Final answer:
The energy of a particle in a quantum system is proportional to the square of its quantum number (n) according to Schrödinger's wave equation, where the allowed energy states are quantized and related to Planck's constant.
Step-by-step explanation:
The energy levels of a particle in a quantum system such as an electron in a hydrogen atom can be found using the Schrödinger's wave equation. According to the equation, the allowed energy states of electrons in an atom are quantized, which means that the energy of an electron is proportional to the square of its quantum number (n), generally represented as E ∝ n^2. This implies that the energy (E) of the particle is directly related to certain quantized levels, and these levels increase with the square of the quantum number. This relationship comes from solving the Schrödinger equation for the hydrogen atom, which involves variables such as the mass of the electron and Planck's constant (h).
The formula E = hf relates the energy of a photon to its frequency (f), where h is Planck's constant. Additionally, when considering the wavelength (λ) of a photon, E = hc/λ can be used, where c is the speed of light and h remains Planck's constant. The allowed energy states for the electron in a hydrogen atom must satisfy the condition where the de Broglie wavelength fits an integer number of wavelengths around the orbit, leading to the quantization of angular momentum and subsequently to the quantization of energy levels proportional to n^2.