TISE: Solution Eigenfunctions of various Potentials PHYS 520 Group 4: Connor Johnstone, Jacob Hempel, and Amber Moore

Abstract Preliminaries Infinite Square Well Applet Harmonic Oscillator Applet Delta-Function Applet Finite Square Well Applet

Preliminaries: Time Independent Schӧdinger Equation: − ℏ 2 2𝑚 ∗ 𝑑 2 Ψ x 𝑑 𝑥 2 +𝑉 𝑥 Ψ 𝑥 = 𝐸 𝑛 Ψ n 𝑥 Step 1 Write down with respect to potential of interest Step 2 Ψ 𝑥 continuous everywhere 𝑑Ψ 𝑑𝑥 continuous everywhere except where potential is infinite Step 3 Normalize to find any coefficients Step 4 Write final solution

Preliminaries: Potentials Infinite Square Well 𝑉 𝑥 = 0, ∞, 0≤𝑥≤𝑎 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Harmonic Oscillator 𝑉 𝑥 = 1 2 𝑚 𝜔 2 𝑥 2 Delta-Function 𝑉 𝑥 =−𝛼𝛿 𝑥 Finite Square Well 𝑉 𝑥 = − 𝑉 0 , 0, −𝑎≤𝑥≤𝑎 𝑥 >𝑎

Example: Delta-Function Scattering States − ℏ 2 2𝑚 𝑑 2 𝜓 𝑥 𝑑 𝑥 2 −𝛼𝛿 𝑥 𝜓(𝑥)=𝐸𝜓(𝑥) Boundary conditions: 𝐴+𝐵=𝐹 𝐴−𝐵−𝐹= 2𝛼𝑚 ℏ 2 𝐴+𝐵 Transmission and Reflection coefficients: 𝑅= 1 1+ 2 ℏ 2𝐸 𝑚 𝛼 2 = 𝐵 2 𝐴 2 T = 1 1+𝑚 𝛼 2 𝑒 ℏ 2 𝐸 = 𝐶 2 𝐴 2 ∴ 𝜓 1 𝑥 = 𝑒 −𝑖𝑘𝑥 +𝑅 𝑒 𝑖𝑘𝑥 𝜓 2 𝑥 =𝑇 𝑒 −𝑖𝑘𝑥 = 1−R e −ikx

Applet Links Infinite Square Well: 520 Infinite Well.nb Harmonic Oscillator: Project Applet SHO Final.nb Delta-Function: delta_function_solution_manipulationplot (1).nb Finite Square Well: 520 Finite Well.nb, 520 Finite Well 2.nb

Works Cited Griffiths, David J. Introduction to Quantum Mechanics. Pearson, 2005. Print.