Eigenfunction

Definition

An eigenfunction is a special type of function that remains unchanged, except for a scalar factor, when acted upon by a linear operator. This property makes eigenfunctions crucial in the study of quantum mechanics, as they are directly related to measurable physical quantities or observables. When an observable is measured, the system is found to be in an eigenstate associated with that observable, revealing deep connections between mathematical operators and physical measurements.

5 Must Know Facts For Your Next Test

1. Eigenfunctions are solutions to the eigenvalue equation, expressed as $$ ext{A} ext{f} = ext{λf}$$, where A is an operator, f is the eigenfunction, and λ is the eigenvalue.
2. The set of all eigenfunctions corresponding to a particular operator forms a complete basis for the space of functions being considered, enabling the representation of any arbitrary function as a combination of these eigenfunctions.
3. In quantum mechanics, each observable has a corresponding set of eigenfunctions and eigenvalues, which represent possible outcomes of measurements.
4. When measuring an observable in quantum mechanics, the system collapses into one of its eigenstates (eigenfunctions), and the associated eigenvalue gives the result of that measurement.
5. Different operators can share common eigenfunctions, leading to simultaneous observables that can be measured without affecting each other's values.

Review Questions

• How do eigenfunctions relate to the concept of operators and observables in quantum mechanics?
  • Eigenfunctions are directly connected to operators and observables since they provide specific solutions when an operator acts on them. Each observable corresponds to an operator that operates on a state vector or wave function. The resulting eigenfunctions reveal the possible states of the system that yield measurable values (eigenvalues) when that observable is measured. This relationship underscores how mathematical formalism describes physical reality in quantum mechanics.
• Discuss the significance of measuring observables in terms of their corresponding eigenfunctions and how this affects a quantum system.
  • Measuring an observable collapses a quantum system into one of its eigenstates described by its eigenfunctions. This process impacts the system by determining its state based on the associated eigenvalue. For example, if we measure position, the system will collapse into a specific position eigenfunction, leading to a well-defined value for position while potentially losing information about other observables due to uncertainty principles. This illustrates the fundamental nature of measurement in quantum mechanics.
• Evaluate the implications of having common eigenfunctions among different operators in quantum mechanics and how it affects simultaneous measurements.
  • Having common eigenfunctions for different operators indicates that those observables can be measured simultaneously without interference or uncertainty. This property signifies compatibility between observables and reveals foundational aspects of quantum systems. For instance, if position and momentum share common eigenfunctions (which they typically do not due to Heisenberg's uncertainty principle), it would allow precise knowledge of both quantities at once. Such scenarios challenge our understanding of measurement and state preparation in quantum mechanics.

Related terms

Operator:

An operator is a mathematical entity that acts on a function to produce another function, often representing physical observables in quantum mechanics.

Observable: An observable is a physical quantity that can be measured, such as position, momentum, or energy, typically represented by operators in quantum mechanics.

Eigenvalue:

An eigenvalue is the scalar factor associated with an eigenfunction when acted upon by an operator, indicating the measurement outcome of the corresponding observable.