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Quantum Mechanics: Wave Equation Probability

  1. Apr 5, 2015 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

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


    3. The attempt at a solution

    I am wondering if I did this correctly?

    Normalization:

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

    $$\langle p|\psi\rangle = \psi(p) = \frac{1}{\sqrt{2\pi \hbar}}\int^{\infty}_{-\infty}e^{-ipx/\hbar}\psi(x)dx = \frac{N}{\sqrt{2\pi \hbar}} \left[\int^{\infty}_0e^{-ipx/\hbar}e^{-kx}dx + \int^0_{-\infty}e^{-ipx/\hbar}e^{kx}dx\right].$$
     
  2. jcsd
  3. Apr 6, 2015 #2

    vela

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    The normalization integral should be from ##-\infty## to ##\infty##. Your setup for the second part is fine.
     
  4. Apr 6, 2015 #3
    So it should be like $$1 = \int^{\infty}_{-\infty}|\psi(x)|^2dx = |N|^2\int^{\infty}_{0} e^{-2kx}dx + |N|^2\int^{0}_{-\infty} e^{2kx}dx?$$
     
  5. Apr 6, 2015 #4

    vela

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    Yup.
     
  6. Apr 6, 2015 #5
    Thank you very much!
     
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