Understanding the Quantum Mechanical Model for V = Infinity and V = kx2/2

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SUMMARY

The discussion focuses on solving the quantum mechanical model where the potential V is defined as infinity for x ≤ 0 and V = kx²/2 for x ≥ 0. The key equations referenced include the energy formula for a 1-D particle in a box, E = (h²/8m)(n²/L²), and the harmonic oscillator energy, E = (v + 1/2)(h/2π)ω. The solution involves solving the Schrödinger equation separately in both regions and ensuring continuity at the boundary x = 0.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically the Schrödinger equation.
  • Familiarity with potential energy functions, particularly harmonic oscillators.
  • Knowledge of boundary conditions in quantum systems.
  • Basic grasp of wave functions and their significance in quantum mechanics.
NEXT STEPS
  • Study the solutions to the Schrödinger equation for piecewise potential functions.
  • Learn about the mathematical treatment of harmonic oscillators in quantum mechanics.
  • Explore the concept of wave function normalization and continuity conditions.
  • Investigate the implications of infinite potential wells in quantum systems.
USEFUL FOR

Students of quantum mechanics, particularly those tackling problems involving potential energy functions and wave functions, as well as educators seeking to clarify concepts related to the Schrödinger equation.

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Homework Statement


What are the energies and wave functions for the quantum mechanical model where V = infinity for x less than or equal to 0 and V = kx2/2 for x greater than or equal to 0?


Homework Equations


1-D particle in a box E = (h2/8m)(n2/L2)

Harmonic Oscillator E = (v + 1/2)(h/2*pi)w


The Attempt at a Solution


Honestly, I had no clue where to start. This was the last problem on our first exam. I tried drawing what it would represent graphically but I'm not really sure what I was supposed to answer... Do I add the energies together? Wave functions? I'm not even sure what format our professor was looking for he didn't tell us...
 
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Solve the Schrödinger equation in each region, x<0 and x>0, and then join the solutions at x=0.
 

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