Operator →
An operator is defined as a mathematical term that is used in the operation of a function so that this function may or may not be transformed into another function.
Operators of Quantum Mechanics →
There are the following quantum mechanical operators which are used in the wave function of particles:-
Momentum Operator
Kinetic Energy Operator
Total Energy Operator (Hamiltonian Operator)
Total Energy Operator in terms of the differential with respect to time
Momentum Operator →
The wave function for a free particle moving along the position $x$-direction is
$\psi(x,t)=A e^{\frac{i}{\hbar}(P_{x}x-Et)}$
Differentiate the above equation with respect to $x$ then we get
$\frac{\partial \psi}{\partial x}= A e^{\frac{i}{\hbar}(P_{x}x-Et)} \frac{i}{\hbar} P_{x} $
$\frac{\partial \psi}{\partial x}= \psi \frac{i}{\hbar} P_{x} $
$ P_{x} \psi = \frac{\hbar}{i}\frac{\partial \psi}{\partial x}$
$ P_{x} = \frac{\hbar}{i}\frac{\partial}{\partial x}$
For three dimensional:-
$\overrightarrow{P}= \frac{\hbar}{i} \overrightarrow{\nabla}$
Kinetic Energy Operator →
We know that the momentum operator
$ P_{x} \psi = \frac{\hbar}{i}\frac{\partial \psi}{\partial x} \qquad(1)$
Differentiate the above equation $(1)$ with respect to $x$ then we get
$ P_{x} \frac{\partial \psi}{\partial x} = \frac{\hbar}{i}\frac{\partial^{2} \psi}{\partial x^{2}} \qquad(2)$
Now substitute the value of $\frac{\partial \psi}{\partial x}$ from equation $(1)$ to equation $(2)$
$\frac{\hbar}{i}\frac{\partial^{2} \psi}{\partial x^{2}} = P_{x} \frac{i}{\hbar} P_{x} \psi$
$\frac{\hbar^{2}}{i^{2}}\frac{\partial^{2} \psi}{\partial x^{2}} = P_{x}^{2} \psi$
$ -\hbar^{2}\frac{\partial^{2} \psi}{\partial x^{2}} = P_{x}^{2} \psi \qquad (\because i^{2}=-1)$
$ -\frac{\hbar^{2}}{2m}\frac{\partial^{2} \psi}{\partial x^{2}} = \frac{P_{x}^{2}}{2m} \psi \qquad {3}$
$ -\frac{\hbar^{2}}{2m}\frac{\partial^{2} \psi}{\partial x^{2}} = K \psi \qquad (\because \frac{P_{x}^{2}}{2m} = K)$
$ K \psi = -\frac{\hbar^{2}}{2m}\frac{\partial^{2} \psi}{\partial x^{2}} $
$ K = -\frac{\hbar^{2}}{2m}\frac{\partial^{2} }{\partial x^{2}} $
For three dimensions:-
$ K = -\frac{\hbar^{2}}{2m}\nabla^{2} $
Total Energy Operator (Hamiltonian Operator) →
The total energy of the particle moving along $x4-aix is given by
$E=\frac{P_{x}^{2}}{2m} + V(x) \qquad(1)$
Where V(x) → Potential Energy
We know that the kinetic energy operator
$ K = -\frac{\hbar^{2}}{2m}\frac{\partial^{2} }{\partial x^{2}} $
$ \frac{P_{x}^{2}}{2m} = -\frac{\hbar^{2}}{2m}\frac{\partial^{2} }{\partial x^{2}} \qquad (\because K=\frac{P_{x}^{2}}{2m})$
Now substitute the value of $ \frac{P_{x}^{2}}{2m}$ in equation$(1)$
$E= -\frac{\hbar^{2}}{2m}\frac{\partial^{2} }{\partial x^{2}} + V(x)$
Multiply $\psi$ on the both side of above equation
$E \psi= -\frac{\hbar^{2}}{2m}\frac{\partial^{2} \psi }{\partial x^{2}} + V(x) \psi$
$E \psi= \left [ -\frac{\hbar^{2}}{2m}\frac{\partial^{2} }{\partial x^{2}} + V(x) \right ] \psi$
$E \psi= \hat{H} \psi$
So the total energy operator
$ \hat{H} = \left [ -\frac{\hbar^{2}}{2m}\frac{\partial^{2} }{\partial x^{2}} + V(x) \right ] $
For three dimensions:-
$\hat{H} = \left [ -\frac{\hbar^{2}}{2m}\nabla^{2} + V(x) \right ] $
The total energy operator is denoted by $\hat{H}$ and called the Hamiltonian Operator.
Total Energy Operator in terms of the differential with respect to time →
We know that the wave function
$\psi= A e^{\frac{i}{\hbar}}\left( P_{x}x - Et \right)$
Differentiate the above equation $(1)$ with respect to $t$ then we get
$\frac{\partial \psi}{\partial t}= A e^{\frac{i}{\hbar}(P_{x} x -Et)} \frac{i}{\hbar} (-E) $
$\frac{\partial \psi}{\partial t}= - \frac{i}{\hbar} E \psi $
$E \psi= -\frac{\hbar}{i} \frac{\partial \psi}{\partial t}$
$E \psi= i^{2} \frac{\hbar}{i} \frac{\partial \psi}{\partial t} \qquad (\because i^{2}=-1)$
$E \psi= i \hbar \frac{\partial \psi}{\partial t}$
This energy operator is denoted by $E$ so
$E = i \hbar \frac{\partial }{\partial t}$
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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