Quantization of Energy

Planck's explanation of the ultraviolet catastrophe in the classical calculations of black body radiation involved the assumption of quantized oscillators. He postulated that the electromagnetic radiation was emitted by oscillators with quantized energy jumps

$\Delta$E = h $\cdot$ $\nu$

where $\nu$ is the frequency and

h = 6.626$\cdot$ 10^-34 J$\cdot$ s

The quantity h is known as Planck's constant. Later it was shown that the energy of these oscillators must be an integral multiple, n, of hn, and that in fact these oscillators corresponded to real quanta of electromagnetic radiation called photons that had energy

E_oscillator = n$\cdot$ h$\cdot$ $\nu$.

Note that $\nu$ is the classical frequency of the oscillator. For example, a mass on a spring would have $\nu$ = (1/2$\pi$)(k/m)^1/2.

We can see that the last formula indeed gives the right separation between neighboring levels, $\Delta$E, of the first equation:

$\Delta$E = n$\cdot$ h$\cdot$ $\nu$ - (n-1)$\cdot$ h$\cdot$ n = h$\cdot$ $\nu$

Question:
Crystals of lithium fluoride vibrate with a resonant frequency of 1.90$\cdot$ 10¹³ Hz. What is the separation of the energy levels, and what is the energy of the 200^th level? The usual units are eV; 1 eV = 1.60$\cdot$ 10^-19 J.

Answer:

Separation of two neighboring energy levels:
$\Delta$E = h$\nu$ with h = 6.626$\cdot$ 10^-34 J$\cdot$ s
= (6.626$\cdot$ 10^-34 J$\cdot$ s) $\cdot$ (1.90$\cdot$ 10¹³ s^-1)
= 1.259$\cdot$ 10^-20 J
= 0.0787 eV.
Energy of the 200^th level:
E = 200 h$\cdot$ $\nu$ = 200 $\Delta$E = 200$\cdot$ 0.079 eV = 15.7 eV