# According to Bohr's model of an atom, which of the following is/are quantized? (a) The total energy of electron is quantized. (b) Angular momentum of electron is quantized. (c) both (a) and (b). (d) None of the above.

Jun 7, 2017

$\left(a\right)$ and $\left(b\right)$.

Both energy and angular momentum are observables that correspond to so-called eigenvalues. Eigenvalues are the values that describe a result that occurs consistently, brought about by an observation.

All energies $E$ in a quantum mechanical system correspond to eigenvalues that are dependent on a particular quantum number.

An example of atomic energies is the hydrogen atom in the Rydberg equation:

$\Delta E = - \text{13.6 eV} \left(\frac{1}{n} _ {f}^{2} - \frac{1}{n} _ {i}^{2}\right)$

where:

• ${n}_{i}$ and ${n}_{f}$ are the initial and final quantum numbers $n$ for the energy levels across which an energy transition occurs.
• $\Delta E$ is the energy gap for that transition in units of $\text{eV}$ ($1.602 \times {10}^{- 19}$ $\text{J}$ $=$ $\text{1 eV}$).

$n = 1 , 2 , 3 , . . .$ is the principal quantum number, indicating each energy level, corresponding to eigenvalues ${E}_{n}$.

Since $n$ is quantized, it goes in integer steps, and thus the energy is quantized as well.

The angular momentum of the electron, corresponding to the "shape" of an orbital (not necessarily a thing for Bohr's model, which pretends there are orbits instead), has eigenvalues dependent on the quantum number, $l$:

$l = 0 , 1 , 2 , . . . , n - 1$ is the angular momentum quantum number, corresponding to the eigenvalue l(l+1)ℏ^2 of the squared angular momentum, ${L}^{2}$.

Clearly, $l$ is going in integer steps, so angular momentum is quantized as well.

What about ${L}_{z}$, the z-angular momentum, which depends on ${m}_{l}$, the magnetic quantum number? Is the z-angular momentum quantized too?