Harmonic oscillator

[greisen]

Hi,

I am approximating a proton transfer from one water molecule to another. I would to have a quantum mechanical description of the proton transfer as a wavefunction. So I have approximated a "transition state" and use this as a harmonic potential. Then I get some energy values around this potential - the problem is that the best fit is still a quadratic function - so it should be harmonic but in my analytic eigenfunction I have x^2 - is it possible to use a*x^2 + b*x + c instead ?
Any help or advice appreciated.
Thanks in advance

Best regards

[nrqed]

Hi,

I am approximating a proton transfer from one water molecule to another. I would to have a quantum mechanical description of the proton transfer as a wavefunction. So I have approximated a "transition state" and use this as a harmonic potential. Then I get some energy values around this potential - the problem is that the best fit is still a quadratic function - so it should be harmonic but in my analytic eigenfunction I have x^2 - is it possible to use a*x^2 + b*x + c instead ?
Any help or advice appreciated.
Thanks in advance

Best regards

You can always reexpress a potential of the form V(x)= a x^2 + bx + c as a quadratic function the following way V(x) = a(x+ \frac{b}{2 a})^2 - \frac{ b^2}{4 a} + c

Now just shift the variable x' \equiv x - \frac{b}{2 a} and you get V(x') = a (x')^2 - \frac{ b^2}{4 a} + c Shifting th epotential by the constant piece leaves you with a harmonic potential in the new variable.

Hope this helps

Patrick