Eigenvalue

Definition

An eigenvalue is a special scalar associated with a linear transformation represented by a matrix, indicating how much a corresponding eigenvector is stretched or compressed during that transformation. In quantum mechanics, eigenvalues play a critical role in measurement, as they represent the possible outcomes when measuring an observable, linking mathematical operations to physical results.

5 Must Know Facts For Your Next Test

1. Eigenvalues are obtained by solving the characteristic polynomial of a matrix, which determines the values for which the matrix does not have an inverse.
2. In quantum mechanics, when a measurement is made on a system, the result corresponds to one of the eigenvalues of the observable's operator.
3. The probability of obtaining a specific eigenvalue as a measurement result is determined by the square of the amplitude of the corresponding eigenvector in the state vector.
4. Eigenvalues can be real or complex numbers, but when they correspond to observable measurements in quantum mechanics, they are always real and non-negative.
5. The concept of eigenvalues is crucial for understanding phenomena like quantum states collapsing to specific values upon measurement and how quantum systems evolve over time.

Review Questions

• How do eigenvalues relate to the measurement process in quantum mechanics?
  • Eigenvalues are fundamental to the measurement process in quantum mechanics because they represent the possible outcomes when measuring an observable. When an observable is measured, the system's state collapses into an eigenstate corresponding to one of these eigenvalues. Thus, understanding eigenvalues helps explain how measurements yield specific values and the probabilistic nature of quantum systems.
• Discuss the significance of eigenvalues in determining the behavior of quantum systems under measurement.
  • Eigenvalues are significant because they indicate what values can actually be observed when measuring an observable in a quantum system. Each eigenvalue corresponds to an eigenstate that reflects possible configurations of the system. The relationship between these values and their respective probabilities gives insight into how quantum systems behave and transition from one state to another during measurements.
• Evaluate how understanding eigenvalues enhances our comprehension of quantum entanglement and superposition.
  • Understanding eigenvalues deepens our comprehension of quantum entanglement and superposition by illustrating how states can exist in combinations that yield specific measurement results. When multiple particles are entangled, their collective state can be described using joint eigenvalues, allowing us to predict correlated measurement outcomes despite spatial separation. This framework not only clarifies individual particle behaviors but also explains complex interactions inherent in quantum systems through superposition and entangled states.

Related terms

Eigenvector: An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied to it, corresponding to its associated eigenvalue.

Observable: In quantum mechanics, an observable is a physical quantity that can be measured, represented mathematically by an operator with eigenvalues representing the potential measurement outcomes.

State Vector:

A state vector describes the state of a quantum system in a Hilbert space and is used to calculate probabilities of measurement outcomes related to eigenvalues.