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Finding the Momentum Eigenfunction
1. Momentum Eigenvalue Equation
The momentum eigenvalue equation is given by:
\[
\widehat{P} ~ \psi_p(x) = p ~ \psi_p(x)
\]
where \( \widehat{P} \) is the momentum operator and \( p \) is the momentum eigenvalue.
2. Momentum Operator
The momentum operator in the position representation is:
\[
\widehat{P} = -i \hbar \frac{\partial}{\partial x}
\]
3. Applying the Operator
Applying the momentum operator to the eigenfunction \( \psi_p(x) \) gives:
\[
\widehat{P} ~\psi_p(x) = -i \hbar ~ \frac{\partial \psi_p(x)}{\partial x}
\]
According to the eigenvalue equation:
\[
-i \hbar \frac{\partial \psi_p(x)}{\partial x} = p ~\psi_p(x)
\]
4. Solving the Differential Equation
Rearranging the equation:
\[
\frac{\partial \psi_p(x)}{\partial x} = -\frac{ip}{\hbar} \psi_p(x)
\]
This is a first-order linear differential equation. The solution is:
\[
\langle x | \psi \rangle = \psi_p(x) = A ~ e^{ipx/\hbar}
\]
where \( A \) is a normalization constant. Note that \( p \) is not quantised and can take any real value.
5. Normalization
To normalize \( \psi_p(x) \), we use:
\[
\int_{-\infty}^{\infty} |\psi_p(x)|^2 \, dx = 1
\]
For the plane wave solution, this leads to:
\[
|\psi_p(x)|^2 = \frac{1}{2\pi \hbar}
\]
Thus, the normalized momentum eigenfunction is:
\[
\psi_p(x) = \frac{1}{\sqrt{2\pi \hbar}} e^{ipx/\hbar}
\]
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