Operators change form for density matrix equations?

[Agrippa]

Imagine applying an operator to a wave-function:


$\psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||}$


Where $\psi _t(x_1, x_2, ..., x_n)$ is initial system state vector, denominator is normalization factor, and L_n(x) is a linear operator equal to:


$L_n(x) = \frac{1}{(\pi r^2_c)^{3/4}}e^{-(q_n - x)^2 / 2r^2_c}$


So, position wave-function is multiplied by a Gaussian (with width $r_c$; $q_n$ is position operator for nth particle, and x is random spatial variable where Gaussian multiplication is centred).

When this exact same equation is presented in a master equation for density matrix we get:


$\frac{d}{dt}\rho(t) = -\frac{i}{\hbar}[H, \rho(t)] - T[\rho(t)]$


where H is standard quantum Hamiltonian and T[] represents effect of the operator. In position representation:


$<x|T[\rho(t)]|y> = [1 - e^{-(x - y)^2 / 4r^2_c}]<x|T[\rho(t)]|y>$


Clearly, form of Gaussian function has changed, but why? Standard presentations (e.g. pp.30-33) never explain the change.
Is there anyone out there who knows the math well enough to be able to explain why all the changes occur e.g. why do we replace the initial fraction with "1 - "? And why replace $2r^2_c$ with $4r^2_c$?


What would go wrong if we simply replaced $1 - e^{-(x - y)^2 / 4r^2_c}$ with $\frac{1}{(\pi r^2_c)^{3/4}}e^{-(q_n - x)^2 / 2r^2_c}$ ?

 

[jvicens]

it is important to understand the underlying mathematical principles and concepts behind any equation or formula. In this case, it is important to understand the concept of the density matrix and its relation to the wave-function.

The density matrix is a mathematical tool used in quantum mechanics to describe the state of a quantum system. It contains information about the probabilities of different states that the system can be in. The density matrix is defined as the outer product of the wave-function with itself:

\rho (t) = |\psi_t><\psi_t|

This means that the density matrix is a function of the wave-function, and any operator acting on the wave-function will also affect the density matrix.

In the given equation, the operator L_n(x) is acting on the wave-function, and therefore, it will also affect the density matrix. The reason for the changes in the Gaussian function is because the operator L_n(x) is a position operator, and as we know, position and momentum are conjugate variables in quantum mechanics. This means that when one is measured with high precision, the other becomes uncertain.

In the given equation, the Gaussian function is modified to take into account this uncertainty. The factor of "1 - " is a normalization factor, which ensures that the density matrix remains traceless (i.e. the sum of the diagonal elements of the matrix is equal to 1). This factor is necessary because the operator L_n(x) introduces additional uncertainty in the position of the particle, and hence, the density matrix needs to be adjusted accordingly.

Similarly, the factor of 4 in the denominator of the exponential term is also a result of the uncertainty in position. The uncertainty in position is related to the width of the Gaussian function, which is given by 2r_c. This width is then squared to account for the uncertainty in both the position of the particle and the random spatial variable x.

If we simply replaced the modified Gaussian function with the original one, it would not accurately represent the uncertainty in position introduced by the operator L_n(x). This could lead to incorrect predictions or results when solving the master equation.

In conclusion, the changes in the Gaussian function in the master equation are necessary to accurately represent the uncertainty in position introduced by the operator acting on the wave-function. It is important to understand the mathematical reasoning behind these changes in order to correctly use and interpret the master equation in quantum mechanics.