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Simple Harmonic(Quantum) Oscillator

  1. Jan 1, 2005 #1
    Using the normalization condition, show that the constant [tex]A[/tex] has the value [tex](\frac{m\omega_0}{\hbar\pi})^{1/4}[/tex].

    I know from text the text book that

    [tex]\psi(x)=Ae^{-ax^2}[/tex]

    where [tex]A[/tex] is the amplitdue and [tex]a=\frac{\sqrt{km}}{2\hbar}[/tex]

    Here is my working:

    Because the motion of the particle is confined to -A to +A, so the probability of finding the particle in the interval of -A to +A must be 1. Therefore the normalization condition is

    [tex]\int_{-A}^{A}|\psi(x)^2| dx = 1[/tex]
    [tex]A^2\int_{-A}^{A} e^{-2ax^2} dx = 1 [/tex]

    Here's where I'm stuck, this equation cannot be solved via integration techniques, it can only be solved using by numerical methods. I only know the "trapezium rule" and the "Simpson's Rule", I tried both of methods but nothing came up. Does this problem require some other numerical methods or is my normalization condition incorrect?
     
    Last edited: Jan 1, 2005
  2. jcsd
  3. Jan 1, 2005 #2

    Dr Transport

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    Normalization is over the entire region, i.e. [tex] -\infty [/tex] to [tex] +\infty [/tex], the integral is then solved very easily.

    [tex]A^2\int_{-\infty}^{\infty} e^{-2ax^2} dx = 1 [/tex] for [tex] a = 1 [/tex] do a change of variables to get the correct answer, hence the normalization factor.
     
  4. Jan 1, 2005 #3

    Tide

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    Have you stated the problem correctly? It appears your limits of integration are the same as the amplitude which doesn't make sense because they have different units.
     
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