Skip to main content
Chemistry LibreTexts

432: Variation Method for a Particle in a Finite 3D Spherical Potential Well

  • Page ID
    136254
  • This problem deals with a particle of unit mass in a finite spherical potential well of radius 2 ao and well height 2 Eh. The trial wave function is given below.

    \[ \psi (r, \beta ) := ( \frac{2 \beta}{ \pi})^{ \frac{3}{4}} exp(- \beta r^2)\]

    \[ T = - \frac{1}{2r} \frac{d^2}{dr^2} (r \blacksquare )\]

    \( V(r) := if [(r \leq 2), 0, 2]\)

    Screen Shot 2019-02-12 at 12.23.13 PM.png

    a. Demonstrate that the wave function is normalized.

    \[ \int_{0}^{ \infty} \psi (r, \beta )^2 4 \pi r^2 dr |_{simplify}^{assume,~ \beta > 0} \rightarrow 1\]

    b. Evaluate the variational integral.

    \[ E( \beta ) := \int_{0}^{ \infty} \psi (r, \beta ) [- \frac{1}{2r} \frac{d^2}{dr^2} (r \psi (r, \beta ))] 4 \pi r^2 dr ... |_{simplify}^{assume,~ \beta > 0} + \int_{2}^{ \infty} 2 \psi (r, \beta )^2 4 \pi r^2 dr\]

    \[ E( \beta ) := \frac{1}{2} \frac{3 \pi^{ \frac{1}{2}} \beta + 4 \pi ^{ \frac{1}{2}} + 16 exp(-8 \beta) 2^{ \frac{1}{2}} \beta^{ \frac{1}{2}}-4 \pi ^{\frac{1}{2}} erf((2) 2^{ \frac{1}{2}} \beta^{ \frac{1}{2}}) }{ \pi ^{ \frac{1}{2}}}\]

    c. Minimize the energy with respect to the variational parameter \( \beta\).

    \( \beta\) := 5 \( \beta\) := Minimize (E, \beta ) \( \beta\) = 0.381 \( E ( \beta ) = 0.786\)

    d. Calculate the average value of r.

    \( \int_{0}^{ \infty} r \psi (r, \beta )^2 4 \pi r^2 dr = 1.293\)

    e. Calculate the kinetic and potential energy.

    Potential energy:

    \( \int_{2}^{ \infty} r \psi (r, \beta )^2 4 \pi r^2 dr = 0.215\)

    Kinetic energy:

    \( E( \beta ) - 0.215 = 0.571\)

    f. Calculate the probability that the particle is in the barrier.

    \( 1 - \int_{0}^{2} \psi (r, \beta )^2 4 \pi r^2 dr = 0.107\)

    g. Plot the wavefunction on the same graph as the potential energy.

    Screen Shot 2019-02-12 at 12.27.48 PM.png