Why do collapse functions change form in master equations?

[Agrippa]

In the GRW spontaneous collapse model (for example) the wave-function evolves by linear Schrödinger equation, except, at random times, wave-function experiences a jump of the form:

\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 system state vector prior to jump 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 at random times, position wave-function is multiplied by a Gaussian (with width $r_c$; $q_n$ is position operator for nth particle, and x is where collapse occurs); so far so good. But now we move to 1-particle GRW master equation for density matrix:

\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 spontaneous collapse. In position representation:

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

where $\lambda$ represents temporal distribution of collapses such that probability per second for collapse is $10^{-16}sec^{-1}.$

Clearly, form of collapse 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 $\frac{1}{(\pi r^2_c)^{3/4}}e^{-(q_n - x)^2 / 2r^2_c}$ ?

 

[Agrippa]

The question, I think, concerns the difference between operators on the wave-function and operators on density matrices. Why does the same operator change mathematical form depending on whether we are applying it to the wave function of state S or the density matrix for that same state S? What are the general principles that tell us how to transform the operators?