Which of the following statement does not form part of Bohr's model of the hydrogen atom ? (a) Energy of the electrons in the orbit is quantized (b) The electron in the orbit nearest the nucleus has the lowest energy (c) Electrons revolve in different orbits around the nucleus (d) The position and velocity of the electrons in the orbit cannot be determined simultaneously

The concept of quantization of energy is a core aspect of Bohr's model of the hydrogen atom. According to this revolutionary idea introduced by Niels Bohr in 1913, energy is not continuous but rather exists in discrete packets known as 'quanta'. In the context of the hydrogen atom, this means that electrons can only occupy orbits with specific, quantized energy levels.

The electrons cannot exist between these levels; they can only 'jump' from one energy level to another. These jumps, or transitions, result in the absorption or emission of photons with energy corresponding to the difference between the initial and final energy levels of the electron. Mathematically, this can be written as: \(E_{photon} = E_{final} - E_{initial}\) where \(E_{photon}\) is the energy of the emitted or absorbed photon. This concept fundamentally contradicts the classical view that energy is a continuum, and it lays the groundwork for understanding atomic spectra and the emission lines of atoms.

Real-World Implications

For instance, the unique spectral lines emitted by hydrogen — the spectral fingerprint of the element — can be explained by electron transitions between quantized energy levels.

Within Bohr's model, the electron orbits around the nucleus are not arbitrary paths; they are well-defined circular tracks where the electron's angular momentum is quantized. This means that the angular momentum of an electron in a particular orbit is an integer multiple of the reduced Planck constant \(\frac{h}{2\pi}\). The formula for this relationship is expressed as: \(L = n\frac{h}{2\pi}\), where \(L\) is the angular momentum and \(n\) is the principal quantum number, an integer representing the quantized orbit which the electron occupies.

These orbits, often called 'stationary states', are stable configurations where the electron does not radiate energy. An electron in a lower-energy orbit (closer to the nucleus) indeed has less energy compared to one in a higher-energy orbit, which explains why the electron in the orbit nearest to the nucleus has the lowest energy. Each orbit corresponds to a specific quantized energy level, further reinforcing the idea of energy quantization.

Heisenberg's Uncertainty Principle is a foundational principle in quantum mechanics, postulated by Werner Heisenberg in 1927, which essentially states that it is fundamentally impossible to know both the position and the momentum of a particle with infinite precision at the same time. The more accurately we know one value, the less accurately we can know the other. This principle can be mathematically represented by the inequality: \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\), where \(\Delta x\) and \(\Delta p\) are the uncertainties in position and momentum, respectively, and \(h\) is Planck's constant.

Contrary to classical physics, where precise simultaneous measurements of both quantities are theoretically possible, Heisenberg's Uncertainty Principle highlights a fundamental limit to our ability to make such measurements at the quantum level. This principle is not reflected in Bohr's model of the atom, as Bohr's model predates quantum mechanics and treats electron orbits in a more classical sense. The inclusion of this principle in modern atomic theory marks a significant departure from the more deterministic descriptions by Bohr, emphasizing the probabilistic nature inherent in quantum systems.