Some experimental observations cannot be explained by classical physics, but can be explained using quantum theory, such as

• black-body/cavity radiation curves (ultraviolet catastrophe)

  → The theoretical black-body radiation curve tended towards infinity for shorter wavelengths which is known as the ultraviolet catastrophe.

  

• the formation of emission and absorption spectra

  

• the photoelectric effect

  → If light were a wave, electrons would be emitted for all frequencies at a high enough intensity and there would be a time delay before electrons are emitted. However, neither of these things are observed.

  

The following relationship can be used to solve problems involving photon energy and frequency.

$$ E=hf $$

The Bohr model of the atom consists of electrons in energy levels, orbiting around a positive nucleus.

In this model, the electrons can orbit in these permitted stable orbits without emitting radiation — the angular momentum of the electrons in these orbits is quantised — i.e., so long as $L$ is a multiple of $\frac{h}{2\pi}$, the electron orbits are stable.

This constant comes from de Broglie’s ideas on wave-particle duality — if electrons are treated as stationary waves, then the smallest allowed circumference of a allowable orbit can be taken to correspond to $\lambda$. The next allowable orbit corresponds to $2\lambda$, $3\lambda$, and so on. The orbit will be stable when the circumference corresponds to $n\lambda$.

→ Stationary waves do not transmit energy, so the classical problem of emitting radiation does not occur.

→ $n\lambda=2\pi r$ and $\lambda=\dfrac{h}{p}$

$\dfrac{nh}{mv}=2\pi r$

⇒ $mvr=L=\dfrac{nh}{2\pi}$

→ $n$ is known as the principle quantum number.

The Bohr model of the atom explains the characteristics of atomic spectra as the discrete lines observed correspond to specific energy transitions by electrons between the different quantum energy levels caused by electrons absorbing or emitting a photon.