# 397: Particle in a Gravitational Field

- Page ID
- 135865

## The Unhindered Quantized Bouncing Particle

- Integration limit: \( z_{max} = 3\)
- Mass: \( m = 2\)
- Acceleration due to gravity: \(g = 1\)

The first 10 roots of the Airy function are as follows:

a_{1} = 2.33810 |
a_{2} = 4.08794 |
a_{3} = 5.52055 |
a_{4} = 6.78670 |
a_{5} = 7.94413 |

a_{6} = 8.02265 |
a_{7} = 10.04017 |
a_{8} = 11.00852 |
a_{9} = 11.93601 |
a_{10} = 12.82877 |

Calculate energy analytically by selecting the appropriate Airy function root:

i = 1 E = \( \frac{mg^{2}}{2}^{ \frac{1}{3}} a_{1}\) E = 2.338

Generate the associated wavefunction numerically: Potential energy: \(V(z) = mgz\)

Given \( \frac{-1}{2 \cdot m} \frac{d^{2}}{dz^{2}} \psi (z) + V (z) \psi (z) \equiv E \psi (z)\)

\( \psi (0.0) = 0.0\)

\( \psi '(0.0) = 0.1\)

Given, \( \psi = Odesolve (z, z_{max})\)

Normalize wavefunction: \( \psi (z) = \frac{ \psi (z)}{ \sqrt{ \int_{0}^{z_{max}} \psi (z)^{2} dz}}\)