Eigenvector

Definition

An eigenvector is a non-zero vector that, when a linear transformation is applied to it, results in a scalar multiple of itself. This concept is crucial in quantum mechanics as it helps describe the states of quantum systems, revealing how certain properties remain unchanged when measured, reflecting their physical significance in the context of operators and observables.

5 Must Know Facts For Your Next Test

1. Eigenvectors correspond to specific states in quantum mechanics where certain physical properties remain invariant under measurement.
2. Each eigenvector has a corresponding eigenvalue, which indicates the factor by which the eigenvector is scaled during the linear transformation.
3. In quantum mechanics, operators act on wave functions, and the eigenvectors of these operators represent the possible states of the system.
4. When measuring a quantum observable, the system collapses into one of its eigenstates (the eigenvector), which provides a definite outcome related to the measurement.
5. The set of all eigenvectors corresponding to a particular operator forms a basis for the space of states that allows for complete descriptions of quantum systems.

Review Questions

• How does an eigenvector relate to the measurement process in quantum mechanics?
  • An eigenvector plays a vital role in the measurement process in quantum mechanics because it represents a possible state of a system associated with a specific observable. When an observable is measured, the system collapses into one of its eigenstates, which are the eigenvectors of the corresponding operator. This collapse leads to definite measurement outcomes, illustrating how eigenvectors provide essential information about the physical properties of quantum systems.
• Discuss the relationship between eigenvectors and linear transformations in the context of quantum operators.
  • Eigenvectors are intimately connected to linear transformations as they demonstrate how these transformations affect specific vectors within a vector space. In quantum mechanics, operators act as linear transformations on wave functions. Eigenvectors of these operators remain unchanged except for a scaling factor (the eigenvalue), which highlights their significance in determining the stability and characteristics of quantum states under such transformations.
• Evaluate how understanding eigenvectors can enhance our grasp of quantum states and their evolution over time.
  • Understanding eigenvectors significantly enhances our grasp of quantum states and their evolution because they encapsulate critical information about the system's behavior. By knowing the eigenstates (eigenvectors) associated with an operator, we can predict how a quantum system will respond to measurements and how it evolves under time-dependent conditions. This knowledge allows us to formulate precise predictions about physical phenomena and contributes to foundational concepts such as superposition and entanglement in quantum mechanics.

Related terms

Eigenvalue: An eigenvalue is a scalar that represents how much an eigenvector is stretched or compressed during a linear transformation.

Linear Transformation: A linear transformation is a mathematical operation that maps vectors to other vectors while preserving the operations of vector addition and scalar multiplication.

Observable: In quantum mechanics, an observable is a physical quantity that can be measured, often represented by an operator whose eigenvectors correspond to possible measurement outcomes.