- #1
Hermite
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1D Time-Indep Sch Equation Solutions...
I'm trying to write a C++ program which displays a 'moveable' graph showing the solution (wavefunctions) to the Schrodinger Equation for a Simple Harmonic Oscillator. I need a little help with the physics of it.
The idea is that (using the mouse) the graph can be slid up and down to be at different energy levels. When the graph is at an energy level corresponding to a non-integer quantum number (which of course doesn't really exist) then the user will see the wavefunction behaving badly - going up to infinity. But, when the graph is at an energy level corresponding to integer quantum numbers then the wavefunction behaves properly.
I've worked through Appenix I of Eisberg & Resnick's textbook and 'understand' how to get general solutions involving Hermite polynomials which can be plotted very nicely. But I don't know how to display the solutions where n (the quantum number) is a non-integer. Obviously what happens is the series cannot be made to terminate and so the wavefunction goes to infinity but I can't plot an infinite sum of terms.
So, my question is: how do I plot the wavefunction as a function of x for continuous n (i.e. for any value of n)? Perhaps you know of some other approach - I'd appreciate any help.
Thanks
I'm trying to write a C++ program which displays a 'moveable' graph showing the solution (wavefunctions) to the Schrodinger Equation for a Simple Harmonic Oscillator. I need a little help with the physics of it.
The idea is that (using the mouse) the graph can be slid up and down to be at different energy levels. When the graph is at an energy level corresponding to a non-integer quantum number (which of course doesn't really exist) then the user will see the wavefunction behaving badly - going up to infinity. But, when the graph is at an energy level corresponding to integer quantum numbers then the wavefunction behaves properly.
I've worked through Appenix I of Eisberg & Resnick's textbook and 'understand' how to get general solutions involving Hermite polynomials which can be plotted very nicely. But I don't know how to display the solutions where n (the quantum number) is a non-integer. Obviously what happens is the series cannot be made to terminate and so the wavefunction goes to infinity but I can't plot an infinite sum of terms.
So, my question is: how do I plot the wavefunction as a function of x for continuous n (i.e. for any value of n)? Perhaps you know of some other approach - I'd appreciate any help.
Thanks