Eigenstate

Definition

An eigenstate is a specific quantum state of a system that corresponds to a definite value, or eigenvalue, of an observable quantity when measured. When a measurement is performed on a system in an eigenstate, the result will always yield that particular eigenvalue. This concept is essential in understanding how operators act on quantum states and how measurements affect systems, particularly in the context of angular momentum and spin.

5 Must Know Facts For Your Next Test

1. Eigenstates are the solutions to the eigenvalue equation \( \hat{A}\psi = a\psi \), where \( \hat{A} \) is an operator, \( a \) is the eigenvalue, and \( \psi \) is the eigenstate.
2. When an observable is measured, if the system is in an eigenstate of that observable, it will collapse to that eigenstate and the measurement will yield the corresponding eigenvalue.
3. In quantum mechanics, different observables can have their own distinct sets of eigenstates and eigenvalues.
4. The concept of eigenstates extends to systems with angular momentum and spin, where specific quantized values correspond to different states.
5. An eigenstate can be part of a larger superposition of states, but upon measurement, it will yield one specific eigenvalue corresponding to one of its constituent eigenstates.

Review Questions

• How do eigenstates relate to operators and observables in quantum mechanics?
  • Eigenstates are directly linked to operators and observables in quantum mechanics through the eigenvalue equation. When an observable is represented by an operator and acts on its eigenstate, it yields a specific measurable value known as the eigenvalue. This means that if a system is in an eigenstate of an observable operator, measuring that observable will always return the corresponding eigenvalue without any uncertainty.
• Discuss the significance of eigenstates in the context of angular momentum and spin for quantum systems.
  • In quantum mechanics, angular momentum and spin are represented by specific operators that have their own sets of eigenstates. These eigenstates represent quantized values of angular momentum or spin, crucial for understanding phenomena such as atomic structure and electron configurations. For instance, when measuring the spin of an electron, its state will collapse into one of its spin eigenstates, yielding either spin-up or spin-down with definite values, providing insights into the behavior of particles at the quantum level.
• Evaluate how the concept of superposition interacts with the idea of eigenstates in quantum mechanics.
  • Superposition allows a quantum system to exist in multiple states at once until measured, which means a state could be represented as a combination of several eigenstates. When a measurement occurs, this superposition collapses into one specific eigenstate corresponding to a definite measurement outcome. This interplay illustrates the probabilistic nature of quantum mechanics; while a system may be described by multiple possible states beforehand, the act of measurement forces it into one distinct eigenstate with a well-defined value for the observable being measured.

Related terms

Observable: A physical quantity in quantum mechanics that can be measured, typically represented by an operator.

Operator: A mathematical object in quantum mechanics that acts on the state of a system to extract physical information, such as momentum or position.

Superposition: A principle in quantum mechanics where a system can exist simultaneously in multiple states until measured, often leading to a combination of eigenstates.