Eigenstate

Definition

An eigenstate is a specific quantum state of a system that is associated with a definite value of an observable when measured. When a quantum mechanical operator acts on an eigenstate, the result is the same eigenstate multiplied by a constant, known as the eigenvalue. This concept is fundamental in understanding how quantum systems behave under measurements and how operators relate to physical observables.

5 Must Know Facts For Your Next Test

1. Eigenstates are crucial for understanding the postulates of quantum mechanics, particularly in relation to measurement and the behavior of quantum systems.
2. When a measurement is made on a system in an eigenstate, the outcome will always correspond to the eigenvalue associated with that state.
3. Not all states are eigenstates; many states can be expressed as superpositions of multiple eigenstates, leading to different measurement outcomes.
4. The set of all eigenstates corresponding to a particular operator forms a complete basis for the space of states for that observable.
5. The concept of eigenstates extends beyond quantum mechanics and has applications in various fields, including linear algebra and engineering.

Review Questions

• How do eigenstates relate to observables and measurements in quantum mechanics?
  • Eigenstates are directly related to observables in quantum mechanics as they represent the specific states in which measurements yield definite outcomes. When an observable is measured, if the system is in an eigenstate corresponding to that observable, the result will be the associated eigenvalue. This means that understanding eigenstates helps clarify how quantum measurements work and how they produce predictable results.
• Discuss how operators function in relation to eigenstates and provide examples of common operators used in quantum mechanics.
  • Operators act on quantum states to extract measurable quantities, and when they are applied to eigenstates, they return the same eigenstate multiplied by the corresponding eigenvalue. Common operators include the position operator ( ext{X}) and momentum operator ( ext{P}), which correspond to position and momentum observables, respectively. Understanding this relationship between operators and eigenstates is essential for solving problems in quantum mechanics.
• Evaluate the significance of superposition in relation to eigenstates and how it impacts measurement outcomes.
  • Superposition plays a critical role in quantum mechanics, especially when considering how states can exist as combinations of multiple eigenstates. A state that is not an eigenstate can yield different measurement outcomes depending on its composition. This concept illustrates the inherent uncertainty in quantum systems until a measurement collapses the superposition into one of the possible eigenstates, leading to a specific measurement result. Thus, recognizing how superposition interacts with eigenstates deepens our understanding of quantum behavior.

Related terms

Observable: An observable is a physical quantity that can be measured in a quantum system, such as position, momentum, or energy.

Operator: In quantum mechanics, an operator is a mathematical entity that represents an observable and acts on the state of a quantum system.

Eigenvalue:

An eigenvalue is the constant that results from the operation of an operator on its corresponding eigenstate, representing the measurable value of the observable.