EQUATIONS (CO)INVENTED
by Lajos Diósi
- 1977: Binomial Moments from Counters of Varying Efficiency [
1 ,
2]
Nucl. Instr. Meth. 140, 533-636 (1977): Eq.(19)
$$
B_k=\left\langle\sum_{\left({\bar n}\atop k\right)}
W_{i_1}W_{i_2}\dots W_{i_k}\right\rangle
$$
- 1984: Schrödinger-Newton Equation, Ground State Soliton Widths [
9]
Phys. Lett. A105, 199-202 (1984): Eqs.(5,15,18)
$$
i\hbar{\partial\psi(x,t)\over\partial t}=
-\frac{\hbar^2}{2M}\Delta\psi(x,t)-
\left(GM^2\int{\vert\psi(x',t)\vert^2\over\vert x'-x\vert}d^3x'\right)\psi(x,t)
$$
$$
a_0\approx\frac{\hbar^2}{GM^3} \mbox{(pointlike mass)};~~
a_0^{(R)}\approx a_0^{1/4}R^{3/4} \mbox{(spherical mass)}
$$
- 1985: Jump Quantum Trajectory of a Simple Open System [
15]
Phys. Lett. A112, 288-292 (1985); Eqs.(22-24)
$$
{d\hat\rho\over dt}=
-{i\over\hbar}[\hat H,\hat\rho]-{\gamma\over 2}[\hat q,[\hat q,\hat\rho]]
$$
$$
\frac{d\psi}{dt}=-{i\over\hbar}\hat H\psi
-\frac{\gamma}{2}[(\hat q -\langle\hat q\rangle)^2-(\Delta q)^2]\psi
$$
$$
\psi\rightarrow\frac{\hat q -\langle\hat q\rangle}{\Delta q}\psi~~\mbox{at rate}~~\gamma(\Delta q)^2
$$
- 1987: Master Equation of Gravity-Related Decoherence [
25,
33 ]
Phys. Lett. A120, 377-381 (1987): Eq.(11)
$$
{d\hat\rho\over dt}=-{i\over\hbar}[\hat H,\hat\rho]
-{G\over2\hbar}\int\int{d^3rd^3r'\over\vert r-r'\vert}
[\hat f(r),[\hat f(r'),\hat\rho]]
$$
- 1988: Stochastic Master Equation of Continuous Quantum Measurement [
31,
32 ]
Phys. Lett. A129, 419-423 (1988): Eqs.(2.6-7)
$$
{d\hat\rho\over dt}=
-{i\over\hbar}[\hat H,\hat\rho]-{\gamma\over 2}[\hat q,[\hat q,\hat\rho]]
+\sqrt{\gamma}w\{\hat q -\langle\hat q\rangle,\hat\rho\}
$$
$$
\bar q = \langle\hat q\rangle + {1\over 2\sqrt{\gamma}}w
$$
- 1990: Relativistic Master Equation of QED Electrons-Positrons [
36]
Found. Phys. 20, 63-70 (1990): Eq. (7)
$$
\hat\rho(t)=T\exp\left\{\frac{i}{2}\!\!\int\!\!\!\!\!\int_{x_0,y_0\langle t}\!\!\!\!\!\!\!\!\!\!\!dxdy\;
\begin{array}{c}\left[D^{(F)}(x-y)\hat J_+(x)\hat J_+(y)
+D^{(\bar{F})}(x-y)\hat J_-(x)\hat J_-(y)\\
-D^{(+)}(x-y)\hat J_+(x)\hat J_-(y)
-D^{(-)}(x-y)\hat J_-(x)\hat J_+(y)\right]\end{array}
\right\}\hat\rho(-\infty)
$$
- 1995: Master Equation of Quantum Brownian Particle in Gas [
49]
Europhys. Lett. 30, 63-68 (1995): Eqs.(19-20)
$$
{d\hat\rho\over dt}=
n_0\int dE d\Omega_i d\Omega_f k^2 {d\sigma(\theta,E)\over d\Omega_f}
\rho^{\cal E}(k_i)\times
\Bigl(\hat V_{{\bf k}_f{\bf k}_i}\hat\rho\hat V_{{\bf k}_f{\bf k}_i}^\dagger
-{1\over2}\{\hat V_{{\bf k}_f{\bf k}_i}^\dagger\hat V_{{\bf k}_f{\bf k}_i},
\hat\rho\}\Bigr)
$$
$$
\hat V_{{\bf k}_f{\bf k}_i}=\left(1-{\beta\over2M}{\bf k}_i{\bf\hat p}\right)
\exp[-i{\bf k}_{fi}{\bf\hat q}]
$$
- 1997: Non-Markovian Stochastic Schrödinger Equation of Open Systems [
56,
57 ,
61 ,
64 ]
Phys. Lett. 235A, 569-573 (1997): Eqs.(2-4)
$$
{d\psi_z(t)\over dt}=-i\hat H\psi_z(t)+i\hat q z(t)\psi_z(t)
-i\hat q\int_0^t ds\alpha(t,s){\delta\psi_z(t)
\over\delta z(s)}
$$
$$
M[z^*(t)z(s)]=\alpha(t,s);~M[z(t)z(s)]=0;~M[z(t)]=0
$$
$$
\hat H_{tot}=\hat H - \hat q \hat F_{res};~~~~
\alpha(t,s)=\langle\hat F(t)\hat F(s)\rangle_{res}
$$
- 2001: One-to-one parametrization of all diffusive quantum trajectories [
70]
Chem. Phys. 268, 91-104 (2001): Eqs.(4.8-12)
$$
{d(\vert\psi\rangle\langle\psi\vert)\over dt}={d\hat P\over dt}=
-{i\over\hbar}[\hat H,\hat P]+\hat c_k\hat P\hat c_k^\dagger-\frac{1}{2}\{\hat c_k^\dagger\hat c_k,\hat P\}
+[w_k^\ast(\hat c_k-\langle\hat c_k\rangle)\hat P + \mathrm{h.c.}]
$$
$$
M[w_j^\ast(t)w_k(s)]=\delta_{jk}\delta(t-s),~~~~
M[w_j(t)w_k(s)]=u_{jk}\delta(t-s),~~~\Vert u \Vert\leq1
$$
- 2006: Stochastic Master Equation of Continuous Quantum State Estimation [
84]
J.Phys. A39, L575-L581 (2006): Eq.(8) {cf.: γ=4γ(1988)}
$$
{d\hat\rho^e\over dt}=
-{i\over\hbar}[\hat H,\hat\rho^e]-{\gamma\over 8}[\hat q,[\hat q,\hat\rho^e]]
+{\gamma\over 2}(\bar q -\langle\hat q\rangle^e)\{\hat q -\langle\hat q\rangle^e,\hat\rho^e\}
$$
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