Let us consider a particle of mass m moving with a velocity v associated with a de-Broglie wave. Let \( \psi \) be the wave function of the matter wave describing the particle.
Let us consider the simple form of a progressive wave:
\[ \psi = \psi_0 \sin(wt - kx) \hspace{5cm} (1) \]
Where \( \psi \) is a function of \( (x,t) \) and \( \psi_0 \) is the amplitude. Differentiating with respect to \( x \) twice:
\[ \frac{\partial \psi}{\partial x} = -k \psi_0 \cos(wt - kx) \]
\[ \frac{\partial^2 \psi}{\partial x^2} = -k^2 \psi_0 \sin(wt - kx) \]
Using Eqn (1),
\[ \frac{\partial^2 \psi}{\partial x^2} = -k^2 \psi \]
\[ \frac{\partial^2 \psi}{\partial x^2} + k^2 \psi = 0 \hspace{5cm} (2) \]
Here \( k \) represents the wave number and \( k = \frac{2\pi}{\lambda} \). Substituting this value in the Eqn (2):
\[ \frac{\partial^2 \psi}{\partial x^2} + \frac{4\pi^2}{\lambda^2} \psi = 0 \]
Substituting the de-Broglie equation, \( \lambda = \frac{h}{mv} \),
\[ \frac{\partial^2 \psi}{\partial x^2} + \frac{4\pi^2 m^2 v^2}{h^2} \psi = 0 \hspace{5cm} (3) \]
If the total energy of the system is \( E \), which is the sum of kinetic energy and potential energy \( V \):
\[ E = \frac{1}{2} mv^2 + V \]
\[ 2(E - V) = mv^2 \]
\[ m^2 v^2 = 2m(E - V) \]
Using the above expression in Eqn (3), we get:
\[ \frac{\partial^2 \psi}{\partial x^2} + \frac{8\pi^2 m(E-V)}{h^2} \psi = 0 \]
Let's introduce a parameter \( \hbar \); \( \quad \hbar = \frac{h}{2\pi} \),
\[ \frac{\partial^2 \psi}{\partial x^2} + \frac{2 m(E-V)}{\hbar^2} \psi = 0 \hspace{5cm} (4)\]
Equation (4) is the time-independent Schrödinger equation in one dimension.
In three dimensions, we get:
\[ \frac{\partial^2 \psi}{\partial x^2} +\frac{\partial^2 \psi}{\partial y^2} +\frac{\partial^2 \psi}{\partial z^2} + \frac{2 m(E-V)}{\hbar^2} \psi = 0 \]
\[ \nabla^2 \psi + \frac{2 m(E-V)}{\hbar^2} \psi = 0 \]
Where \( \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2 }{\partial y^2} + \frac{\partial^2 }{\partial z^2} \), is called the Laplacian operator.