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Why is it that eigenfunctions of different excited states for 1 atom have to be orthogonal?
An eigenfunction is a mathematical function that satisfies a specific equation, known as the eigenvalue equation. It is associated with a particular eigenvalue, which is a constant that scales the eigenfunction. Eigenfunctions are important in the study of quantum mechanics, where they represent the possible states of a physical system.
Excited states refer to the energy levels of a quantum system that are higher than the ground state, which is the lowest energy level. These states are typically reached when the system absorbs energy, such as through a photon of light. In the context of eigenfunctions, excited states refer to the eigenfunctions associated with these higher energy levels.
In quantum mechanics, the wave function of a physical system is described as a superposition of all possible eigenfunctions. For a system in an excited state, this means that its wave function is composed of multiple eigenfunctions. In order for this superposition to accurately represent the system, the eigenfunctions must be orthogonal, meaning they are perpendicular to each other in a mathematical sense. This ensures that each eigenfunction contributes to the total wave function in a unique and independent way.
The uncertainty principle states that it is impossible to measure both the position and momentum of a particle with absolute certainty. This is due to the wave-like nature of particles described by quantum mechanics. The orthogonality of eigenfunctions plays a crucial role in this principle, as it allows for the simultaneous existence of multiple possible states for a particle, while still maintaining the overall probability distribution described by the wave function.
Eigenfunctions of excited states are used in various practical applications, particularly in the fields of quantum computing and spectroscopy. In quantum computing, they are used as a basis for representing information and performing calculations. In spectroscopy, they are used to analyze the energy levels of atoms and molecules, providing valuable insights into their electronic structures and chemical properties.