Time independent Schrodinger wave equation:
We know the time dependent Schrodinger wave equation:
$i \hbar \frac{\partial \psi(x,t)}{\partial t}= -\frac{\hbar^{2}}{2m} \frac{\partial^{2} \psi(x,t)}{\partial x^{2}}+ V(x) \psi(x,t) \qquad(1)$
The wave function $\psi(x,t)$ is the product of space function $\psi(x)$ and time function $\psi(t)$. So
$\psi(x,t)=\psi(x) \psi(t) \qquad (2)$
Now apply the wave function form of equation$(2)$ to time dependent Schrodinger wave equation $(1)$
$i \hbar \psi(x) \frac{d \psi(t)}{d t}= -\frac{\hbar^{2}}{2m} \psi(t) \frac{d^{2} \psi(x)}{d x^{2}}+ V(x) \psi(x) \psi(t) \qquad(3)$
In the above equation $(3)$ ordinary derivatives is used in place of partial derivatives because each of function $\psi(x)$ and $\psi(t)$ depends on only one variable.
Now divide the above equation $(3)$ by $\psi(x)\psi(t)$ so
$i \hbar \frac{1}{\psi(t)} \frac{d \psi(t)}{d t}= -\frac{\hbar^{2}}{2m} \frac{1}{\psi(x)} \frac{d^{2} \psi(x)}{d x^{2}}+ V(x) \qquad(4)$
The above equation is known as the separation of time-independent part and time-independent part of the wave equation. The time-independent part is known as the energy function operator. i.e
$E=i \hbar \frac{1}{\psi(t)} \frac{d \psi(t)}{d t} \qquad(5)$
So from equation $(4)$ and equation$(5)$
$E= -\frac{\hbar^{2}}{2m} \frac{1}{\psi(x)} \frac{d^{2} \psi(x)}{d x^{2}}+ V(x)$
$E \psi(x)= -\frac{\hbar^{2}}{2m} \frac{d^{2} \psi(x)}{d x^{2}}+ V(x) \psi(x)$
$\frac{d^{2} \psi(x)}{d x^{2}}+\frac{2m}{\hbar^{2}}(E-V)\psi(x)=0$
This is time-independent Schrodinger wave equation.
Now for a free particle i.e, there is no force acting on the particle then the potential energy of a particle will be zero i.e. $V(x)=0$. Therefore time independent Schrodinger equation can be written as:
$\frac{d^{2} \psi(x)}{d x^{2}}+\frac{2mE}{\hbar^{2}}\psi(x)=0$
This is time-independent Schrodinger wave equation for a free particle.
Angle of Acceptance → If incident angle of light on the core for which the incident angle on the core-cladding interface equals the critical angle then incident angle of light on the core is called the "Angle of Acceptance. Transmission of light when the incident angle is equal to the acceptance angle If the incident angle is greater than the acceptance angle i.e. $\theta_{i}>\theta_{0}$ then the angle of incidence on the core-cladding interface will be less than the critical angle due to which part of the incident light is transmitted into cladding as shown in the figure below Transmission of light when the incident angle is greater than the acceptance angle If the incident angle is less than the acceptance angle i.e. $\theta_{i}<\theta_{0}$ then the angle of incidence on the core-cladding interface will be greater than the critical angle for which total internal reflection takes place inside the core. As shown in the figure below Transmission of lig
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