Operators and Eigenvalues (Worksheet)
- Page ID
- 39216
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Name: ______________________________
Section: _____________________________
Student ID#:__________________________
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
Suppose
\[\hat{A} = \dfrac{d}{dx}\]
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:
- \(\hat{A} (f( x)+ g (x)) = \hat{A}f (x )+ \hat{A}g(x)\) (the operator is distributive)
- \(\hat{A} cf (x) = c \hat{A}f (x)\) (\(c\) is a real, imaginary or complex constant)
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?