Quantum Mechanics: Wave Equation Probability

[Robben]

Homework Statement

Normalize the wave function $$ \langle x|\psi\rangle = \left\{ \begin{array}{l l} Ne^{-kx} & \quad x>0\\
Ne^{kx} & \quad x<0 \end{array} \right..$$
Determine the probability that a measurement of the momentum p finds the momentum between $p$ and $p + dp$ for this wave function.

Homework Equations

$\langle p|\psi\rangle = \frac{1}{\sqrt{2\pi \hbar}}\int^{\infty}_{-\infty}e^{-ipx/\hbar}\psi(x)dx.$

The Attempt at a Solution

I am wondering if I did this correctly?

Normalization:

$$1 = \int^{\infty}_{0}|\psi(x)|^2dx = |N|^2\int^{\infty}_{0} e^{-2kx}dx \implies N = \sqrt{2k}.$$

$$\langle p|\psi\rangle = \psi(p) = \frac{1}{\sqrt{2\pi \hbar}}\int^{\infty}_{-\infty}e^{-ipx/\hbar}\psi(x)dx = \frac{N}{\sqrt{2\pi \hbar}} \left[\int^{\infty}_0e^{-ipx/\hbar}e^{-kx}dx + \int^0_{-\infty}e^{-ipx/\hbar}e^{kx}dx\right].$$

 

[vela]

The normalization integral should be from $-\infty$ to $\infty$. Your setup for the second part is fine.

 

[Robben]

The normalization integral should be from $-\infty$ to $\infty$. Your setup for the second part is fine.

So it should be like $$1 = \int^{\infty}_{-\infty}|\psi(x)|^2dx = |N|^2\int^{\infty}_{0} e^{-2kx}dx + |N|^2\int^{0}_{-\infty} e^{2kx}dx?$$

 

[vela]

Yup.

 

[Robben]

Yup.

Thank you very much!