Eigenstates

Definition

Eigenstates are specific quantum states of a system that correspond to a particular eigenvalue of an observable operator, representing a measurable physical quantity. When a measurement is made, the system collapses into one of its eigenstates, yielding the associated eigenvalue as the result. Understanding eigenstates is crucial for analyzing how quantum systems respond to perturbations and how they evolve in quantum mechanics.

5 Must Know Facts For Your Next Test

1. In non-degenerate perturbation theory, eigenstates are used to describe how a system's energy levels change under small perturbations, allowing for predictions of new energy eigenvalues.
2. For systems with discrete energy levels, each eigenstate corresponds to a unique energy level, making it possible to understand how external influences can modify these states.
3. The creation and annihilation operators act on eigenstates in quantum mechanics, transforming them into other eigenstates or superpositions of states, which is essential in describing particle behavior.
4. Eigenstates are often normalized, meaning their total probability density equals one, ensuring that when a measurement is made, the result adheres to the probabilistic nature of quantum mechanics.
5. Understanding eigenstates allows physicists to predict measurement outcomes and analyze phenomena like tunneling and transitions between states in various quantum systems.

Review Questions

• How do eigenstates relate to the concept of measurements in quantum mechanics?
  • Eigenstates are directly tied to measurements because when an observable is measured, the system collapses into one of its eigenstates. Each eigenstate corresponds to a specific eigenvalue, which represents the measurement outcome. Therefore, understanding how these states function helps us comprehend how quantum systems behave under observation and how they yield particular results based on their inherent properties.
• Discuss the role of eigenstates in non-degenerate perturbation theory and how they help predict changes in energy levels.
  • In non-degenerate perturbation theory, eigenstates serve as the foundation for analyzing how slight changes in a system affect its energy levels. When a perturbation is applied, the original eigenstates are perturbed slightly, leading to new energy eigenvalues and adjusted states. This approach allows physicists to make accurate predictions about the behavior of quantum systems under external influences by examining how these altered eigenstates reflect changes in energy and dynamics.
• Evaluate the impact of creation and annihilation operators on the eigenstates of a quantum harmonic oscillator.
  • Creation and annihilation operators play a significant role in manipulating the eigenstates of a quantum harmonic oscillator by allowing transitions between these states. The creation operator increases the quantum number associated with an eigenstate, effectively moving it to a higher energy state. Conversely, the annihilation operator lowers the energy state. This interplay allows us to understand processes like photon emission and absorption in quantum optics, demonstrating how changes in energy states influence particle behavior and interactions.

Related terms

Observable: An observable is a physical quantity in quantum mechanics that can be measured, such as position or momentum, and is represented mathematically by an operator.

Eigenvalue:

An eigenvalue is a scalar associated with an eigenstate, representing the outcome of measuring the corresponding observable for that state.

Superposition: Superposition refers to the principle that a quantum system can exist in multiple states simultaneously until it is measured, leading to different possible outcomes.