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3A: Operators and Eigenvalues (Worksheet)

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    93311
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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.

    An operator is a function over a space of physical states to another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are a very useful tool in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

    1: Operators

    Suppose

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

    1. What is the expression for \(\hat{A} f (x)\), that is this operator operating on an arbitrary function?
    2. If \(f (x) = \frac{1}{x^2} \) what is \(\hat{A}f (x )\) ?

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

    • \(\hat{A}( f( x)+ g (x) )= \hat{A}f (x )+ \hat{A}g(x)\) (the operator is distributive)
    • \(\hat{A} cf (x) = c \hat{A}f (x)\) (\(c\) is a real, imaginary or complex constant)
    1. Is the operator \(\hat{A} = \frac{d}{dx}\) linear? Why or why not.
    2. What is an example of \(\hat{A}\) that is linear?
    3. What is an example of \(\hat{A}\) that is not linear?
     

    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 and it may be real, imaginary or complex numerical constant (i.e., scalar) and \(f(x)\) is an eigenfunction (there are typically many and often an infinite number possible for any operator).

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

    1. What is the eigenvalue of \(\hat{A}\) operating on \(f(x)\)?
    2. For \(\hat{A} = \frac{d}{dx}\), can any mathematical function, \(g(x)\) serve as the eigenfunctions of \(\hat{A}\)? Give a examples of \(g(x)\) that would not work?
    3. Suppose \(\hat{B} = \dfrac{d^2}{dx^2}\). What function could be an eigenfunction of \(\hat{B}\) ?
    4. What is the eigenvalue for the eigenfunction you chose?
     

    3: The Order of Operations

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

    Suppose that

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

    and

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

    1. For any function \(f(x)\), what is \(\hat{A}f(x)\)?
    2. What is \(\hat{B}f(x)\)?
    3. What is \(\hat{A}\hat{B}f(x)\)?
    4. What is \(\hat{B}\hat{A}f(x)\)?
    5. Is \(\hat{A}\hat{B}f(x)=\hat{B}\hat{A}f(x)\)? Why?
    6. 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?
     

    4: The Order of Operations Redux

    Suppose that

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

    and

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

    1. What is \(\hat{A}f(x)\)?
    2. What is \(\hat{B}f(x)\) ?
    3. What is \(\hat{A}\hat{B}f(x)\) (i.e., \(\hat{B}\) operating first on \(f(x)\), then \(\hat{A}\) operating on the result)?
    4. What is \(\hat{B}\hat{A}f(x)\) (i.e., \(\hat{A}\) operating first on \(f(x)\), then \(\hat{B}\) operating on the result)?
    5. Do \(\hat{A}\) and \(\hat{B}\) commute?