Stationary States in Quantum Mechanics

Fundamental theory in physics describing the properties of nature on an atomic scale.

In quantum mechanics, the concept of stationary states is of fundamental importance. These states, also known as energy eigenstates, are solutions to the time-independent Schrödinger equation. They represent quantum states with definite energy, and their properties provide deep insights into the behavior of quantum systems.

Definition and Properties of Stationary States

A stationary state is a state of a quantum system that does not change with time. More specifically, the probability density associated with a stationary state is time-independent. This means that if a system is in a stationary state, measurements of its properties will yield the same results at any point in time.

Mathematically, stationary states are represented by solutions to the time-independent Schrödinger equation:

Hψ = Eψ

where H is the Hamiltonian operator (representing the total energy of the system), ψ is the wave function of the state, and E is the energy eigenvalue associated with the state.

The Concept of Energy Eigenstates

The term "energy eigenstate" is often used interchangeably with "stationary state". An energy eigenstate is a state that yields a definite value (the eigenvalue) when the energy operator (the Hamiltonian) is applied to it. In other words, if a system is in an energy eigenstate, a measurement of its energy will always yield the same result.

The energy eigenvalues are of particular importance because they correspond to the possible outcomes of energy measurements. The associated eigenstates provide a complete description of the quantum system.

Solving the Time-Independent Schrödinger Equation

To find the stationary states of a quantum system, one must solve the time-independent Schrödinger equation. This involves finding the wave functions ψ that satisfy the equation for a given Hamiltonian.

The solutions depend on the specific form of the Hamiltonian, which in turn depends on the potential energy function of the system. For simple systems, such as a particle in a box or a harmonic oscillator, the equation can be solved exactly. For more complex systems, approximate methods may be required.

Superposition of Stationary States

One of the most intriguing aspects of quantum mechanics is the principle of superposition, which states that a quantum system can exist in a combination of different states at the same time. This principle applies to stationary states as well.

A general state of a quantum system can be expressed as a superposition of stationary states, each with its own amplitude. The probabilities of measuring different energy values are given by the squared magnitudes of these amplitudes.

In conclusion, understanding stationary states is crucial for understanding quantum mechanics. These states provide a complete description of a quantum system, and their superpositions allow us to describe the system's evolution over time.