# Symbolic perturbation expansion for quantum mechanics using Hellmann-Feynman derivaties

I am interested in some quantum mechanical perturbation expansion for energies. Actually I want to implement these terms $$E_n^{(k)}$$. As is stated below one can do that using CAS. I would be interested how to implement this in Mathematica. In particular I am working on this system.

I could imagine there are even some codes around in the net or published, since these are very common procedures in particle physics.

I am pretty much Mathematica beginner and I yet have no concrete idea how to do that.

I could imagine one could go for a two-step procedure first to derive the symbolic terms e.g. $$E_n^{(2)} = \frac{|V_{nk_2}|^2}{E_{nk_2}}$$ then one generates the explicit forms like

E[n,2]=Sum[Integrate[psi[n,x] V[x] psi[k2,x],{x,0,π}],{k2,1,Infinity}]/(E[n,0]-E[k2,0]


Which would be the exact correspondence of the symbolic term from above. On a more concrete level my problem is a very simple case, where I get on an explicit level

Integrate[psi[n,x] V[x] psi[k2,x],{x,0,π}],{k2,1,Infinity}] = v0 2/π Sin[n π/2] Sin[k2 π/2]


and

E[k2,0]=k2^2


But I am basically clueless at the moment how to exactly do that. Any help would be greatly appreciated.

So the first part of the question is basically a physical one:

How to generate the symbolic representations of the QM energy perturbation expansion in Mathematica? I still want to pose it here, since I suppose that this has been already implemented quite often by someone in Mathematica. This is even mentioned in the Wikipedia page:

"Both energy derivatives and state derivatives will be involved in deduction. Whenever a state derivative is encountered, resolve it by inserting the complete set of basis, then the Hellmann-Feynman theorem is applicable. Because differentiation can be calculated systematically, the series expansion approach to the perturbative corrections can be coded on computers with symbolic processing software like Mathematica. "

The second part of the question would be, how one could implement the actual expressions

E[n,2]=Sum[Integrate[psi[n,x] V[x] psi[k2,x],{x,0,π}],{k2,1,Infinity}]/(E[n,0]-E[k2,0]


given the symbolic terms:

$$E_n^{(2)} = \frac{|V_{nk_2}|^2}{E_{nk_2}}$$

• May you be more specific as to your desired output? Just as a heads up, some of your links simply go to the top of the Wikipedia page, rather than the specific area which I guess you want to focus on. Are you trying to focus on the second and third energy corrections, symbolically as is described in the Wiki page? It seems like one could easily recreate these terms. wikiwand.com/en/Perturbation_theory_(quantum_mechanics)#/… – Brad Mar 9 at 16:30
• Hi @Brad thanks for the in interest and the hint. Could it be that the links open differently on different system? In my system I end exactly were your link points to as well (however with the difference that it does so in the primary window without opening a second tab :-( so I am really confused now what to do. Very specifially I want the expressions in the yellow field (hope its yellow on everyones browser). – Rudi_Birnbaum Mar 9 at 17:58
• @Rudi_Bimbaum - I don't think so, since I used the hyperlinks in the Wiki page to direct you to the specific subsection I thought you were needed. If you follow my link above, I don't know if that big list of expressions for E is what you're looking for. If you just wanted to symbolically but automatically recreate the series of expressions for E in terms of the V's then that certainly seems doable. I'm simply unclear as to what you need; in your yellow box, you seem to want to explicitly calculate the separate values. Do you have the expressions for V[x] and all of the Psi as well? – Brad Mar 9 at 18:07
• @Brad: I have E and V as well they are very simple: $$E_{jk} = j^2 - k^2$$ and $$V_{ij}=sin(i π/2)sin(j π/2)$$. So I most strongly desire the final Sum[ ...] expressions, but I would also be very very happy about the symbolic expressions, which are exactly what you are referring to (big list of expressions for E). – Rudi_Birnbaum Mar 9 at 18:59
• What is to stop you simply defining something like E[j_,k_]:=E0*(j^2-k^2) and V[i_,j_]:= Sin[i* Pi/2]*Sin[j *Pi/2], and then substituting them into your general formula? (I am unclear what the modulus achieves in this case, and also what the general formula to actually calculate for E is). Note that subscripts in real applications can be converted to arguments in functions for MMA. – Brad Mar 9 at 19:07