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Quantum Mechanics: Wave Mechanics in One Dimension

  1. Mar 30, 2015 #1
    1. The problem statement, all variables and given/known data

    Let ##\langle\psi| = \int^{\infty}_{-\infty}dx\langle\psi|x\rangle\langle x|.## How do I calculate ##\langle\psi|\psi\rangle?##

    2. Relevant equations

    ##\int^{\infty}_{-\infty}dxf(x)\delta(x-x_0)=f(x_0)##

    3. The attempt at a solution

    ##\langle\psi|\psi\rangle = \int\int dx'dx\langle \psi|x\rangle\langle x|x'\rangle\langle x'|\psi\rangle = \int\int dx'dx\langle\psi|x\rangle\delta(x-x')\langle x'|\psi\rangle## but then how does that equal to ##\int dx \langle\psi|x\rangle\langle x|\psi\rangle?##
     
  2. jcsd
  3. Mar 31, 2015 #2
    <x|x`> is delta function and x` is as x in above.
     
  4. Mar 31, 2015 #3
    I am not sure what you mean?
     
  5. Mar 31, 2015 #4
    Above in OP is equation in 'Relevant equations'
     
  6. Mar 31, 2015 #5

    BvU

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    Since ##\langle \psi|x\rangle## does not depend on ##x'## you can take it out of the integral over ##dx'##:$$
    \int\int dx'dx\langle \psi|x\rangle\langle x|x'\rangle\langle x'|\psi\rangle =\int dx \langle \psi|x\rangle\ \left ( \int dx'\langle x|x '\rangle\langle x'|\psi\rangle \right )
    $$
     
  7. Mar 31, 2015 #6
    But how does $$\left ( \int dx'\langle x|x '\rangle\langle x'|\psi\rangle \right ) = \langle x|\psi\rangle?$$
     
  8. Mar 31, 2015 #7

    Dick

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    It's what abbas_majidi wrote in post #2. ##<x|x'>=\delta(x-x')##.
     
  9. Apr 1, 2015 #8
    How does ##\delta(x-x')## act on ##\langle x'|\psi\rangle## in order for it to equal ##\langle x|\psi\rangle##, i.e. $$\int dx'\delta(x-x')\langle x'|\psi\rangle?$$
     
  10. Apr 1, 2015 #9

    BvU

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    You can apply your equation
    $$
    \int^{\infty}_{-\infty}dxf(x)\delta(x-x_0)=f(x_0)
    $$to function values, but also to functions. So at each ##x'## in ##\langle x'|\psi \rangle = \psi(x')## is "replaced by ##x##" .
     
  11. Apr 1, 2015 #10
    I see. This is my first time using dirac delta function and i was confused. My book didn't do a good job in explaining. Thank you all!
     
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