How to create numerically and symbolically creation/annihilation operator?

Hi everyone, I am attempting to create creation and annihilation operator used in 2nd quantization of quantum mechanics. I want to do both numerically and symbolically. Those who are not familiar here is a simple math to illustrate idea:

n= (cd)(c)

where (cd) is the creation operator or c dagger and (c) is annihilation operator or c basic rules

[(cd),(c)] = (cd)(c)+(c)(cd) = 1

[(cd),(cd)] = (cd)(cd)+(cd)(cd) = 0

[(c),(c)] = (c)(c)+(c)(c) = 0

Below is my listing for numerical approach:

 {clear all;
close all;
clc;
matdimension = 3;
tempvector   = 0:1:matdimension
tempvector = sqrt(tempvector)
tempmatrix = diag(tempvector);
creation = circshift(tempmatrix,-1);
annihilation = creation';
% creation = circshift(diag(sqrt(0:1:mat_dim)),-1);
H1=creation*annihilation;
H2=annihilation*creation;
H3= H1 + H2
H4= H2 + H1;
H5= creation*creation + creation*creation;}

And the output is :

 {tempvector = 0   1   2   3
  tempvector = 0.00000   1.00000   1.41421   1.73205
  H3 =
   1.00000   0.00000   0.00000   0.00000
   0.00000   3.00000   0.00000   0.00000
   0.00000   0.00000   5.00000   0.00000
   0.00000   0.00000   0.00000   3.00000}

1) Obviously something is wrong with my code since answer H3 should be equal to 1 which is not. What I am missing here?

2)Is it possible to create new math rules in matlab? ( I have commutation in mind [A,B]= ) I want to create the above code but do the symbolic calculation as well. If yes, is there a simple example to illustrate the method?

3)I managed to copy dagger symbol and paste it on matlab script, however the script will not compile. Is there a work around this?

Answer 1

[Arif Ullah Khan]

Please read this article , this might be helpful https://arxiv.org/pdf/1603.04167.pdf