What is Uncertainity Principle? Estimate ground state energy of Hydrogen atom using it.

ANSWER : The uncertainty principle is a fundamental concept in quantum mechanics, stating that certain pairs of physical properties, like position and momentum, cannot be measured with absolute precision simultaneously. In simpler terms, the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.

The uncertainty principle was formulated by the German physicist Werner Heisenberg in 1927. Heisenberg's work on quantum mechanics, including the uncertainty principle, was a significant contribution to the development of the field and has since become one of the key principles in understanding the behavior of particles at the quantum level.

To estimate the ground state energy of the hydrogen atom using the uncertainty principle, we consider the uncertainties in position and momentum of the electron.

The uncertainty principle is expressed as :

\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]

where :

- \(\Delta x\) is the uncertainty in position,

- \(\Delta p\) is the uncertainty in momentum, and

- \(\hbar\) is the reduced Planck's constant (\(\approx 1.054 \times 10^{-34}\) Js).

For the hydrogen atom, the ground state is the lowest energy state that an electron can occupy. To estimate its ground state energy, we assume that the uncertainty in position (\(\Delta x\)) is approximately the size of the smallest orbit, which is the Bohr radius (\(a_0 \approx 5.29 \times 10^{-11}\) meters).

Now, let's consider the uncertainty in momentum (\(\Delta p\)). The momentum of the electron in the hydrogen atom is related to its reduced mass (\(m\)) and its velocity (\(v\)) as follows:

\[ \Delta p = m \cdot \Delta v \]

The velocity of the electron in the ground state can be approximated using the electron's speed in the Bohr orbit, given by :

\[ v = \frac{e^2}{2a_0 \cdot \hbar} \]

where :

- \(e\) is the elementary charge (\(\approx 1.602 \times 10^{-19}\) C).

Now, we can calculate the uncertainty in momentum (\(\Delta p\)).

Finally, using the uncertainty principle, we can estimate the ground state energy (\(E\)) of the hydrogen atom by equating the product of uncertainties to \(\frac{\hbar}{2}\):

\[ \Delta x \cdot \Delta p = \frac{\hbar}{2} \]

Substituting the values, we find :

\[ \Delta x \cdot (m \cdot \Delta v) = \frac{\hbar}{2} \]

And now, we can calculate \(E\) using the ground state energy formula for the hydrogen atom:

\[ E = -\frac{{m \cdot e^4}}{{2 \cdot (4 \pi \varepsilon_0)^2 \cdot \hbar^2}} \]

where :

- \(\varepsilon_0\) is the vacuum permittivity (\(\approx 8.854 \times 10^{-12}\) F/m).

After performing the calculations, we get an estimated ground state energy of the hydrogen atom:

\[ E \approx -2.18 \times 10^{-18}\, \text{J} \]

It's important to note that this is just an approximation. The actual ground state energy of the hydrogen atom is obtained more accurately through quantum mechanical methods, but this estimation gives us a rough idea of its value.

Also Read : State and Derive the Maxwell-Boltzmann's Canonical Distribution Law.

Also Read : Classify the particles as Fermion or Boson : α-particle, electron, H₂ molecule.

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