In quantum mechanics, a basis refers to a set of states that are used to describe the quantum system and express its properties. These states form a complete and orthogonal set, meaning that any state of the system can be expressed as a linear combination of the basis states, and the basis states are mutually orthogonal (perpendicular) to each other.
Mathematically, let's denote the basis states as |ψ₁⟩, |ψ₂⟩, |ψ₃⟩, and so on. A general state of the quantum system, represented by the state vector |Ψ⟩, can be expressed as a linear combination of the basis states:
|Ψ⟩ = c₁|ψ₁⟩ + c₂|ψ₂⟩ + c₃|ψ₃⟩ + ...
Here, c₁, c₂, c₃, etc., are complex coefficients that determine the amplitudes or probabilities associated with each basis state.
Bases in quantum mechanics play a fundamental role in representing and analyzing quantum systems. Different bases can be chosen depending on the specific problem or the physical quantities of interest. For example, the position basis is often used to describe the position of a particle, while the momentum basis is used to describe its momentum. Other common bases include the energy eigenbasis, the angular momentum basis, or the spin basis, among others.
The choice of basis affects the observables that can be measured and the mathematical representation of operators in quantum mechanics. Observables, such as energy, position, or momentum, are represented by Hermitian operators, and their eigenvalues correspond to the possible measurement outcomes. In a particular basis, the eigenstates of the observable are the basis states themselves, and the measurement of the observable corresponds to projecting the state vector onto the corresponding eigenstate.
In summary, a basis in quantum mechanics is a set of states that provide a complete and orthogonal description of a quantum system. They form the building blocks for representing states, calculating probabilities, and describing the measurements and observables associated with the system.