldots \label{EQ03} \end{equation}

4. Commutation Relations
The operators corresponding to the generalized coordinates and momenta $\{q_k, p_k\}$ of a classical system satisfy \begin{eqnarray} \hat{q}_i \hat{q}_j  - \hat{q}_j \hat{q}_i = 0 \label{EQ04}\\ \hat{p}_i \hat{p}_j - \hat{p}_j \hat{p}_i = 0 \label{EQ05}\\ \hat{q}_i \hat{p}_j- \hat{p}_j \hat{q}_i = i\hbar \delta_{ij}\label{EQ06} \end{eqnarray} The above relations are called {\sf canonical commutation relations

5. Equation of motion
The time development of a system is governed by the Schrodinger equation \begin{equation} i \hbar \dd{t} \ket{\psi t} = \hat{H} \ket{\psi t} \end{equation} where $\ket{\psi t}$ is the state vector of the system at time $t$ and $\hat{H}$ is the operator representing the Hamiltonian of the system. }

6. Symmetrization postulate for identical particles

For a system of identical particles, the state of the system remains unchanged under exchange of a pair of particles, it should be either symmetric or antisymmetric under an exchange of all the variables of the two identical particles.