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Chemistry LibreTexts

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:

    a1 = 2.33810 a2 = 4.08794 a3 = 5.52055 a4 = 6.78670 a5 = 7.94413
    a6 = 8.02265 a7 = 10.04017 a8 = 11.00852 a9 = 11.93601 a10 = 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: \( \psi (z) = \frac{ \psi (z)}{ \sqrt{ \int_{0}^{z_{max}} \psi (z)^{2} dz}}\)

    Screen Shot 2019-02-06 at 1.03.04 PM.png