Momentum operator in quantum mechanics
Definition
In quantum mechanics, the momentum operator is as follows:
$$ P = \frac{\hbar}{\i}\frac{\partial}{\partial x} = -\i\hbar \dfrac{\partial }{\partial x} $$
Description
The momentum operator is a function that allows one to calculate the momentum of a wave function. When a wave function with momentum $p = \hbar k$ is substituted, it satisfies the following equation.
$$ P \psi = p \psi $$
In the case of dimensions higher than two, the momentum operator in each direction is as follows:
$$ P_{x} = -\i\hbar\frac{\partial}{\partial x},\quad P_{y} = -\i\hbar\frac{\partial}{\partial y},\quad P_{z} = -\i\hbar\frac{\partial}{\partial z} $$
Therefore, the momentum operator in three dimensions is,
$$ P = -\i\hbar\nabla $$
Derivation
Let’s consider the one-dimensional case. The operator we seek is one that satisfies the following equation for a wave function with momentum $p$.
$$ P \psi = p \psi $$
In quantum mechanics, since the momentum and energy are $p = \hbar k$ respectively, the wave function is,
$$ \psi (x,t) = e^{i(px - \hbar \omega t)/\hbar} $$
To obtain the momentum $p$ from here, we differentiate using $x$.
$$ \dfrac{\partial }{\partial x} e^{\i(px - \hbar \omega t)/\hbar} = \dfrac{\i}{\hbar} p e^{\i(px - \hbar \omega t)/\hbar} $$
Therefore,
$$ \begin{align*} && \dfrac{\hbar}{\i}\dfrac{\partial }{\partial x} e^{\i(px - \hbar \omega t)/\hbar} &= p e^{\i(px - \hbar \omega t)/\hbar} \\ \implies && \dfrac{\hbar}{\i}\dfrac{\partial }{\partial x} \psi (x, t) &= p \psi (x, t) \\ \implies && P = \dfrac{\hbar}{\i}\dfrac{\partial }{\partial x} \end{align*} $$
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