Eigenvalue

Definition

An eigenvalue is a special scalar associated with a linear transformation represented by an operator, which indicates how much a corresponding eigenvector is stretched or shrunk during that transformation. In quantum mechanics, eigenvalues are particularly significant because they represent measurable quantities, or observables, of a physical system, helping to define the state of the system in a vector space. They are closely related to the concepts of operators and their actions on vectors, as well as fundamental properties like spin angular momentum.

5 Must Know Facts For Your Next Test

1. Eigenvalues can be found by solving the characteristic equation $$ ext{det}(A - ext{λ}I) = 0$$ where A is the operator and I is the identity operator.
2. In quantum mechanics, each observable corresponds to a Hermitian operator, and the possible measurement outcomes are given by its eigenvalues.
3. The eigenvalues of an operator can provide insights into the stability and dynamics of a quantum system.
4. The process of diagonalizing an operator involves finding its eigenvalues and corresponding eigenvectors, simplifying many calculations in quantum mechanics.
5. For spin systems, the eigenvalues can represent quantized spin states, showing how angular momentum behaves in discrete units.

Review Questions

• How do eigenvalues relate to observables in quantum mechanics?
  • Eigenvalues are crucial because they correspond to the possible measurement outcomes of an observable when a quantum system is described by a Hermitian operator. When an operator acts on its eigenvector, the result is simply the eigenvalue multiplied by that eigenvector. This relationship shows that measuring an observable will yield one of the eigenvalues, providing key information about the state of the system.
• Discuss the significance of Hermitian operators in relation to eigenvalues and physical measurements.
  • Hermitian operators are vital because they guarantee that all eigenvalues are real, which is necessary for physical measurements since measurable quantities must be real numbers. The eigenvectors of Hermitian operators are also orthogonal, meaning they represent distinct states in quantum mechanics. This allows for a clear interpretation of measurements: each unique eigenvalue corresponds to a different physical state or outcome when measuring an observable.
• Evaluate how understanding eigenvalues enhances our ability to analyze quantum systems and their dynamics.
  • Understanding eigenvalues allows physicists to simplify complex problems involving quantum systems by reducing them to diagonal forms through diagonalization. This not only makes calculations more manageable but also reveals intrinsic properties like energy levels or angular momentum states. By analyzing how these eigenvalues change under different conditions or perturbations, researchers can predict the behavior of systems over time and gain insights into stability and transitions between different quantum states.

Related terms

Eigenvector:

An eigenvector is a non-zero vector that, when an operator acts on it, only changes in scale (not direction) by a scalar known as the eigenvalue.

Linear Operator: A linear operator is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Hermitian Operator:

A Hermitian operator is a specific type of linear operator that has real eigenvalues and orthogonal eigenvectors, making it crucial for measuring physical observables in quantum mechanics.