Helium atom, variation method and virial theorem

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• Aleolomorfo
In summary, the conversation discusses the use of the variational method to calculate the energy of the ground state of a helium atom, with a specific wave function and effective charge as the variational parameter. The Hamiltonian of the system is also mentioned, along with the expectation value of the interaction term. The conversation then delves into the calculation of the first two terms, with the use of the virial theorem. However, a mistake is found in the result and a solution is suggested from a book by Griffiths.
Aleolomorfo
I need to calculate the energy of the ground state of a helium athom with the variational method using the wave function:
$$\psi_{Z_e}(r_1,r_2)=u_{1s,Z_e}(r1)u_{1s, Z_e}(r2)=\frac{1}{\pi}\biggr(\frac{Z_e}{a_0}\biggr)^3e^{-\frac{Z_e(r_1+r_2)}{a_0}}$$
with ##Z_e## the effective charge considered as a variational parameter.
The Hamiltonian of the system is:
$$\hat{H}=\hat{H_1}+\hat{H_2} + \frac{e^2}{4\pi\epsilon_0|\vec{x_1}-\vec{x_2}|}$$
The first two terms are idrogen Hamiltonians for the two electrons and the third term is the interaction term.
The expection value of the interaction term is ##\frac{5}{4}Z_eRy## with ##Ry= 13.6 \space eV##.
For the first two term:
$$<\psi_{Z_e}|\hat{H_1}+\hat{H_2}|\psi_{Z_e}>$$
$$<u_{1s,Z_e}(r1)u_{1s,Z_e}(r2)|\hat{H_1}+\hat{H_2}|u_{1s,Z_e}(r1)u_{1s,Z_e}(r2)>$$
$$2<u_{1s,Z_e}(r1)|\hat{H_1}|u_{1s,Z_e}(r1)>$$
Since ##\hat{H_1}=\hat{T}+U## I can calculate:
$$<U> = <u_{Z_e}|U|u_{Z_e}>=-\frac{Ze^2}{4\pi\epsilon_0}<u_{Z_e}|\frac{1}{r}|u_{Z_e}>$$
$$<U>= \frac{Z}{4\pi\epsilon_0a_0} = -2ZZ_{e}Ry$$
Now I have used the virial theorem ##2<\hat{T}> + <U> = 0##, so:
$$<\hat{T}> = ZZ_{e}Ry$$
But this is wrogn because it should be ##Z_{e}^2Ry## and I do not see where is the mistake. Thank you in advance!

It's a know problem. You can find a cool explanation/solution in "Introduction to Quantum Mechanics" by Griffiths Chap 7 (p. 264). Here's an extract:

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1. What is a helium atom?

A helium atom is the smallest and simplest atom, with an atomic number of 2 and the symbol He. It is composed of two protons, two neutrons, and two electrons.

2. What is the variation method in relation to the helium atom?

The variation method is a mathematical technique used to approximate the energy of a system, such as the helium atom. It involves choosing a trial function and adjusting its parameters to find the lowest possible energy.

3. How does the variation method work in determining the energy of a helium atom?

The variation method works by comparing the trial function to the exact wave function of the system and minimizing the difference between the two. This results in an approximation of the lowest possible energy of the system.

4. What is the significance of the virial theorem in studying helium atoms?

The virial theorem is a mathematical relationship that connects the average kinetic and potential energies of a system. In the case of the helium atom, it allows us to understand the relationship between the electrons and the nucleus, and how changes in one can affect the other.

5. How does the virial theorem relate to the stability of helium atoms?

The virial theorem states that for a stable system, the average kinetic energy is equal to the negative of the average potential energy. This is important for understanding the stability of helium atoms, as it shows that the attractive forces between the nucleus and electrons are balanced with the repulsive forces between the electrons themselves, resulting in a stable system.

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