# Projection and number operator

asked by Saverio Ricci (2020/04/16 03:33)

Dear all, I want to calculate the projection of whatever wavefunction to a specific target. For example: i have a system described by an hamiltonian and calculate the eigenstates. Than I want to project one of such states in real orbitals of d shell (like the d_xy or d_z^2). how is it possible to do it? I tried to create a specific d orbitals with the command NewWavefunction but it seems that this kind of orbital and those calculated as eigenstates are not compatible.

Thank you for the help

## Answers

You can define a creation operator, for example, a^+_xy and annihilation operator a_xy, and act them on your wavefunction via a^+_xy*a_xy*wavefunction. You shall get what you want.

Hi Ricci,

A wave-function in Quanty describes a state with $N$ electrons. It is the many particle wave function of the entire system. If you define a new wave function with one electron and calculate $\langle \psi_1 | \psi_N \rangle$, with $\psi_1$ a one particle wave function and $\psi_N$ an $N$ particle function that the results is undefined as these two states belong to a different Hilbert space. The dot product integrates over all electron coordinates wich is not the same for the two functions.

If you want to know the occupation of an orbital you can calculate the one particle density operator. This you generate, as Chang-Yang states by $a^{\dagger}_{\tau} a_{\tau'}$ with $\tau$ and $\tau'$ the orbitals of interest.

Best wishes, Maurits

Thank you so much!!!! It worked