Eigenstates

Definition

Eigenstates are specific states of a quantum system that correspond to definite values (eigenvalues) of an observable, represented mathematically by operators acting on the state vector in Hilbert space. They play a crucial role in quantum mechanics as they help define the possible outcomes of measurements and can also be utilized in semiclassical approximations and symmetry operations.

5 Must Know Facts For Your Next Test

1. Eigenstates must satisfy the time-independent Schrödinger equation, where the Hamiltonian operator acts on the eigenstate to yield an eigenvalue that corresponds to energy levels.
2. When a measurement is made on a quantum system, it collapses into one of its eigenstates, which determines the result based on the probabilities derived from its wavefunction.
3. In semiclassical methods like WKB approximation, eigenstates can be used to approximate the solutions of quantum systems in regions where classical mechanics applies.
4. Symmetries in quantum mechanics can lead to conservation laws, and eigenstates often exhibit transformation properties under discrete symmetries like parity and time reversal.
5. The completeness of eigenstates in Hilbert space allows any arbitrary quantum state to be expressed as a superposition of these states, facilitating calculations and predictions in quantum mechanics.

Review Questions

• How do eigenstates relate to the measurement process in quantum mechanics, particularly when discussing observable quantities?
  • In quantum mechanics, eigenstates are directly linked to the measurement process because they represent the possible outcomes when measuring an observable. When a measurement occurs, the quantum state collapses into one of its eigenstates, corresponding to a definite value (eigenvalue) for that observable. The probabilities of obtaining each eigenvalue are given by the square of the coefficients of the wavefunction expansion in terms of those eigenstates.
• Discuss how the WKB approximation utilizes eigenstates in semiclassical methods to analyze quantum systems.
  • The WKB approximation is a technique used in semiclassical physics where eigenstates help bridge classical and quantum descriptions. In this method, one assumes that the wavefunction can be expressed in terms of classical paths, leading to solutions that closely resemble those found using classical mechanics. By finding these approximate eigenstates, physicists can analyze systems where classical mechanics predominates while still accounting for quantum effects, enhancing our understanding of particle behavior in potential wells.
• Evaluate the significance of discrete symmetries such as parity and time reversal on the properties of eigenstates and their implications for conservation laws.
  • Discrete symmetries like parity and time reversal have profound implications for the properties of eigenstates in quantum mechanics. Eigenstates can transform under these symmetries, influencing conservation laws such as parity conservation in certain interactions. For example, if a system's Hamiltonian is invariant under parity transformations, then its eigenstates will also respect this symmetry. This analysis helps identify conserved quantities and allows predictions about particle interactions and decays within quantum systems.

Related terms

Eigenvalues: Eigenvalues are the scalar values associated with eigenstates when a linear operator acts on them, representing measurable quantities in quantum systems.

Wavefunction: The wavefunction is a mathematical function describing the quantum state of a particle or system, which is directly related to eigenstates through the solutions of the Schrödinger equation.

Observable:

An observable is a physical quantity in quantum mechanics that can be measured, represented by Hermitian operators whose eigenstates correspond to possible measurement outcomes.