Module 1 : Quantum Mechanics
Chapter 3: Elements of Quantum Mechanics
 
  with $n$ being an integer. But the observation is that the pattern remains unchanged when the intensity of the beam is varied, indeed the same pattern is observed even when the particles appear only one at a time. This of course implies that the wave properties are associated with the particles, and the intensity implies a statistical probability for the particle to be observed at various positions.

For developing general wave equations for particles, one observes that we have the wave equation and wave function for radiation which we can associate with photon particles. For the radiation one has

  \begin{displaymath} \big(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\b... ...vec E=0, \qquad \vec E=\vec E_0\;e^{2\pi i(-\nu t+x/\lambda)}, \end{displaymath} (3.3)
  which follows from Maxwell's equations. For the corresponding photons we have  
  \begin{displaymath} E^2-p^2 c^2=0,\; \Rightarrow \; p=E/c=h\nu/c=h/\lambda, \;\vec E=\vec E_0 e^{i(-Et+px)/\hbar}. \end{displaymath} (3.4)
  We can write  
 
$\displaystyle (E^2-p^2c^2)\vec E_0e^{i(-Et+px)/\hbar}=0$
$\displaystyle \Rightarrow\big[\big(i\hbar\frac{\partial}{\partial t}\big)^2 -c^2\big(-i\hbar\vec \nabla\big)^2\big]\vec E_0e^{i(-Et+px)/\hbar}=0.$
(3.5)
  The equation can be obtained by starting with the particle relation $E^2-p^2c^2=0$ and converting it into a wave equation
by replacing $E\rightarrow i\hbar \partial/\partial t$, $\vec p\rightarrow -i\hbar \vec \nabla$ and operating on the required wave function. Here it may be noted that $(E,\vec pc)$ transform as a Lorentz 4-vector and $(\partial/\partial(ct), -\vec\nabla)$ also transform as a Lorentz 4-vector. Therefore it is reasonable to relate these with appropriate coefficients. We generalise this to other systems. For this, we first obtain a relation for $E$ in terms of $\vec p$ and $\vec r$, and replace $E$ by $i\hbar \partial/\partial t$, $\vec p$ by $-i\hbar \vec \nabla$, and operate on the wave function. For the free particle one gets
  \begin{displaymath} E=\frac{p^2}{2m} \quad \Rightarrow \quad i\hbar\frac{\partial \psi}{\partial t}=\frac{1}{2m}(-i\hbar\vec \nabla)^2\psi, \end{displaymath} (3.6)