topic "Eigenvalue spectroscopy." How do you like this method? in Science Questions and Collaboration
https://communities.acs.org/t5/Science-Questions-and/quot-Eigenvalue-spectroscopy-quot-How-do-you-like-this-method/m-p/8149#M307
<HTML><HEAD></HEAD><BODY><P>Assume that you have two computer programs, one being for solving the initial value problem for one-dimensional quantum mechanical system, the other being for doing Fourier analysis. Assume an appropriate static potential function like a parabolic function V(x)=x^2. And set an appropriate initial wave function PSI(x,0) for the initial value problem program.</P><P></P><P>Now, let's go! First you let the initial value problem program run for a certain period, during which you also record the real part value PSIR(xs,t) and imaginary part value PSII(xs,t) of the wave function at a particular fixed sampling point xs. So you can obtain a couple of sequential data PSIR(t) and PSII(t). Next you perform the Fourier analysis on these data, and you draw a Fourier transform power function FR(w)^2+FI(w)^2 (w for Greek lower case omega, the angular frequency.)</P><P></P><P>Can you predict the result of the Fourier analysis?</P><P>Hey, presto! It's the eigenvalue spectrogram!</P><P></P><P>Of course, the result depend on what function you set as the initial wave function PSI(x,0). If you choose an even fucntion like cos(3x)exp(-x^2/s^2) as the initial function, then you will get only even order peaks in the spectrogram (see the attached file EVENCASE.docx.) Conversely if you choose an odd function like sin(3x)exp(-x^2/s^2), then you will get only odd order peaks(file ODDCASE.docx.)</P><P></P><P>Thus far, I have tried only a few examples. I cannot still prove the mechanism of this method mathematically. Can anybody prove this?</P><P></P><P>Thank you for reading</P></BODY></HTML>Tue, 03 Jun 2014 08:34:59 GMTMitsuruYamada2014-06-03T08:34:59Z"Eigenvalue spectroscopy." How do you like this method?
https://communities.acs.org/t5/Science-Questions-and/quot-Eigenvalue-spectroscopy-quot-How-do-you-like-this-method/m-p/8149#M307
<HTML><HEAD></HEAD><BODY><P>Assume that you have two computer programs, one being for solving the initial value problem for one-dimensional quantum mechanical system, the other being for doing Fourier analysis. Assume an appropriate static potential function like a parabolic function V(x)=x^2. And set an appropriate initial wave function PSI(x,0) for the initial value problem program.</P><P></P><P>Now, let's go! First you let the initial value problem program run for a certain period, during which you also record the real part value PSIR(xs,t) and imaginary part value PSII(xs,t) of the wave function at a particular fixed sampling point xs. So you can obtain a couple of sequential data PSIR(t) and PSII(t). Next you perform the Fourier analysis on these data, and you draw a Fourier transform power function FR(w)^2+FI(w)^2 (w for Greek lower case omega, the angular frequency.)</P><P></P><P>Can you predict the result of the Fourier analysis?</P><P>Hey, presto! It's the eigenvalue spectrogram!</P><P></P><P>Of course, the result depend on what function you set as the initial wave function PSI(x,0). If you choose an even fucntion like cos(3x)exp(-x^2/s^2) as the initial function, then you will get only even order peaks in the spectrogram (see the attached file EVENCASE.docx.) Conversely if you choose an odd function like sin(3x)exp(-x^2/s^2), then you will get only odd order peaks(file ODDCASE.docx.)</P><P></P><P>Thus far, I have tried only a few examples. I cannot still prove the mechanism of this method mathematically. Can anybody prove this?</P><P></P><P>Thank you for reading</P></BODY></HTML>Tue, 03 Jun 2014 08:34:59 GMThttps://communities.acs.org/t5/Science-Questions-and/quot-Eigenvalue-spectroscopy-quot-How-do-you-like-this-method/m-p/8149#M307MitsuruYamada2014-06-03T08:34:59Z