Postulates of Quantum Mechanics

1. The state of a quantum mechanical system is completely specified by the wavefunction $\Psi({\bf r}, t)$.
2. To every observable in classical mechanics, there corresponds a linear, Hermitian operator in quantum mechanics. For example, in coordinate space, the momentum operator ${\hat P}_x$ corresponding to momentum $p_x$ in the $x$ direction for a single particle is $-i \hbar \frac{\partial}{\partial x}$.
3. In any measurement of the observable associated with operator ${\hat A}$, the only values that will ever be observed are the eigenvalues $a$ which satisfy ${\hat A} \Psi = a \Psi$. Although measurements must always yield an eigenvalue, the state does not originally have to be in an eigenstate of ${\hat A}$. An arbitrary state can be expanded in the complete set of eigenvectors of ${\hat A}$ ( ${\hat A} \psi_i = a_i \psi_i$) as $\Psi = \sum_i c_i \psi_i$, where the sum can run to infinity in principle. The probability of observing eigenvalue $a_i$ is given by $c_i^* c_i$.
4. The average value of the observable corresponding to operator ${\hat A}$ is given by
   

   $$\langle A \rangle = \frac{\int_{-\infty}^{\infty} \Psi^* ... ...} \Psi d \tau} {\int_{-\infty}^{\infty} \Psi^* \Psi d \tau}.$$ (3)

   
   

5. The wavefunction evolves in time according to the time-dependent Schrödinger equation
   

   $${\hat H} \Psi({\bf r}, t) = i \hbar \frac{\partial \Psi}{\partial t}$$ (4)

   
   

6. The total wavefunction must be antisymmetric with respect to the interchange of all coordinates of one fermion with those of another. Electronic spin must be included in this set of coordinates. The Pauli exclusion principle is a direct result of this antisymmetry principle.