# 5.E: Translational States (Exercises)

## Q5.1

Write the Schrödinger equation for a free particle in three-dimensional space.

## Q5.2

Solve the Schrödinger equation to find the wavefunctions for a free particle in three-dimensional space.

## Q5.3

Show that these functions are eigenfunctions of the momentum operator in three-dimensional space.

## Q5.4

If you have not already done so, use vector notation for the wave vector and position of the particle.

## Q5.5

Write the wavefunctions using vector notation for the wave vector and the position.

## Q5.6

Write the momentum operator in terms of the del-operator, which is defined as $$\hat {\nabla} = \vec {x} \frac {\partial}{\partial x} + \vec {y} \frac {\partial}{\partial y} + \vec {z} \frac {\partial}{\partial z}$$ where the arrow caps on x, y, and z designate unit vectors.

## Q5.7

Write the Laplacian operator in terms of partial derivatives with respect to x, y, and z. The Laplacian operator is defined as the scalar product of del with itself, $$\hat {\partial} ^2 = \hat {\partial} \cdot \hat {\partial}$$.

## Q5.8

Write the kinetic energy operator in terms of the Laplacian operator.

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")