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Chemistry LibreTexts

411: Particle in a Box with Multiple Internal Barriers

  • Page ID
    137740
  • Integration limit: xmax = 1

    Effective mass: \( \mu\) = 1

    Barrier height: V0 = 100

    Potential energy:

    \[V(x) = \bigg|^{V_0~if~(x \geq .185)(x \leq .215) + (x \geq .385)(x \leq .415) + (x \geq .585) (x \leq .615) + (x \geq .785) (x \leq .815)}_{0~otherwise}\]

    Numerical integration of Schrödinger's equation:

    Given

    \[ \frac{-1}{2 \mu} \frac{d^2}{dx^2} \psi (x) + V(x) \psi (x) = E \psi (x)\]

    \[ \psi (0) = 0\]

    \[ \psi ' (0) = 0.1\]

    \[ \psi = Odesolve (x, x_{max}\]

    Normalize wave function:

    \[ \psi (x) = \frac{ \psi (x)}{ \sqrt{ \int_{0}^{x_{max}} \psi (x)^2 dx}}\]

    Enter energy guess: E = 18.85

    Screen Shot 2019-02-19 at 11.23.40 AM.png

    Calculate kinetic energy:

    \[ T = \int_{0}^{1} \psi (x) \frac{-1}{2} \frac{d^2}{dx^2} \psi (x) dx = 5.926\]

    Calculate potential energy:

    \[ V = E - T = 12.924\]

    Tunneling probability:

    \[ \frac{V}{V_0} \times 100 = 12.924\]