10.1: Trial Wavefunctions for Various Potentials 10.2: Energy Minimization - Four Methods Using Mathcad 10.3: The Variation Theorem in Dirac Notation 10.4: A Rudimentary Model for Alpha Particle Decay Simple models for the potential energy experienced by an alpha particle in a nucleus have the form shown below. In the interest of mathematical simplicity we will not attempt to model any particular alpha emitter, but just try to capture the essentials of the quantum mechanical tunneling mechanism for alpha decay that was formulated by Gamov, Gurney and Condon in 1928. 10.5: Variational Method for a Particle in a Finite Potential Well 10.6: Variation Method for a Particle in a Symmetric 1D Potential Well 10.7: Variation Method for the Rydberg Potential 10.8: Variation Method for the Quartic Oscillator 10.9: Momentum-Space Variation Method for the Quartic Oscillator 10.10: Variation Method for a Particle in a Gravitational Field 10.11: Linear Variational Method for a Particle in a Slanted 1D Box 10.12: Variation Method for a Particle in a Semi-Infinite Potential Well 10.13: Variation Method for a Particle in a Box with an Internal Barrier 10.14: Variation Method for a Particle in a 1D Ice Cream Cone 10.15: Variation Method for a Particle in an Ice Cream Cone 10.16: Variation Method for a Particle in a Finite 3D Spherical Potential Well 10.17: Variation Method for the Harmonic Oscillator This exercise deals with a variational treatment for the ground state of the simple harmonic oscillator which is, of course, an exactly soluble quantum mechanical problem. 10.18: Trigonometric Trial Wave Function for the Harmonic Potential Well 10.19: Trigonometric Trial Wave Function for the 3D Harmonic Potential Well 10.20: Gaussian Trial Wavefunction for the Hydrogen Atom A Gaussian function is proposed as a trial wavefunction in a variational calculation on the hydrogen atom. 10.21: Variation Calculation on the 1D Hydrogen Atom Using a Trigonometric Trial Wave Function 10.22: Variation Calculation on the 1D Hydrogen Atom Using a Gaussian Trial Wavefunction 10.23: Variational Calculation on the Two-dimensional Hydrogen Atom 10.24: Variational Calculation on Helium Using a Hydrogenic Wavefunction 10.25: Gaussian Trial Wave Function for the Helium Atom 10.26: Trigonometric Trial Wavefunction for the Helium Atom 10.27: Trigonometric Trial Wavefunction for the Hydrogen Atom 10.28: Hydrogen Atom Calculation Assuming the Electron is a Particle in a Sphere of Radius R 10.29: Electronic Structure - Variational Calculations on the Lithium Atom 10.30: The Variation Method in Momentum Space 10.31: Momentum-Space Variation Method for Particle in a Gravitational Field 10.32: Momentum-Space Variation Method for the Abs(x) Potential 10.33: Variational Method for the Feshbach Potential 10.34: Numerical Solution for the Feshbach Potential 10.35: First Order Degenerate Perturbation Theory - the Stark Effect of the Hydrogen Atom 10.36: Variational Calculation for the Polarizability of the Hydrogen Atom In this exercise the polarizability of the hydrogen atom is calculated. 10.37: Hybrid Variational Calculation for the 1D Hydrogen Atom with Delta Function Potential This hybrid variational calculation on a one‐dimensional model of the hydrogen atom using a delta function potential interaction between the proton and electron yields the correct ground state energy. The model is composed of two peculiar elements: a trial wave function that has discontinuity at x = 0 and a potential energy interaction that is zero everywhere except at x = 0. 10.38: Variation Method Using the Wigner Function- Finite Potential Well 10.39: 455. Variation Method Using the Wigner Function- V(x) = |x| 10.40: Variation Method Using the Wigner Function- The Harmonic Oscillator 10.41: Variation Method Using the Wigner Function - The Quartic Oscillator 10.42: Variation Method Using the Wigner Function - The Feshbach Potential