Expectation Value of a Potential Using Radial Solution of Hydrogen Atom

Question

I am trying to input $R_{nl}$, which is the radial solution of the hydrogen atom and I would like to obtain an expectation value of particular potential. This is my code:

R[n_, l_, r_] := 
  Sqrt[4*Z^3*((n - l - 1)!/(Subscript[a, \[Mu]]^3*n^4*(n + l)!))]*
   Exp[(-Z)*(r/(n*Subscript[a, \[Mu]]))]*(2*Z*(r/(n*Subscript[a, \[Mu]])))^
    l*LaguerreL[n - l - 1, 2*l + 1, 2*Z*(r/(n*Subscript[a, \[Mu]]))]

This is the integration I would like to perform:

 FullSimplify[Integrate[r^2*R[n, l, r]^2*(Exp[(-s)*(r - R) - 1]/r), 
   {r, R, Infinity}], Assumptions -> 
   {n, l, R, Z, s, Subscript[a, \[Mu]]}*\[Epsilon]*Positive]

Mathematica only returns the input for me. Any suggestions that I can fix this?

Answer 1

Note that your code uses $R$ as both a function and a constant in the potential and in the limits of integration. Also, the *ϵ*Positive doesn't do what you want it to do. Further, MMA usually is not able to perform symbolic integration with functions like LaguerreL. We must usually give numeric values for $n, l$.

For the ground state, we could write

ClearAll["Global`*"]

ψr[n_, l_, r_] := 
 Sqrt[4*Z^3*((n - l - 1)!/(Subscript[a, μ]^3*n^4*(n + l)!))]*
  Exp[(-Z)*(r/(n*Subscript[a, μ]))]*(2*
     Z*(r/(n*Subscript[a, μ])))^l*
  LaguerreL[n - l - 1, 2*l + 1, 2*Z*(r/(n*Subscript[a, μ]))]

With[{n = 1, l = 0},
 Integrate[r^2*ψr[n, l, r]^2*(Exp[(-s)*(r - R) - 1]/r),
  {r, R, Infinity}, Assumptions -> {0 < {Subscript[a, μ], Z, s}}]
 ]

$$\frac{4 Z^3 e^{-\frac{2 R Z}{a_{\mu }}-1} \left(a_{\mu } (R s+1)+2 R Z\right)}{a_{\mu }^2 \left(s a_{\mu }+2 Z\right){}^2}$$