Energy Eigenvalue

Definition

An energy eigenvalue is a specific value of energy that corresponds to a quantum state of a system as described by the Schrödinger equation. In quantum mechanics, energy eigenvalues arise when solving the time-independent Schrödinger equation, yielding discrete energy levels for bound states in a potential. These values play a crucial role in determining the allowed energy states of a particle and can be thought of as the quantized energies that the system can occupy.

5 Must Know Facts For Your Next Test

1. Energy eigenvalues are found by applying the Hamiltonian operator to the wavefunction and solving the time-independent Schrödinger equation.
2. In bound systems, energy eigenvalues are quantized, meaning they can only take on specific discrete values rather than any continuous value.
3. The set of energy eigenvalues forms a spectrum which can be discrete for bound states or continuous for free states.
4. Each energy eigenvalue is associated with an eigenstate, which describes the spatial distribution and behavior of a quantum particle in that energy state.
5. Measuring the energy of a quantum system will yield one of its energy eigenvalues with a probability given by the square of the amplitude of the wavefunction.

Review Questions

• How do energy eigenvalues relate to the solutions of the time-independent Schrödinger equation?
  • Energy eigenvalues arise directly from solving the time-independent Schrödinger equation. When this equation is solved using appropriate boundary conditions for a given potential, it yields discrete energy levels corresponding to stable quantum states. Each solution results in an energy eigenvalue, which signifies that these energies are allowed for particles in a bound state within that potential.
• Discuss the implications of quantization in relation to energy eigenvalues in quantum systems.
  • The quantization of energy eigenvalues implies that particles can only exist in specific energy states rather than any arbitrary value. This leads to phenomena such as electron configurations in atoms, where only certain energy levels are permissible. The implications extend to understanding chemical bonding, spectral lines, and other properties observed at atomic and molecular levels, all influenced by these quantized energies.
• Evaluate how changes in potential energy affect the energy eigenvalues in a quantum system.
  • Changes in potential energy significantly influence the calculated energy eigenvalues for a quantum system. If the potential is modified, such as through external fields or changes in geometry, the solutions to the time-independent Schrödinger equation will yield different sets of eigenvalues. For instance, deepening a potential well typically increases the separation between energy levels while also raising their values, leading to a shift in observable phenomena like absorption or emission spectra. This evaluation highlights how dynamic systems are shaped by their potentials and allows for predictions about transitions between states.

Related terms

Wavefunction: A mathematical function that describes the quantum state of a particle or system, containing all the information about the system's behavior and properties.

Hamiltonian Operator:

An operator corresponding to the total energy of a system, used in quantum mechanics to obtain energy eigenvalues and eigenstates from the Schrödinger equation.

Quantum State:

A complete description of a quantum system, represented by a wavefunction, which encompasses all observable properties, including energy.