# Group Work 3: Operators

• • 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.

## 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?