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Worksheet 4B Solutions

  • Page ID
    283126
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    Symmetry in 1-D

     

    Q1

    Even. Symmetric across the y-axis.

    Q2

    Odd. Symmetric across the origin.

    Q3

    Depends on a.

    a small:

    clipboard_e2576258670e7ee88229530e4cd321bbe.png

    to a large.

     clipboard_ea7a0b1714e0e20af7a6dffd60914294a.png

    This function is even. 

    The integral of \( f(x) = e^{-x^2/2a^2} \) is \( 2 \left( \frac{\pi}{4a} \right)^{1/2} \)

     

    Q4

    The requested integral is double the given integral.

    The integral of \( \infty_{-\infty}^{\infty} x^3 is 0, which can be seen from the symmetry of the plot.

    f(x) =x is odd and shown below.

    clipboard_e7f2e607afe9286068ac7b7a855e3b49f.png

    The integral of an odd function over an even interval will be zero as it cancels itself out.

    Is the product of two even functions even or odd?

    Q5

    \( f(x) = x e^{-x^2/2a^2} \) is odd. 

    For a small

    clipboard_ef4dc403caea40eb692c115bde1993b69.png

    For a large

    clipboard_eaf588fc6c532643c177f796711707a99.png

    This is an odd function. 

    The product of two even functions is even.

    The product of two odd or two even functions is always even.

    Q6

    Which levels are even functions?

    • 0,2

    Which are odd functions?

    • 1,3

    Q7

    The probability densities for the lowest four levels of the harmonic oscillator (v=1−4(v=1−4 are below.

    Which levels are even functions?

    • 0,1,2,3

    Which levels are odd functions?

    • None

    How does this compare to your earlier statements about products of functions?

    • Follows from above.

    Consider the graphs of the harmonic oscillator probability densities. Without calculating an integral, what is ⟨x⟩⟨x⟩?

    • 1/2

    How do you explain your answer in terms of the probability density graphs?

    • Symmetric.

    How do you explain your answer in terms of the even/odd character of the probability densities?

    • All probability densities are even.

    Worksheet 4B Solutions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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