Stationary states

Definition

Stationary states refer to specific solutions of the Schrödinger equation that describe quantum systems in which the probability distribution of a particle's position does not change over time. These states are characterized by a fixed energy and are represented by wave functions that remain unchanged, except for a phase factor. The significance of stationary states lies in their ability to describe the behavior of quantum systems at equilibrium, making them fundamental to understanding quantum mechanics.

5 Must Know Facts For Your Next Test

1. Stationary states correspond to energy eigenvalues of a quantum system, meaning they represent stable configurations where energy remains constant over time.
2. In stationary states, the expectation values of observables do not change with time, making them essential for predicting long-term behavior in quantum systems.
3. The time-dependent wave function for a stationary state can be expressed as a product of a time-independent part and a time-dependent exponential factor.
4. Examples of stationary states include bound states in atoms, such as electrons in hydrogen, where each state corresponds to a specific energy level.
5. Stationary states play a crucial role in quantum mechanics because they form the basis for more complex behaviors and interactions within quantum systems.

Review Questions

• How do stationary states relate to the solutions of the Schrödinger equation and what significance do they hold in quantum mechanics?
  • Stationary states are specific solutions to the Schrödinger equation that describe systems where probability distributions remain constant over time. This means that once a system is in a stationary state, its properties can be predicted accurately without change. The significance lies in their representation of stable configurations, allowing physicists to understand and analyze quantum systems at equilibrium.
• Describe how the concept of energy eigenstates is connected to stationary states and provide examples from quantum mechanics.
  • Energy eigenstates are a subset of stationary states that correspond to definite energy values within a quantum system. They arise from solving the time-independent Schrödinger equation, resulting in wave functions that represent stable energy levels. For instance, electrons in atoms exist in various energy eigenstates, such as those found in hydrogen, where each energy level corresponds to a unique stationary state.
• Evaluate the implications of stationary states for predicting the behavior of quantum systems under measurement conditions.
  • Stationary states have profound implications for understanding how quantum systems behave when subjected to measurement. Since these states have fixed energies and unchanging probability distributions, measuring an observable associated with a stationary state will yield predictable outcomes. This reliability allows scientists to build models based on these stable configurations, ultimately helping to uncover deeper insights into complex quantum interactions and phenomena.

Related terms

wave function:

A mathematical function that describes the quantum state of a system, containing all the information about the system's properties and probabilities.

quantum superposition:

The principle that allows a quantum system to exist simultaneously in multiple states until it is measured, collapsing into one of the possible states.

energy eigenstates: Specific stationary states associated with definite energy values, where the wave function satisfies the time-independent Schrödinger equation.