More split op updates (#306)

* modifying julia split_op code with imaginary time evolution.

* adding largescale updates to quantum sysmtes and split_op

* adding plotting intructions to julia file.

* revisions to the bulk text

* adding images and fixing typos.

* smallscale changes to a large number of things.

* updating real time gif.

* formatting LaTeX

* fixing notation in split_op.

Author: leios

contents/split-operator_method/code/julia/split_op.jl

@@ -33,8 +33,8 @@ end
 
 mutable struct Operators
     V::Vector{Complex{Float64}}
-    PE::Vector{Complex{Float64}}
-    KE::Vector{Complex{Float64}}
+    R::Vector{Complex{Float64}}
+    K::Vector{Complex{Float64}}
     wfc::Vector{Complex{Float64}}
 
     Operators(res) = new(Vector{Complex{Float64}}(res),
@@ -49,11 +49,11 @@ function init(par::Param, voffset::Float64, wfcoffset::Float64)
     opr.V = 0.5 * (par.x - voffset).^2
     opr.wfc = exp.(-(par.x - wfcoffset).^2/2)
     if (par.im_time)
-        opr. = exp.(-0.5*par.k.^2*par.dt)
-        opr.PE = exp.(-0.5*opr.V*par.dt)
+        opr.K = exp.(-0.5*par.k.^2*par.dt)
+        opr.R = exp.(-0.5*opr.V*par.dt)
     else
-        opr. = exp.(-im*0.5*par.k.^2*par.dt)
-        opr.PE = exp.(-im*0.5*opr.V*par.dt)
+        opr.K = exp.(-im*0.5*par.k.^2*par.dt)
+        opr.R = exp.(-im*0.5*opr.V*par.dt)
     end
 
     return opr
@@ -64,19 +64,19 @@ function split_op(par::Param, opr::Operators)
 
     for i = 1:par.timesteps
         # Half-step in real space
-        opr.wfc = opr.wfc .* opr.PE
+        opr.wfc = opr.wfc .* opr.R
 
         # fft to momentum space
         opr.wfc = fft(opr.wfc)
 
         # Full step in momentum space
-        opr.wfc = opr.wfc .* opr.
+        opr.wfc = opr.wfc .* opr.K
 
         # ifft back
         opr.wfc = ifft(opr.wfc)
 
         # final half-step in real space
-        opr.wfc = opr.wfc .* opr.PE
+        opr.wfc = opr.wfc .* opr.R
 
         # density for plotting and potential
         density = abs2.(opr.wfc)

contents/split-operator_method/split-operator_method.md

@@ -103,7 +103,7 @@ Regardless, we first need to set all the initial parameters, including the initi
 
 As a note, when we generate our grid in momentum space `k`, we need to split the grid into two lines, one that is going from `0` to `-kmax` and is then discontinuous and goes from `kmax` to `0`.
 This is simply because the FFT will naturally assume that the `0` in our grid is at the left side of the simulation, so we shift k-space to match this expectation.
-Also, for this code we will be using the notation from above: `opr.R` will be the real space operators and `opr.M` will be the momentum space operators.
+Also, for this code we will be using notation to what we used above: `opr.R` will be the real space operators and `opr.K` will be the momentum space operators.
 There is another boolean value here called `im_time`, which is for imaginary time evolution.
 
 Afterwards, we turn them into operators: