wavepacket2D: ordering in defining the potential V(x,y) about quantumoptics.jl-examples

[david-pl]

Hi there!

Thanks for pointing that out. You are right, the example is wrong. Like you said, due to the symmetry that was quite easy to miss. Sorry about that.

You also were right about the solution to the problem: the nesting of the loops should be reversed:

V = diagonaloperator(b_comp_x, [potential(x, y) for y in ysample for x in xsample])

It's easy to check that this is correct in the example. In order to ensure that things come out correct, we can break the symmetry (that checks out even though there's an error) by simply changing the potential function to

potential(x,y) = exp(-x^2/30.0-y^2/10.)

The Gaussian potential function can separated, i.e. e^(-x^2/30-y^2/10)=e^(-x^2/30)*e^(-y^2/10). Due to this special property, the potential operator can be written as a tensor product, like so

potential_x(x) = exp(-x^2/30.0)
potential_y(y) = exp(-y^2/10.0)
Vx = potentialoperator(b_position, potential_x)
Vy = potentialoperator(b_positiony, potential_y)
Vcomp = Vx ⊗ Vy

We can then compare the Vcomp operator to V which tells us that they are identical (up to rounding errors).

It might not be as easy to check this when the potential function is non-separable. Let me know if you are still having trouble in your use-case.

In the meantime I will correct the example.

[liuzheng1990]

Right. Simply swapping the two for-loops will do. Right, usually the potential is not separable, but the "diagonaloperator" trick should work in general, as long as we are careful of this ordering. Thanks very much!

[david-pl]

Agreed. Actually I think you shouldn't be bothered with keeping this in mind. This is why I implemented the potentialoperator for composite bases. You can check out the master branch to use it.

Now you can just do

potential(x,y) = exp(-(x^2 + y^2)/30.0)
V = potentialoperator(b_comp_x, potential)

The only thing to keep in mind here is that the potential function has to accept as many arguments as there are bases constituting the composite basis b_comp_x and that the arguments must have the same order as the tensor product. So, in the example since b_comp_x = b_position ⊗ b_positiony the function has to be potential(x, y) and not potential(y, x) (of course in the wavepacket2D example this wouldn't matter due to symmetry).

[liuzheng1990]

Great. This change is definitely more friendly to our intuition. Now everything about the lazy operators should be easy to use. I'll go on playing with them. Thanks very much, David!

[david-pl]

The updated example and the new API docs for the potentialoperator function are now on-line in the documentation so I'm closing this.