Quantum Mechanical Operators

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

$E = i \hbar \frac{\partial }{\partial t}$