Eigenstate
Introduction
An eigenstate is a term used in quantum mechanics to describe a specific state of a quantum system. The term "eigenstate" is derived from the German word "eigen", which means "own" or "inherent". In the context of quantum mechanics, an eigenstate is a state that is defined by a particular set of observable quantities, such as energy, momentum, or spin.
Quantum Mechanics and Eigenstates
Quantum mechanics is a branch of physics that describes the behavior of particles on the smallest scales, such as atoms and subatomic particles. One of the key principles of quantum mechanics is the concept of superposition, which states that a quantum system can exist in multiple states at once. These states are known as eigenstates.
An eigenstate is a state of a quantum system that is associated with a specific value of an observable quantity. For example, in the case of an electron in an atom, the eigenstates could be the different energy levels that the electron can occupy. When a measurement is made on the system, the system is found to be in one of these eigenstates, and the value of the observable is the eigenvalue associated with that eigenstate.
Mathematical Representation of Eigenstates
In quantum mechanics, eigenstates are represented mathematically by wave functions. A wave function is a mathematical function that describes the quantum state of a system. The wave function can be thought of as a probability distribution that describes the likelihood of finding the system in a particular state.
The eigenstates of a quantum system are the solutions to the Schrödinger equation, which is the fundamental equation of quantum mechanics. The Schrödinger equation describes how the wave function of a quantum system evolves over time. The solutions to the Schrödinger equation are the possible wave functions of the system, and each of these wave functions corresponds to a different eigenstate.
Eigenstates and Quantum Measurement
One of the most intriguing aspects of quantum mechanics is the role of measurement. According to the Copenhagen interpretation, one of the most widely accepted interpretations of quantum mechanics, a quantum system exists in a superposition of states until a measurement is made. When a measurement is made, the system "collapses" into one of its eigenstates, and the value of the observable is the eigenvalue associated with that eigenstate.
This process is often referred to as "wave function collapse", and it is one of the most mysterious and debated aspects of quantum mechanics. Despite its strangeness, this interpretation has been extremely successful in predicting the outcomes of experiments, and it is a fundamental part of our understanding of the quantum world.
Eigenstates in Quantum Computing
The concept of eigenstates is also central to the field of quantum computing. In a quantum computer, information is stored in quantum bits, or qubits, which can exist in a superposition of states. Each state of a qubit is an eigenstate, and the information is encoded in these eigenstates.
Quantum computers can perform calculations by manipulating these eigenstates, allowing them to solve certain problems much more efficiently than classical computers. The ability to manipulate eigenstates is one of the key advantages of quantum computing, and it is one of the reasons why this field is the subject of intense research and development.