Question

TEXT BOOK: INTRODUCTORY NUCLEAR PHYSICS, Kenneth S. Krane

2.5 Find the solution to the “half” harmonic oscillator:

               v ( x ) = ∞               x < o

                                          = 12kx²                   x > 0

Compare the energy values and wave functions with those of the full

harmonic oscillator. Why are some of the full solutions present and some

missing in the “half” problem?

Answer 1

The wavefunction of the normal linear harmonic oscillator is

and this is derived by considering the general boundary condition

Energy eigenvalue is

But in the half potential

the boundary condition is

while considering this, the Hermite polynomial will not vanish for an even number of n at x=0

as per quantum mechanics, the wave function must vanish in the boundary.

So the energy eigenvalue of the half LHO is

otherwise, it follows the same solution as the general LHO.