What does it mean when we say that the energy of something is "quantized"?

When we say that the energy of something is "quantized," it means that the energy of the system can only take specific, discrete values, rather than taking any value across a continuous range. In quantum systems, these energy values are usually determined by the quantized states (or wavefunctions) that particles can occupy.

The concept of quantized energy first appeared in the early 20th century, with the development of quantum mechanics. Physicist Max Planck proposed that energy levels of electromagnetic radiation are quantized to explain the behavior of black body radiation. Later, Niels Bohr incorporated the idea of quantized energy levels in his model of the hydrogen atom, where electrons could only occupy specific orbits with discrete energy levels.

In atomic systems, electrons occupy specific energy levels around the nucleus. These energy levels are quantized, meaning that electrons can only exist in certain orbits with specific energy values. When an electron moves from one energy level to another (either by absorbing or emitting a photon), it can only change its energy by the exact difference between the two allowed energy levels.

Let's consider the hydrogen atom as an example. In the Bohr model of the hydrogen atom, the electron orbits the nucleus at specific radii, to have quantized angular momentum. The energy levels of the electron in a hydrogen atom can be calculated using the formula: E_n = -\frac{13.6 eV}{n^2} where E_n is the energy of the electron in the nth energy level, n is an integer (1, 2, 3,...), and eV stands for electronvolt, a unit of energy. The negative sign in the formula means that the electron is in a bound state in the atom. In this model, the energy levels of the electron in hydrogen are quantized, and the electron can only occupy these specific energy levels. In summary, when we say that the energy of something is "quantized," it means that the energy can only take specific, discrete values in the system. This concept originated with the development of quantum mechanics and is an essential property of atomic systems, like electrons in an atom.