Ground State Energy of a Helium Atom using the Uncertainty Principle:

The ground state energy of a Helium atom can be calculated using the uncertainty principle. The uncertainty principle states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position and momentum, can be known simultaneously. This principle was first formulated by Werner Heisenberg in 1927.

1. Uncertainty Principle:

The uncertainty principle can be mathematically expressed as:
Δx * Δp ≥ h/4π

where Δx is the uncertainty in the position of the particle, Δp is the uncertainty in the momentum of the particle, and h is the Planck's constant.

2. Applying the Uncertainty Principle to the Helium Atom:

In the case of a Helium atom, we consider the uncertainty in the position of the electron and the uncertainty in its momentum. The ground state of a Helium atom is when both electrons are in the lowest energy level, which is the 1s orbital.

3. Uncertainty in the Position:

The 1s orbital of a Helium atom is spherically symmetric, and therefore, the uncertainty in the position of the electron can be considered to be equal in all directions. Let's assume Δx is the uncertainty in the position of the electron.

4. Uncertainty in the Momentum:

The momentum of an electron in the 1s orbital can be calculated using the de Broglie wavelength:
λ = h / p

where λ is the de Broglie wavelength, h is the Planck's constant, and p is the momentum. Since the electron is in the 1s orbital, its momentum can be approximated as:

p = mv = h / λ

where m is the mass of the electron and v is its velocity.

5. Calculating the Ground State Energy:

Using the uncertainty principle, we can write:

Δx * Δp ≥ h/4π

Δx * (h / Δx) ≥ h/4π

Δp = h / Δx ≥ h/4π

The ground state energy of a Helium atom is given by:

E = p^2 / (2m) = (h^2 / Δx^2) / (2m)

Substituting the value of Δp from the uncertainty principle:

E = (h^2 / (h^2 / Δx^2)) / (2m)

E = Δx^2 / (2m)

6. Using Known Values:

To calculate the ground state energy, we need to know the values of Δx and m. The mass of an electron is approximately 9.10938356 x 10^-31 kilograms.

Assuming Δx to be the size of the 1s orbital, which is approximately 5.3 x 10^-11 meters, we can calculate the ground state energy using the equation:

E = Δx^2 / (2m)

E = (5.3 x 10^-11)^2 / (2 x 9.10938356 x 10^-31)

E ≈ -10.34 eV

Therefore, the ground state energy of a Helium atom, calculated using the uncertainty principle, is approximately -