- #1

Dario56

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Quantum states can also be described by the density matrices (operators). For a pure state, density matrix is defined by the outer product of the system's wavefunction, ##\Psi##: $$ \gamma = |\Psi\rangle \langle\Psi| \tag{1} $$

In the textbook: Density Functional Theory; Parr, Yao; Chapter 2.2 Density Operators, it is written that any element of this matrix is obtained as follows: $$ \gamma(x_1^{'}x_2^{'}\dots x_n^{'} , x_1x_2\dots x_n) = \Psi(x_1^{'}x_2^{'}\dots x_n^{'})\Psi^*(x_1x_2\dots x_n) \tag{2} $$

where ##x_1^{'}x_2^{'}\dots x_n^{'}## denote that if we set ##x_i^{'} = x_i## for every ##i## than we get diagonal element of the density matrix. Namely, ##\Psi(x_1x_2\dots x_n) \Psi^*(x_1x_2\dots x_n)##

I have two questions:

1. What do variables such as ##x_1^{'}x_2^{'}\dots x_n^{'}## actually represent and what it is their difference compared to ## x_1x_2\dots x_n ##? In the textbook, no good explanation is given and I couldn't find any source which explains it better.

2. Since density matrix is defined as the outer product of the wavefunction ##\Psi##, what do variables ##x_1^{'}x_2^{'}\dots x_n^{'}## and ##x_1x_2\dots x_n## have to do with the entry values of the matrix? If we want to calculate the outer product, we need to represent wavefunction ##|\Psi\rangle## in a certain basis set with coefficients forming a column vector which we use to calculate the outer product. Therefore, entries of the density matrix have to do with the basis set we use to represent the wavefunction ##|\Psi\rangle## and not with the values of ##x_1^{'}x_2^{'}\dots x_n^{'}## and ##x_1x_2\dots x_n##.