I would like to create a notebook for calculations in supersymmetric quantum mechanics. The basic building blocks are two functions (or maybe better: operators):

A := (W[#1] #2 + D[#2, #1]) &;
B := (W[#1] #2 -D[#2, #1]) &;

With these definitions you can apply these operators to a pair of a variable x and a function f[x] like A[x,f[x]] and one obtains f[x] W[x] + Derivative[1][f][x].

Now, these functions A and B are embedded into matrices:

Q1 := ({{0, 0}, {A[#1, #2], 0}}) &;
Q2 := ({{0, B[#1, #2]}, {0, 0}}) &;

Again, it is possible to apply these matrices to a variable x and a function f[x] like Q[x,f[x]] and one obtains

{{0, 0}, {f[x] W[x] + Derivative[1][f][x], 0}}

or in traditional form enter image description here

I wonder, if there is a possibility to multiply the matrices Q1 and Q2 in such a way that the result is again a function that expects the two arguments [x,f[x]].

The trouble is, that the internal matrix operations have to distinguish between multiplication and addition of ordinary numbers and functions. I succeeded to implement addition and multiplication that is sensitive to the nature of the operands on the level of A and B,eg.

mul[a_, b_] := If[NumberQ[a] && NumberQ[b], a*b,
If[NumberQ[a], Through[a*b[#1, #2], 0] &,
If[NumberQ[b], Through[b*a[#1, #2], 0] &,
 a[#1, b[#1, #2]] &


add[a_, b_] :=If[NumberQ[a] && NumberQ[b], a + b,
 If[NumberQ[a], a + b[#1, #2] &,
  If[NumberQ[b], b + a[#1, #2] &,
   Through[a[#1, #2] + b[#1, #2], 0] &]]]

I have no idea, how to deal with the matrix multiplication. For instance, how can I get the (2,1) component of Q as a function. If I write Q[[1,2,1]], I get #2 W[#1] but I would need W[#1] #2 + D[#2,#1]&?

  • $\begingroup$ A = W[#1] #2 + HoldForm@D[#2, #1] & ? $\endgroup$
    – Fortsaint
    Jun 17 '19 at 21:20

If you're looking for a generalized matrix product, then Inner will do the trick. This snippet performs a matrix product, where multiplication is replaced with mul and addition with add:


However, Q1 and Q2 are written as functions, rather than matrices with function-valued elements. Using them as written, the matrix product would be implemented like this:

Q = Q1[#1,#2].Q2[#1,#2]&

Then a particular component could be reached with Q[#1,#2][[row,col]]&.

  • $\begingroup$ Thank you so far, but I need the combination of both ideas. Q1[#1,#2].Q2[#1,#2]& works fine, however, it just multiplies the functions of the components of the matrices. In the customized mul, the functions are composed A[#1,A[#1,#2]]&. So, I tried Mmul[a_,b_]:=Inner[mul,a[#1,#2],b[#1,#2],add]&, which doesn't work however. $\endgroup$
    – p6majo
    Jun 18 '19 at 9:08
  • $\begingroup$ Also, just Mmul[a_, b_] :=Inner[mul, a, b, add] does not work. What is the correct syntax for this situation? $\endgroup$
    – p6majo
    Jun 18 '19 at 9:42
  • $\begingroup$ The reason Inner won't work is because Q isn't actually a matrix. Q[x,f[x]] is a matrix with numeric elements. If you define them like Q1 := ({{0, 0}, {A, 0}}), then Inner will work. $\endgroup$
    – Woofmao
    Jun 18 '19 at 13:44
  • $\begingroup$ Fine, this works. Now I can access elements of Q=Inner[mul,Q1,Q2,Plus] via Through[Q[[2, 2]][x, f[x]]] for example. Is there a way to perform this for the full matrix in one step? Just Through[Q[x,f[x]]] doesn't work, since the matrix itself is not a function now but only its components. $\endgroup$
    – p6majo
    Jun 18 '19 at 20:07
  • $\begingroup$ You can use Map[#[x,f]&, Q, {2}] to call the function for each cell of the matrix. $\endgroup$
    – Woofmao
    Jun 19 '19 at 13:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.