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Lunar_Lander

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**"Optimizing" a Wave Function**

## Homework Statement

Consider a Hydrogen Atom, an electron in an attractive Coulomb potential of the form [itex]V(r)=-\frac{e_0^2}{4\pi\epsilon_0r}[/itex], where e

_{0}is the elementary charge. Assume the following wave function for the electron (with α>0):

[itex]\psi(r)=Ae^{-\alpha r}[/itex].

a) Determine the expectation value for the kinetic energy.

b) Determine the expectation value for the potential energy.

c) The expectation value of the total energy seems to be given by the sum of the results of a) and b). Plot the dependency of the kinetic, potential and total energy to α. Minimize the total energy in respect to α. If your calculations are correct, you will obtain the energy of the

*1s*state.

## Homework Equations

[itex]H=\frac{p^2}{2m}+V(r)[/itex]

## The Attempt at a Solution

I am quite sure what to do in c), that it is required to calculate the derivative of the total energy function to alpha and then find the minimum of that. However, I need to get some help with a) and b). I know so far that the expectation value is calculated like this:

[itex]<V>=<\psi|V|\psi>=\int_{-\infty}^{\infty} \psi\cdot V\cdot \psi~dr,[/itex]

and that in a), the Laplace operator needs to be applied in spherical coordinates. But how exactly is the integration done?

I would be grateful if someone could explain that, as I find the idea of integrating from negative to positive infinity somewhat puzzling.