29.1 Quantization of energy  (Page 2/7)

Here $n$ is any nonnegative integer (0, 1, 2, 3, …). The symbol $h$ stands for Planck’s constant    , given by

The equation $E=\left(n+\frac{1}{2}\right)\text{hf}$ means that an oscillator having a frequency $f$ (emitting and absorbing EM radiation of frequency $f$ ) can have its energy increase or decrease only in discrete steps of size

It might be helpful to mention some macroscopic analogies of this quantization of energy phenomena. This is like a pendulum that has a characteristic oscillation frequency but can swing with only certain amplitudes. Quantization of energy also resembles a standing wave on a string that allows only particular harmonics described by integers. It is also similar to going up and down a hill using discrete stair steps rather than being able to move up and down a continuous slope. Your potential energy takes on discrete values as you move from step to step.

Using the quantization of oscillators, Planck was able to correctly describe the experimentally known shape of the blackbody spectrum. This was the first indication that energy is sometimes quantized on a small scale and earned him the Nobel Prize in Physics in 1918. Although Planck’s theory comes from observations of a macroscopic object, its analysis is based on atoms and molecules. It was such a revolutionary departure from classical physics that Planck himself was reluctant to accept his own idea that energy states are not continuous. The general acceptance of Planck’s energy quantization was greatly enhanced by Einstein’s explanation of the photoelectric effect (discussed in the next section), which took energy quantization a step further. Planck was fully involved in the development of both early quantum mechanics and relativity. He quickly embraced Einstein’s special relativity, published in 1905, and in 1906 Planck was the first to suggest the correct formula for relativistic momentum, $p=\text{γmu}$ .

Note that Planck’s constant $h$ is a very small number. So for an infrared frequency of ${\text{10}}^{\text{14}}\text{Hz}$ being emitted by a blackbody, for example, the difference between energy levels is only $\mathrm{\Delta }E=\text{hf}\text{=}(6\text{.}\text{63}\times {\text{10}}^{\text{–34}}\text{J·s})({\text{10}}^{\text{14}}\text{Hz})\text{= 6}\text{.}\text{63}\times {\text{10}}^{\text{–20}}\text{J,}$ or about 0.4 eV. This 0.4 eV of energy is significant compared with typical atomic energies, which are on the order of an electron volt, or thermal energies, which are typically fractions of an electron volt. But on a macroscopic or classical scale, energies are typically on the order of joules. Even if macroscopic energies are quantized, the quantum steps are too small to be noticed. This is an example of the correspondence principle. For a large object, quantum mechanics produces results indistinguishable from those of classical physics.

Atomic spectra

Now let us turn our attention to the emission and absorption of EM radiation by gases . The Sun is the most common example of a body containing gases emitting an EM spectrum that includes visible light. We also see examples in neon signs and candle flames. Studies of emissions of hot gases began more than two centuries ago, and it was soon recognized that these emission spectra contained huge amounts of information. The type of gas and its temperature, for example, could be determined. We now know that these EM emissions come from electrons transitioning between energy levels in individual atoms and molecules; thus, they are called atomic spectra    . Atomic spectra remain an important analytical tool today. [link] shows an example of an emission spectrum obtained by passing an electric discharge through a material. One of the most important characteristics of these spectra is that they are discrete. By this we mean that only certain wavelengths, and hence frequencies, are emitted. This is called a line spectrum. If frequency and energy are associated as $\mathrm{\Delta }E=\text{hf}\text{,}$ the energies of the electrons in the emitting atoms and molecules are quantized. This is discussed in more detail later in this chapter.