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

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

3. 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

4. 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

5. 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= ℏ 2𝐸 𝑚 𝛼 2 = 𝐵 𝐴 2
T = 1 1+𝑚 𝛼 2 𝑒 ℏ 2 𝐸 = 𝐶 𝐴 2
∴ 𝜓 1 𝑥 = 𝑒 −𝑖𝑘𝑥 +𝑅 𝑒 𝑖𝑘𝑥
𝜓 2 𝑥 =𝑇 𝑒 −𝑖𝑘𝑥 = 1−R e −ikx

6. 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

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