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Chemistry LibreTexts

Group Work 3: Operators

[ "article:topic", "authorname:levingern", "showtoc:no" ]
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    31873
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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.

    Preamble

    The order that operators are applied to a function can be very important.

    Q1

    Suppose that

    \[\hat{A}=\dfrac{d}{dx} \tag{W.1}\]

    and

    \[\hat{B}=x^2 \tag{W.2}\]

    For any function \(f(x)\) what is \(\hat{A}f(x)\)?

     

     

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

     

     

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

     

     

    What is \(\hat{B}\hat{A}f(x)\)?

     

     

    Is \(\hat{A}\hat{B}f(x)=\hat{B}\hat{A}f(x)\)? Why?

     

     

    When \(\hat{A}\hat{B}f(x)=\hat{B}\hat{A}f(x)\), the two operators commute. Do \(\hat{A}=\dfrac{d}{dx}\) and \(\hat{B}=x^2\) commute?

     

     

    Q2

    Suppose that

    \[\hat{A}=\dfrac{d}{dx} \tag{W.3}\]

    and

    \[\hat{B}=10 \tag{W.4}\]

    What is  \(\hat{A}f(x)\)?

     

     

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

     

     

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

     

     

    What is \(\hat{B}\hat{A}f(x)\)?

     

     

    Do \(\hat{A}\) and \(\hat{B}\) commute?