Eigenstate

Definition

An eigenstate is a specific quantum state of a quantum system that corresponds to a definite value of an observable, represented by an eigenvalue. When a measurement is performed on a quantum system in an eigenstate, the result will always be the associated eigenvalue. This concept is crucial in understanding how systems transition between states and the probabilities associated with these transitions.

5 Must Know Facts For Your Next Test

1. Eigenstates are solutions to the time-independent Schrödinger equation, providing the possible states of a quantum system at equilibrium.
2. When a quantum system undergoes a transition due to an external interaction, it may change from one eigenstate to another, and selection rules determine the allowed transitions.
3. The probability of finding a system in a particular eigenstate is given by the square of the amplitude of its wave function when expressed in that state.
4. Measurement collapses the wave function of a quantum system into one of its eigenstates, producing a definite result corresponding to the eigenvalue.
5. Different observables can have different sets of eigenstates, and some observables can be incompatible, meaning they cannot have simultaneous eigenstates.

Review Questions

• How do eigenstates relate to measurements in quantum mechanics?
  • Eigenstates are directly tied to measurements because when a quantum system is in an eigenstate of an observable, measuring that observable will yield the corresponding eigenvalue with certainty. This means that knowing the system is in a particular eigenstate allows us to predict the outcome of measurements accurately. In this way, eigenstates help bridge the gap between abstract quantum theory and practical experimental results.
• Discuss how selection rules influence the transitions between eigenstates in quantum systems.
  • Selection rules are criteria derived from symmetry considerations and conservation laws that dictate which transitions between eigenstates are allowed or forbidden. These rules determine the probabilities of transitions when an external influence, such as electromagnetic radiation, interacts with the system. Understanding selection rules helps explain why certain spectral lines are observed while others are not, as they restrict which eigenstates can be connected through these interactions.
• Evaluate the significance of superposition and how it interacts with eigenstates during state transitions.
  • Superposition plays a critical role in quantum mechanics by allowing a system to exist in multiple states simultaneously until a measurement occurs. When considering transitions between eigenstates, superposition describes how a system can evolve into combinations of different eigenstates over time. The interaction of superposition with these states leads to interference effects and complex behaviors in quantum systems, ultimately determining the probabilities of transitioning from one eigenstate to another based on their respective amplitudes.

Related terms

Observable: A physical quantity that can be measured in a quantum system, represented mathematically by an operator.

Eigenvalue:

The value associated with an observable that corresponds to a specific eigenstate, representing the outcome of a measurement.

Superposition: A principle stating that a quantum system can exist simultaneously in multiple states until measured, influencing the transition probabilities between eigenstates.