Eigenvectors

Definition

Eigenvectors are special vectors associated with a linear operator that remain in the same direction when that operator is applied. When a linear operator acts on an eigenvector, the result is simply a scalar multiple of that eigenvector, which is referred to as the eigenvalue. This property makes eigenvectors essential in understanding how observables behave under measurements in quantum mechanics.

5 Must Know Facts For Your Next Test

1. Eigenvectors can be thought of as the 'directions' in which a linear transformation acts simply by scaling.
2. For any given linear operator, there can be multiple eigenvectors corresponding to the same eigenvalue, leading to the concept of eigenspaces.
3. In quantum mechanics, observables correspond to Hermitian operators, and their eigenvectors represent possible states of a system after measurement.
4. Finding eigenvectors often involves solving the characteristic equation derived from the linear operator's matrix representation.
5. The completeness of a set of eigenvectors allows for the expansion of any vector in the space in terms of those eigenvectors, which is fundamental for quantum state representation.

Review Questions

• How do eigenvectors relate to the behavior of observables in quantum mechanics?
  • Eigenvectors play a crucial role in quantum mechanics as they represent the possible states of a system after a measurement of an observable. When a Hermitian operator, which corresponds to a physical observable, acts on an eigenvector, it produces an eigenvalue that reflects the measured value. This connection between eigenvectors and observables helps to establish the foundational principles of measurement and state collapse in quantum mechanics.
• Discuss the significance of eigenspaces in relation to eigenvectors and their respective eigenvalues.
  • Eigenspaces are defined as the collection of all eigenvectors corresponding to a particular eigenvalue, including the zero vector. This means that multiple eigenvectors can share the same eigenvalue, leading to an entire space characterized by that value. Understanding eigenspaces is important because they provide insight into the geometric structure of linear transformations and help simplify complex problems by breaking down vectors into components along these special directions.
• Evaluate how the properties of Hermitian operators affect the nature of their eigenvalues and eigenvectors in quantum mechanics.
  • Hermitian operators have distinct properties that significantly influence their eigenvalues and eigenvectors. The key aspect is that all eigenvalues are real, which ensures that observable quantities measured in quantum systems yield real results. Additionally, the eigenvectors corresponding to different eigenvalues are orthogonal, allowing for a complete basis set in which any state can be expressed. This orthogonality is fundamental for ensuring accurate and unambiguous measurements within quantum mechanics, reinforcing why Hermitian operators are central to understanding quantum observables.

Related terms

Eigenvalues: Eigenvalues are the scalars associated with eigenvectors that represent how much the eigenvector is stretched or compressed during the application of a linear operator.

Linear Operators: Linear operators are mathematical functions that map vectors to other vectors while preserving vector addition and scalar multiplication, crucial for understanding transformations in vector spaces.

Hermitian Operators: Hermitian operators are a class of linear operators in quantum mechanics that have real eigenvalues and orthogonal eigenvectors, which are significant when measuring physical observables.