Energy eigenstate

Definition

An energy eigenstate is a specific quantum state of a system that corresponds to a definite energy value, described mathematically by an eigenfunction of the Hamiltonian operator. In quantum mechanics, these states are crucial because they represent the possible states that a particle can occupy, and each eigenstate is associated with a particular energy level that the system can have when it is not influenced by external forces.

5 Must Know Facts For Your Next Test

1. Energy eigenstates are obtained by solving the time-independent Schrödinger equation for a given Hamiltonian, yielding discrete energy levels for bound systems.
2. Each energy eigenstate has an associated energy eigenvalue that represents the energy measurement outcome when the system is in that state.
3. In perturbation theory, energy eigenstates are modified due to small perturbations in the system, allowing for corrections to be calculated for their energy levels.
4. The principle of superposition allows any quantum state to be expressed as a linear combination of energy eigenstates, leading to complex behavior in quantum systems.
5. The variational principle states that the expectation value of the energy calculated from any trial wavefunction will always be greater than or equal to the ground state energy eigenvalue.

Review Questions

• How do energy eigenstates relate to the Hamiltonian operator in quantum mechanics?
  • Energy eigenstates are directly linked to the Hamiltonian operator because they are defined as solutions to the time-independent Schrödinger equation where the Hamiltonian acts on a wavefunction. The eigenfunctions obtained from this equation correspond to specific energy levels, represented by eigenvalues. This relationship is fundamental in quantum mechanics as it helps describe how systems evolve over time and how they respond to external influences.
• Discuss how perturbation theory affects energy eigenstates and their associated energy levels.
  • In perturbation theory, when a quantum system experiences small changes in its potential energy, the original energy eigenstates can be slightly altered. This method allows us to calculate new approximate energy levels by introducing corrections based on the strength and nature of the perturbation. The adjustments help predict how systems behave under non-ideal conditions and highlight the sensitivity of energy eigenstates to external factors.
• Evaluate the role of energy eigenstates in understanding quantum systems through the variational principle.
  • Energy eigenstates play a critical role in applying the variational principle, which provides insight into estimating ground state energies. By utilizing trial wavefunctions that can be expressed as combinations of energy eigenstates, one can compute expectation values that serve as upper bounds for actual ground state energies. This principle not only helps in approximating energies but also illustrates how closely a chosen trial state resembles true energy eigenstates, providing deeper understanding of complex quantum systems.

Related terms

Hamiltonian: The Hamiltonian is the operator corresponding to the total energy of the system, which includes both kinetic and potential energies.

Wavefunction: A wavefunction is a mathematical description of the quantum state of a particle or system, providing information about the probability distribution of its position and momentum.

Perturbation Theory: Perturbation theory is a method used to approximate the behavior of quantum systems when they are subject to small disturbances or changes in potential energy.