What is the difference between electron density \rho(r) and probability density |\Psi(r1,..rN)|^2? Can someone explain?
How comes \rho(r) integrates to N and |\Psi|^2 integrates to 1? What does it mean?
Do you mean \rho(r) integrates to N?
Because electron density is the sum over N electrons of the probability density of finding an electron at r. The probability integrates to 1, that would make the integration of \rho(r) = N
Yes, I meant \rho(r) integrates to N. I corrected the typo. Your explanation sounds plausible, let me improve it a little:
Because the electron density is defined as the probability density of finding an (any) electron at r, and since we have N electrons, the probability is N over all space.
How can we prove it? Hint: write down the definition of \rho(r) in terms of wave function \Psi(r1,r2...rN).
I guess the process to prove it is similar to that on page138 of Modern Quantum Chemistry by A Szabo and N Ostlund, I have wrote the proof down. Please find it in the attachment.
8168.answer.pdf
I think this is the exercise 3.11(P139) in Ostlund's Quantum Chemistry book. because the density operator \rho(r) is an one-electron operator, its integeral is similar to that of the first item of Hamiltonian O1. Therefore we can use the equation of case 1 in Table 2.5(P72) and replace h with \delta(r-ri). Then it is easy to derive the equation(3.142)P(138).
Great! But what is the relationship between \rho(r) and \psi(r1,r2,...rN)? Can you write down the equation?
\rho(r) = < \psi(r1,r2...rN) | sum_i{delta(r-ri)} | \psi(r1,r2,...rN) > , is it right?
Yes, but I want to have the formula without the density operator. Hint: \rho(r) is the integral of \psi(r1,r2,...rN) ...
rho(r) = N *integral( dr2 dr3 dr4 ... drN | psi(r1,r2,r3,r4,...rN)|**2) leaving out any one r ( dr1 here ), and multiply by N so that when we integrate over the remaining r we get N
-> intergral (rho(r) dr) = N*integral( |psi(r)|**2 dr) = N ?
Yes.
Yes, the answer seems right. But I still feel a little puzzled about this expression. Please refer to my attachment. I really want to know what the meaning of the part2 and part3 of the equation in my attachment is, or, In other words, why we need part2 and part3?
7585.confusion.pdf