Variational Estimate of Hydrogen Atom Ground State Energy

[latentcorpse]

Obtain a variational estimate of the ground state energy of the hydrogen atom by taking as a trial function $\psi_T(r) = \text{exp } \left( - \alpha r^2 \right)$

How does your result compare with the exact result?

You may assume that

$\int_0^\infty \text{exp } \left( - b r^2 \right) dr = \frac{1}{2} \sqrt{ \frac{\pi}{b}}$

and that $1 \text{Ry} = \left( \frac{e^2}{4 \pi \epsilon_0} \right)^2 \frac{m}{2 \hbar^2}$

so i obviously need to minimise

$E[r] = \frac{ \langle \psi \vline \hat{V} \vline \psi \rangle}{ \langle \psi \vline \psi \rangle}$

i think I'm getting the wrong answer because my V is wrong.

so i want a coulomb potential between a proton and a electron surely?

$V=\frac{-e^2}{4 \pi \epsilon_0 r^2}$
clearly I'm going wrong since i have this $r^{-2}$ term that i don't know how to integrate (the only r dependence in hte given integral is in the exponent) and also i haven't used the rydberg thing (although i guess that might not be needed until the end of the question).

thanks for any help.

 

[kuruman]

You need to minimize the expectation value of the whole Hamiltonian, not just the potential energy.

<H>=<T>+<V>

where <T> is the expectation value of the kinetic energy term, so you have to do two integrals. Furthermore, the Coulomb potential goes as 1/r not 1/r².

 

[latentcorpse]

okay but i still get integrals with r terms outside the exponential

$\langle \psi | \hat{H} | \psi \rangle = - \int_V \frac{\hbar^2}{m} \text{exp} ( - 2 \alpha r^2) ( 2 \alpha r^2 - \alpha ) dV - \int_V \frac{e^2}{4 \pi \epsilon_0 r} \text{exp} ( - 2 \alpha r^2 )$

and the normalisation on the denominator works out at $\sqrt{2} \pi^{\frac{3}{2}}$

any idea on how to proceed? thanks.

 

[kuruman]

Are you saying you don't know how to do the integrals? What is dV and what are your limits of integration?

 

[latentcorpse]

$dV=r^2 \sin{\theta} dr d\theta d\phi$

so we can pull out a $4 \pi$ from the theta and phi integrals. then the r integral is from 0 to infinity since dV is all space.

so, for example how do we integrate

$\int_0^\infty -\frac{4 \pi \hbar^2}{m} 2 \alpha r^2 \text{exp} ( - 2 \alpha r^2) dr$

the given formula doesn't apply here, does it?

 

[kuruman]

No it does not. However, before we go into that, can you write the complete expression for the integral giving the expectation value for the kinetic energy?

 

[latentcorpse]

ok.

$\langle \psi | \hat{T} | \psi \rangle = \int_0^\infty \int_0^\pi \int_0^{2 \pi} e^{- \alpha r^2} -\frac{\hbar^2}{2m} (2 \alpha r^2 - \alpha) e^{- \alpha r^2} r^2 \sin{\theta} dr d \theta d \phi = -\frac{\hbar^2}{2m} \int_0^\infty \int_0^\pi \int_0^{2 \pi} r^2 (2 \alpha r^2 - \alpha) e^{- 2 \alpha r^2} \sin{\theta} dr d \theta d \phi$

i can't see why this is wrong...

 

[kuruman]

You are missing the normalization constant, but that can always be added later.
I think the radial Laplacian term is incorrect. Can you show how you derived it?