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Chemistry LibreTexts

Group Work 2: Operators & Eigenvalues

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    Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

    Part 1 - Operators


    \[\hat{A} = \dfrac{d}{dx}\]

    What is \(\hat{A}f (x)\)?

    If \(f (x) = \dfrac{1}{x^2} \) what is \(\hat{A}f (x )\) ?

    In quantum mechanics, we will work only with linear operators. Linear operators follow the two rules:

    1. \(\hat{A} (f( x)+ g (x)) = \hat{A}f (x )+ \hat{A}g(x)\) (the operator is distributive)
    2. \(\hat{A} cf (x) = c \hat{A}f (x)\) (\(c\) is a real, imaginary or complex constant)

    Is the operator \(\hat{A} = \frac{d}{dx}\) linear? Why or why not.

    What is an example of \(\hat{A}\) that is linear?

    What is an example of \(\hat{A}\) that is not linear?

    Part 2 - Eigenvalue problems

    In an eigenvalue problem, an operator applied to a function is equivalent to a constant value multiplied times the function, that is,

    \[\hat{A}f (x) =af (x)\]

    In this equation, a is the eigenvalue; it is just a real, imaginary or complex numerical constant.

    Suppose \(\hat{A} = \frac{d}{dx}\) and \(f(x)=e^{6x}\)

    What is the eigenvalue of \(\hat{A}\) ?

    For \(\hat{A} = \frac{d}{dx}\), can any mathematical function, \(g(x)\) serve as the eigenfunction of \(\hat{A}\) or are there examples of \(g(x)\) that would not work?

    Suppose \(\hat{B} = \dfrac{d^2}{dx^2}\). What function could be an eigenfunction of \(\hat{B}\) ?

    What is the eigenvalue for the eigenfunction you chose?