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I defined a one-dimensional momentum operator $\hat{p}=-i\hbar\frac{\partial}{\partial{x}}$ in Mathematica

p = -I * h * D[#, x]&

and I want to get the kinetic energy operator $\hat{T}=\hat{p}^2/2m=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}$ from it, so

T = p^2/(2m)

but it gives the wrong result

What can I do about it? Do I have to derive the operators by hand?

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You can't just squre an operator since no multiplication of functions (operators) are defined in Mathematica. You have to write something like

T = (p@p@#)/(2 m) &

or

T = p[p[#]]/(2 m) &
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Here is a more generalized method for what you want to do:

Let's define our momentum operator as you did above:

P := -I * h * D[#, x]&

Then we can define the nth power operator in a more general way as:

T[n_] := Nest[P, #, n] &

So for example the Kinetic energy operator (which is P^2 / (2 m)) will be:

T[2] / (2 m)

And we can use it on some function f as follows

(1/ (2 m))T[2]@(E^(I k x))

Which gives:

Mathematica graphics

as expected.

Now if we want P^3 we just use T[3] etc.

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