📂Quantum Mechanics

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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