Eigenstates

Definition

Eigenstates are specific quantum states of a quantum system that correspond to a particular eigenvalue of an observable operator. When a measurement is performed on a quantum system, the result is one of the possible eigenvalues, and the system collapses into the corresponding eigenstate. This connection is crucial for understanding how measurement impacts a system and illustrates the inherent uncertainty present in quantum mechanics.

5 Must Know Facts For Your Next Test

1. Eigenstates are critical for understanding measurement in quantum mechanics because they determine the possible outcomes of an observable when measured.
2. Each observable has a set of eigenstates, and these states form a complete basis for the state space of the quantum system.
3. When a quantum system is in an eigenstate of an observable, measuring that observable will yield the associated eigenvalue with certainty.
4. The concept of eigenstates is intimately linked with the uncertainty principle, as measuring one observable can affect the state and thus the uncertainty of another observable.
5. If a system is not in an eigenstate of an observable at the time of measurement, it will collapse into one of the eigenstates upon measurement, which introduces inherent randomness.

Review Questions

• How do eigenstates relate to measurements in quantum mechanics?
  • Eigenstates are directly related to measurements because they represent the states a quantum system can collapse into when an observable is measured. Each observable has specific eigenstates corresponding to its eigenvalues. When you perform a measurement, if the system is in an eigenstate of that observable, you will get a definite outcome. This relationship illustrates how measurements in quantum mechanics lead to distinct results based on the underlying state of the system.
• Discuss how the uncertainty principle connects with the concept of eigenstates.
  • The uncertainty principle states that certain pairs of observables cannot be precisely measured simultaneously, such as position and momentum. Eigenstates play a key role in this principle because measuring one observable collapses the wavefunction into an eigenstate of that observable, which inherently alters our knowledge about complementary observables. For instance, if you measure position accurately (collapsing into position eigenstates), your certainty about momentum decreases, reflecting the core idea of uncertainty in quantum systems.
• Evaluate the implications of superposition on eigenstates and their role during measurements.
  • Superposition allows a quantum system to exist in multiple states at once, meaning it can be represented as a linear combination of different eigenstates. When a measurement occurs, this superposition collapses into one specific eigenstate based on probabilities determined by the coefficients in the superposition. This highlights not only the probabilistic nature of quantum measurements but also emphasizes how potential outcomes are influenced by the system's state prior to measurement, ultimately shaping our understanding of reality in quantum mechanics.

Related terms

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

Superposition:

A fundamental principle in quantum mechanics where a quantum system can exist in multiple states simultaneously until a measurement collapses it into one eigenstate.

Wavefunction: A mathematical function that describes the quantum state of a system, containing all information about the system and how it evolves over time.