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Particle in a box and the Harmonic Oscillator

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Consider the particle of mass subject to a one-dimensional potential of the following form:

V(x) = 1/2 kx^2 for x>0
V(x) = + infinity for x < 0

This is a combination of the particle in a box and the harmonic oscillator that might be a better model for real diatomic than the standard harmonic oscillator. On the right side of x = 0, the Hamiltonian is exactly the same as a harmonic oscillator Hamiltonian. The hard wall at x = 0, however, introduces a boundary condition. Use what you have learned about both the ordinary harmonic oscillator and particle in a box boundary conditions to answer the following questions.
a) what does this boundary condition require the wave function to do at x = 0?

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a. Boundary condition is that at x = 0, Wave function &#968;(x) = &#968;(0) = ...

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I have provided solution to this problem without the use of tedious mathematical formula. Solution is very well presented with explanations.