Evaluation of the Position-Momentum Commutator
In quantum mechanics, the commutator of the position operator \( \hat{x} \) and the momentum operator \( \hat{p} \) plays a fundamental role. It is defined as:
\( [\hat{x}, \hat{p}] = \hat{x}~ \hat{p} - \hat{p} ~ \hat{x} \)
Step-by-Step Evaluation
To evaluate this commutator, follow these detailed steps:
1. Define the Operators
The position operator \( \hat{x} \) and the momentum operator \( \hat{p} \) in the position representation are:
\(
\begin{align}
\hat{x} & = x \\
\hat{p} & = -i\hbar \frac{d}{dx}
\end{align}
\)
2. Compute the Action of the Operators
First, apply \( \hat{x} \) and \( \hat{p} \) in sequence:
\(
\begin{align}
\hat{x}~\hat{p}~\psi(x) & = \hat{x}\left(-i\hbar \frac{d}{dx}\psi(x)\right) \\
& = -i\hbar x \frac{d}{dx}\psi(x) \\[1mm]
\hat{p}~\hat{x}~\psi(x) & = -i\hbar \frac{d}{dx}(x\psi(x))
\end{align}
\)
3. Apply the Derivative
Apply the derivative in the second term:
\(
\frac{d}{dx}(x~\psi(x)) = \psi(x) + x ~ \frac{d}{dx}\psi(x)
\)
4. Compute the Commutator
Now compute \( \hat{x}\hat{p} - \hat{p}\hat{x} \):
\(
\begin{align}
\hat{x}~\hat{p}~\psi(x) & = -i\hbar ~ x ~ \frac{d}{dx}\psi(x) \\
\hat{p}~\hat{x}~\psi(x) & = -i\hbar \left(\psi(x) + x \frac{d}{dx}\psi(x)\right) \\
& = -i\hbar \psi(x) - i\hbar x \frac{d}{dx}\psi(x) \\[1mm]
[\hat{x}, \hat{p}]~\psi(x) & = \hat{x}~\hat{p}~\psi(x) - \hat{p}~\hat{x}~\psi(x) \\
& = -i\hbar ~x ~\frac{d}{dx}\psi(x) - \left(-i\hbar ~\psi(x) - i\hbar ~x ~\frac{d}{dx}\psi(x) \right) \\
& = i\hbar ~ \psi(x)
\end{align}
\)
5. Result
The commutator is:
\(
[\hat{x}, \hat{p}] = i\hbar
\)