# Group Work 2: Operators & Eigenvalues

• • Contributed by Nancy Levinger
• Professor and University Distinguished Teaching Scholar (Chemistry) at Colorado State University

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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

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:

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?