In the context of quantum mechanics, the term "orthogonal" refers to a property of quantum states or vectors. In quantum mechanics, states are represented as vectors in a complex vector space called a Hilbert space.
Two quantum states are said to be orthogonal if their corresponding vectors are perpendicular to each other in this Hilbert space. Mathematically, the inner product (also known as the dot product) of two orthogonal vectors is zero. Orthogonal states have no overlap or correlation with each other.
The concept of orthogonality is crucial in quantum mechanics for several reasons. One important aspect is the measurement of quantum systems. When measuring a quantum state, the measurement process effectively projects the state onto one of the possible measurement outcomes. Orthogonal states can be distinguished perfectly in a measurement, meaning that if a system is in one orthogonal state, it will not be found in the other orthogonal state upon measurement.
Furthermore, orthogonal states are used as a basis for representing quantum states. An orthogonal basis is a set of states that spans the entire Hilbert space, meaning that any quantum state can be expressed as a linear combination of the basis states. The most commonly used orthogonal basis in quantum mechanics is the set of eigenstates of an observable, which are states corresponding to the possible outcomes of measuring that observable.
Orthogonality is a fundamental concept in quantum mechanics that underlies many key principles, such as superposition, measurement, and the mathematical representation of quantum states. It plays a crucial role in understanding the behavior and properties of quantum systems.