Decomposition of total Wavefunction for N electron problem

[askhetan]

A. Given the exact locations of N electrons - can we find out the total wave function and the individual wave functions by some law of nature?

B. Given the exact total wave function - can we find out the coordinates of each electron and the indiviual wave function of each electron by some law of nature?

C. Given the exact indiviual electron wavefunction AND the exact location of N electrons - can we find out the total wavefunction by some law of nature?

...What is the total wavefunction representative of anyways? The wavefunction of an individual electron named "Mr.X" when squared, gives us the probability density of finding "Mr. X" (please don't come here with other contesting interpretations, that's not what this question is about). BUT what is the implied meaning of the total wavefunction? whose probability density does it represent?? surely not all electrons, that doesn't make a meaningful quantity - "probability density of finding all electrons" - where?.

Also, can someone please direct me to a source where they explain how CI method can lead to an exact solution of SE in principle. I mean I cannot understand what the unoccupied states of a single electron system have got to do with occupied ground states of a N electron System

 

[askhetan]

To be precise - is there a direct explicit physics of how the individual electron wavefunctions combine to give the total wavefunction (whatever that quantity means) ?

 

[DrDu]

To be precise - is there a direct explicit physics of how the individual electron wavefunctions combine to give the total wavefunction (whatever that quantity means) ?

This is not possible as there is no such thing as an "individual electron wavefunction", only the total collective wavefunction.

 

[askhetan]

Thanks DrDru. That is also what my undetstanding was. this now makes it more meaningful for me to ask some more questions:


1. Are we not conceeding here that a formalism// a law/ an equation that distinguishes one particle from another is missing? Please note here, I know what the indistinguishability of fermions means. What I am pointing is that we cannot tell apart the probability distribution of one electron from the other. Effectively we're saying that the total wavefunction now represents the spatial probability distribution of a "lump" of N electrons. And this brings me to my third question again as follows:

2. The Kohn-Sham orbitals and the D(ensity)O(f)S(tate) calculations in DFT, what do they mean? If there are no individual electron wavefunctions then there cannot be electron orbitals/shells.. right?

3. In Configuration interaction, the additional slater determinants are composed of wavefunctions of electrons in unoccupied states. What "in a physical sense" is the relation of those unoccupied states of single electron systems to the total wavefunction which corresponds to the ground state of this multi-electron system

 

[DrDu]

2. Kohn-Sham has at the end some Vodoo component. That makes it so hard to develop correlation functionals.
3. If you are only interested in expectation values of one-electron operators, then it is possible to derive a one-density matrix from the full particle function which allows calculation of all these expectation values. The one-density matrix can be diagonalized in terms of the so-called "natural" orbitals. In an interacting system, infinitely many of these natural orbitals have a non-zero weight ("occupation")

 

[askhetan]

Alright, so the formalism is there. It would be very kind of you to please direct me to some textbook/topicname/source where i can study about this formalism of decomposing the wave function into one electron natural orbitals.

Please correct me if my understanding is wrong - So in CI method, we try to find the best complete wave function possible (depending in the number of slater determinants considered) and from that we have find out that the electrons have fractional occupation numbers in these "natural orbitals" which can be themselves derived from the full wavefunction.

Thanks a lot!

 

[DrDu]

Maybe this article is helpful:
http://rmp.aps.org/abstract/RMP/v44/i3/p451_1
Obviously there are many alternatives to natural orbital analysis. Nevertheless they somehow are the best approximation to one particle properties in manybody problems.