Question

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.

Answer 1

`(a)` and `(b)`.

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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:

`DeltaE = -"13.6 eV"(1/n_f^2 - 1/n_i^2)`

where:

• `n_i` and `n_f` are the initial and final quantum numbers `n` for the energy levels across which an energy transition occurs.
• `DeltaE` is the energy gap for that transition in units of `"eV"` (`1.602 xx 10^(-19)` `"J"` `=` `"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.

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What about `L_z`, the z-angular momentum, which depends on `m_l`, the magnetic quantum number? Is the z-angular momentum quantized too?