![]() ![]() The application of the Schrödinger equation is most easily understood in the case of the hydrogen atom because the one-electron atom allows us to avoid the complex interactions of multiple electrons. The simplest case - a hydrogen atom with a single electron The Ψ 2 term describes how electron density and electron probability are distributed in space around the nucleus of an atom. When Ψ is squared, the resulting solution gives the probability of finding an electron at a particular place in the atom. The challenges raised by the combination of wave particle duality and the Heisenberg Uncertainty Principle are simplified and expressed by Schrödinger’s equation that, when solved, produces wave functions denoted by Ψ. In other words, those that would normally be defined by simple x, y, and z coordinates. Correct! Schrödinger solves the probability questionīecause the electron is not a true particle, we cannot describe its movement or location in traditional terms. the product of uncertainty in position and momentum of an electron cannot be less than the reduced Planck constant. It tells us that mathematically, the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of an electron cannot be less than the reduced Planck constant ℏ/2 (Equation 1).ī. ![]() Werner Heisenberg developed a principle to describe this uncertainty, called appropriately the Heisenberg Uncertainty Principle (Heisenberg, 1927). However, the electron is so tiny that even a single photon will influence its trajectory – thus if we shine a beam of light on it to measure its position, the energy of the photon will affect its momentum, and vice versa. If we shine a light beam on a moving tennis ball, the light has little effect on the tennis ball and we can measure both its position and momentum with a high degree of accuracy. Since we measure the position of objects with light, the small size of the electron introduces a challenge. This module further explores these solutions, the position of electrons, the shape of atomic orbitals, and the implications of these ideas.Īs we saw in earlier reading, the electron is not a true particle, but a wave-particle similar to the photon. Born took the wave functions that Schrödinger produced and said that the solutions to the equation could define the energies and the most probable positions of electrons within atoms, thus allowing us to build a much more detailed description of where electrons might be found within an atom. The Schrödinger equation was seen as a key mathematical link between the theory and the application of the quantum model. In Atomic Theory III: Wave-Particle Duality and the Electron, we discussed the advances that were made by Schrödinger, Born, Pauli, and others in the application of the quantum model to atomic theory. Orbitals can be thought of as the three dimensional areas of space, defined by the quantum numbers, that describe the most probable position and energy of an electron within an atom. Quantum numbers, when taken as a set of four (principal, azimuthal, magnetic and spin) describe acceptable solutions to the Schrödinger equation, and as such, describe the most probable positions of electrons within atoms. The Pauli exclusion principle states that no two electrons with the same spin can occupy the same orbital. The discovery of electron spin defines a fourth quantum number independent of the electron orbital but unique to an electron. The Heisenberg uncertainty principle establishes that an electron’s position and momentum cannot be precisely known together instead we can only calculate statistical likelihood of an electron’s location. The equation allows the calculation of each of the three quantum numbers related to individual atomic orbitals (principal, azimuthal, and magnetic). The Schrödinger equation describes how the wave function of a wave-particle changes with time in a similar fashion to the way Newton’s second law describes the motion of a classical particle. ![]() The wave-particle nature of electrons means that their position and momentum cannot be described in simple physical terms but must be described by wave functions. ![]()
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