Can you please check this translation

[lucagalbu]

Hi!
I have to write a resume of a project in my university and I have to write it in English. My mother language is Italian and I'm not so good in writing in English, so can you please check if my translation is correct? Thank you very much.


Quantum mechanics represented a big revolution in the way we conceive the world and the reality all around us. Nevertheless the Schrodinger equation can be exactly solved only in few situations; it is thus interesting to develop pro- grams in order to numerically determine the evolution of several systems. That was, indeed, the goal of this work, where a C++ code was written to integrate the Schrodinger equation in two and three spatial dimensions and to show the real-time evolution of the wave-packet.
The method used is based on Lie-Trotter formula in order to write the time evolution operator as an infinite product of terms diagonal in coordinate and momentum space. This product is written as a finite number of terms by the discretization of the temporal evolution in small intervals ∆t; the error intro- duced in this way is of the order of (∆t)2). In order to evolve the wave function, first it is multiplied by the factor diagonal on the coordinate basis; then the wave function is written in the basis of the momentum space and it’s multiplied by the factor diagonal in that basis and finally the wave function is written back in the coordinate space and multiplied by the third factor of the split evolution operator. The two basis are linked by the Fourier transform.
If a magnetic field is present, it can be shown that its effect can be accomplished with the introduction of an appropriate phase. Any substantial difficulty is in- troduced if the spin interaction is also considered. It can be shown that the spin coupling with the magnetic field can be achived with a matrix inserted in the evolution operator and considering the wave function as a two-component spinor.
A fundamental part in the development of the code was the validation of the results. To do this, the numerical results have been compared with the analytic solutions for some cases where the exact solution of the Schr ̈odinger equation is available. These include the free particle, the harmonic oscillator, the infinite-height potential barrier and the spin megnetic resonance. In conclusion, the code can manage any initial state and any potential, showing the real-time evolution of the wave function and visually displaying the strictly quantum behavior so distant from our daily experience.

[GinaS]

Not much wrong with your English. I'm assuming the hyphenated words are artifacts of cut and paste. My quantum mechanics and computer programming are a bit rusty, but here goes...

The theories of quantum mechanics resulted in a major revolution in the way we conceive the world and the reality all around us. Nevertheless the Schrödinger equation can be exactly solved only in few situations. The goal of this work was to develop programs (more than one?) in order to numerically determine the evolution of several (several kinds of? multiple?) systems. A C++ code was written to integrate the Schrödinger equation in two and three spatial dimensions and to show the real-time evolution of the wave-packet.

The method used was based on the Lie-Trotter product formula in order to write the time evolution operator as an infinite product of diagonal terms in coordinate and momentum space. This product is written as a finite number of terms by the discretization of the temporal evolution in small intervals ∆t; the error introduced in this way is of the order of (∆t)2) {you have an open parentheses here}. In order to evolve the wave function, first it is multiplied by the factor diagonal on the coordinate basis, then the wave function is written in the basis of the momentum space and is multiplied by the factor diagonal in that basis, and finally the wave function is written back into the coordinate space and multiplied by the third factor of the split evolution operator. The two bases are linked by the Fourier transform.

If a magnetic field is present, it can be shown that its effect can be accomplished with the introduction of an appropriate phase. Any substantial difficulty is {not sure what you want to say here - I guess either Substantial difficulty is introduced (sounds like a problem) or Any substantial difficulty can be introduced (sounds like a useful experimental feature if you want to consider spin interactions, but then "difficulty" sounds odd to me.)} introduced if the spin interaction is also considered. It can be shown that the spin coupling with the magnetic field can be achived with a matrix inserted into the evolution operator and considering the wave function as a two-component spinor.

Fundamental to the development of the code was the validation of the results. To do this, the numerical results have been compared with the analytic solutions for some cases where the exact solution of the Schrödinger equation is available. These include the free particle, the harmonic oscillator, the infinite-height potential barrier and the spin megnetic resonance.

In conclusion, the code can manage any initial state and any potential, showing the real-time evolution of the wave function, and visually displaying the strictly quantum behavior so distant from our daily experience.

[lucagalbu]

Originally Posted by GinaS

I'm assuming the hyphenated words are artifacts of cut and paste.

Yes.. I wrote the post in a rush and so I didn't check it

Originally Posted by GinaS

[color="Gray"]Any substantial difficulty is {not sure what you want to say here - I guess either Substantial difficulty is introduced (sounds like a problem) or Any substantial difficulty can be introduced (sounds like a useful experimental feature if you want to consider spin interactions, but then "difficulty" sounds odd to me.)}

Yes, this part it's a bit confused eheh! I meant that it is not hard at all to take into account also the spin coupling. In effect I have rewritten it in a totally different way!

Can I ask you another favor? I have modified a part of the text, if you have time can you please check it too? Thank you very much!

The evolution at time ∆t can be obtained multiplying the wave function at the initial time in the coordinate representation by the first of the two factors with which the time evolution operator has been represented. By a Fourier transform, the wave function is then written in the momentum representation and it is multiplied by the second factor of the time evolution, diagonal in that space. The inverse Fourier transform allows us to write the wave function back in the coordinate space and from there the procedure is repeated.

[GinaS]

Originally Posted by lucagalbu

Yes, this part it's a bit confused eheh! I meant that it is not hard at all to take into account also the spin coupling. In effect I have rewritten it in a totally different way!

Since you didn't say how you rewrote it, is this what you were trying to say? "If a magnetic field is present, its effects can be shown and any spin interactions can also be incorporated."

Originally Posted by lucagalbu

The evolution at time ∆t can be obtained by multiplying the wave function at the initial time in the coordinate representation by the first of the two factors with which the time evolution operator has been represented. By a Fourier transform, the wave function is written in the momentum representation, and then multiplied by the second factor of the time evolution, diagonal in that space. The inverse Fourier transform allows us to write the wave function back in the coordinate space and from there the procedure is repeated.

I've probably misunderstood this, but I think the red "in" should be "as," since you're transforming the wave function and expressing it as a momentum representation (are you transforming the raw wave function, or the product of the multiplication?).

Does this make sense to you, or is it gibberish: The inverse Fourier transform allows us to recover the wave function within coordinate space and from there the procedure is repeated.

I'm curious as to who this is meant for - it seems too technical for lay people, but your opening and closing sentences seem like they're intended for a general audience, who would be lost at the first mention of Schrödinger's equation. On the other hand, physicists would probably feel like you're talking down to them.

It might not be a bad idea at all to find a physics forum and run this by them too. At least they won't have to ask stupid questions like me, and they'll know if you've made any errors in the science.