I have some problems with the non-dimensionalization of the Hamiltonian of motion in a Coulomb field.The Hamiltonian has a following form:
$$H=-\frac{\hbar^2}{2\mu^*} \Delta_r-\frac{e^2}{\epsilon_0 r}$$
where $\hbar$, $\mu$, $e$, $\epsilon_0$ are Planck's constant, mass, charge and dielectric constant respectively.
I would like to dimensionless the Hamiltonian two ways:
1)$\quad$$E_{01}=\frac{\mu^* e^4}{{\epsilon_0}^2 \hbar^2}$, $\quad$$a_1=\frac{\hbar^2 \epsilon_0}{\mu^* e^2}$, $\quad$$E_{01}=\frac{e^2}{a_1\epsilon_0}$
Now I divided the Hamiltonian by $E_{01}$ and after some simple expression transformations I got:
$$\tilde{H}=-\frac{1}{2} \Delta_\tilde{r}-\frac{1}{ \tilde{r}}$$ where $\tilde{r}=\frac{r}{a_1}$
2)$\quad$$E_{02}=\frac{\mu e^4}{{\epsilon_0}^2 \hbar^2}$, $\quad$$a_2=\frac{\hbar^2 \epsilon_0}{\mu e^2}$, $\quad$$E_{02}=\frac{e^2}{a_2\epsilon_0}$
I again divided the Hamiltonian by $E_{02}$ and after some simple expression transformations I got:
$$\tilde{\tilde{H}}=-\frac{1}{2} \frac{\mu}{\mu^*}\Delta_\tilde{\tilde{r}}-\frac{1}{ \tilde{\tilde{r}}}$$ where $\tilde{\tilde{r}}=\frac{r}{a_2}$
It's known from the theory the eigen wave functions and eigen energies for this problem. Consider ground state, it's have following wave function: $\psi=2e^{-r}$
Now I will give the code in Wolfram Mathematica, where try to calculate energy of ground state:
In the code I use symbol $\mu bar$ instead of $\mu^*$.
ClearAll["Global`*"]hbar = 1054571/1000000*10^(-27);(*Planck constant*)eV = 1602176/1000000*10^(-12);ee = 4803204/1000000*10^(-10);(*e charge*)meV = 10^(-3)*eV;\[Epsilon]0 = 30;(*dielectric constatnt*)\[Mu] = 9.277*^-29;\[Mu]bar = 1.261*^-28;E01 = (\[Mu]bar*ee^4)/(\[Epsilon]0^2*hbar^2);E02 = (\[Mu]*ee^4)/(\[Epsilon]0^2*hbar^2);Psi[r_] := 2 E^-r;(*dimensionless Hamiltonian 1*)(*kinetic energy*)KK1 = -(1/2)* NIntegrate[ Psi[r]*Laplacian[Psi[r], {r, \[Theta], \[Phi]}, "Spherical"]* r^2, {r, 0, \[Infinity]}];(*potential energy*)PP1 = Integrate[Psi[r]*(-1/r)*Psi[r]*r^2, {r, 0, \[Infinity]}];EE1 = KK1 + PP1(*energy in dimensionless units*)Out[620]= -0.5EEE1 = EE1*E01/meV (*energy in meV*)Out[621]= -2.09269(*dimensionless Hamiltonian 2*)(*kinetic energy*)KK2 = -(1/2)*\[Mu]/\[Mu]bar* NIntegrate[ Psi[r]*Laplacian[Psi[r], {r, \[Theta], \[Phi]}, "Spherical"]* r^2, {r, 0, \[Infinity]}];(*potential energy*)PP2 = Integrate[Psi[r]*(-1/r)*Psi[r]*r^2, {r, 0, \[Infinity]}];EE2 = KK2 + PP2(*energy in dimensionless units*)Out[627]= -0.632157EEE2 = EE2*E02/meV (*energy in meV*)Out[628]= -1.94649Please explain to me what am I doing wrong? The energies EEE1 and EEE2 should be the same, because the results should not depend on non-dimensionalization.
