Symbolic Quantum Mechanics

[Anurag Hota]

Hello,

I am Anurag , a 3rd year undergrad from BITS Pilani,India . I am pursuing a dual degree M.Sc Physics and B.E Computer Science there. I wasn't aware of the sympy module till this year but after using it, I found it very effective.Being in 3rd year, I have covered all major Physics courses including Quantum Mechanics 1/2 , Computational Physics, Nuclear, Atomic/Molecular etc. So I have a fair understanding of the quantum theories. I would like to contribute to the open source software and be part of GSoc '14.

The ideas/topics i would like to cover -

1. Abstract Dirac Notation -

   With all the basic notations present such as ket,bra, dagger etc. , we can move on to make it more dynamic instead of having something specific.

   I would like to implement the Perturbation Theory ( time dependent and independent) as well as the Scattering Theory(elastic/inelastic) and s-wave scattering.

2. Specific Operators-

    I tried to look for the specific operators but didn't find one. So we can move on to develop/implement the specific operators like Angular Momentum Operator(L),Lx,Ly,Lz as well       as Spin Operator(S) , Total Momentum Operator (J) and the eigenvalues associated with it/ how it affects the state of the wavefunction.

    Basically we can introduce the ladder operator and see the allowed state transitions. 

3. Stochastic Equations-

    We have generally solved the deterministic equations throughout. We can introduce randomness and solve the stochastic equations using ido operator. This might help in more generalized real time analysis which we can further use in Quantum Finance.

[Anurag Hota]

There are a few problems -

   The Operator has been represented in such a way that for it to act on bra/ket we have to use (*). 

    Instead we can define Operator as a function where we pass the bra/ket .This function can store the values (for calculation) and then we can print the notation too in its symbolic form.

e.g. def Operator( n,m):   #pass the states

     Next problem is when we cant represent an Operator with a superscript form( Atleast I havent figured out yet)

    e.g   right = (Bra('m')*(H**2)*Ket('n'))/(E**n - E**m)

<m|(H)2|n>−Em+En

 represents it as (H) to the power 2(actually evaluates) instead of superscript which a Symbol does( In this case I took 'E' as symbol just for representation)

If I take 'H' as a symbol, it takes 'H' out of the bra/ket notation( assumes as constant)

So we can define an Operator as:

def Operator(Name, n, m): #Name as a symbolic notation of Operator, n= subscript, m = superscript)

We store them for integration of the wavefunction as well as finally print it out in the actual notation form.

@Brian @Ondrej Sir, please elaborate on how to go forward with this.