Eigenvalue Problem

Definition

The eigenvalue problem refers to a mathematical equation where a linear operator acts on a function, producing a scalar multiple of that function. In quantum mechanics, this concept is critical as it relates to finding the allowed states of a quantum system, known as eigenstates, and their corresponding energy values, called eigenvalues. The relationship between operators, eigenstates, and eigenvalues helps define the behavior of physical systems within Hilbert space.

5 Must Know Facts For Your Next Test

1. The eigenvalue problem can be expressed mathematically as $$A\psi = \lambda\psi$$, where A is an operator, $$\psi$$ is an eigenstate, and $$\lambda$$ is the eigenvalue.
2. Eigenvalues can correspond to measurable quantities in quantum mechanics, such as energy or momentum, making them essential for understanding physical systems.
3. In quantum mechanics, solving the eigenvalue problem allows for determining stationary states of systems, which do not change over time apart from a phase factor.
4. Not all operators have eigenvalues and eigenstates; self-adjoint (or Hermitian) operators are crucial for ensuring real eigenvalues, which correspond to observable quantities.
5. The process of finding eigenvalues and eigenstates often involves techniques like separation of variables and the use of boundary conditions to satisfy physical requirements.

Review Questions

• How does the eigenvalue problem relate to the concept of operators in quantum mechanics?
  • The eigenvalue problem is fundamentally tied to operators in quantum mechanics as it describes how these operators act on state vectors. Specifically, when an operator acts on a state vector and results in a scalar multiple of that vector, we identify the state vector as an eigenstate and the scalar as the corresponding eigenvalue. This relationship is essential in determining physical properties of quantum systems and understanding how measurements correspond to specific outcomes.
• In what ways do eigenvalues represent measurable quantities in quantum systems, and why are self-adjoint operators important in this context?
  • Eigenvalues represent measurable quantities such as energy levels or angular momentum in quantum systems because they provide the possible outcomes when a measurement is made. Self-adjoint (or Hermitian) operators ensure that these eigenvalues are real numbers, which is crucial since only real numbers can be observed in experiments. The properties of self-adjoint operators also guarantee that their eigenstates form a complete basis for the Hilbert space, allowing any state vector to be expressed as a combination of these basis states.
• Evaluate how the solutions to the eigenvalue problem contribute to our understanding of stationary states in quantum mechanics.
  • Solutions to the eigenvalue problem are vital for identifying stationary states in quantum mechanics since these states remain unchanged in time except for a phase factor. By solving the eigenvalue equations associated with relevant operators, we obtain both the eigenvalues and their corresponding eigenstates. These results provide insight into the behavior of quantum systems over time and help predict how systems evolve under various conditions, thereby deepening our understanding of quantum dynamics.

Related terms

Operator: An operator is a mathematical object that acts on elements of a vector space, often representing physical observables in quantum mechanics.

Hilbert Space: Hilbert space is a complete vector space equipped with an inner product, providing the framework for quantum mechanics where state vectors reside.

State Vector: A state vector is a mathematical representation of the state of a quantum system in Hilbert space, containing all necessary information about the system's properties.