Stationary state

In the strange world of quantum mechanics, most things are constantly shifting and changing. However, there are special conditions called stationary states where everything remains constant. A stationary state is a quantum state where all observables are independent of time. This means that if you measure a particle's position, velocity, or spin, those properties will not change as time passes. Because these states do not change in an observable way, they are often called energy eigenstates. They are also known as energy eigenvectors, energy eigenfunctions, or energy eigenkets.

To understand how a stationary state works, we must look at its mathematical structure. Even though the state is called "stationary," the underlying wavefunction is not actually constant. Instead, the wavefunction acts like a standing wave. It possesses a part called a complex phase factor that continually changes over time. This phase factor oscillates with a specific frequency. According to the Planck–Einstein relation, if you multiply this oscillation frequency by the Planck constant, you calculate the energy of the state.

Scientists use specific mathematical tools to define these states. A stationary state is a solution to the time-independent Schrödinger equation. In this equation, the Hamiltonian acts as a linear operator on a vector space. The stationary state is an eigenvector of this Hamiltonian, and its corresponding eigenvalue is the energy. If you plug a stationary state into the time-dependent Schrödinger equation, it still works perfectly. As long as the Hamiltonian remains unchanged in time, the state will continue to behave predictably.

There is a subtle distinction between different types of energy levels. In a hydrogen atom, for example, there are many different states like 1s, 2s, and 2p. In simple, nonrelativistic quantum mechanics, all of these are considered stationary states. However, real-world physics is more complex due to quantum field theory. In reality, only the 1s state, known as the ground state, is truly stationary. Higher energy levels, such as 2s or 2p, can experience spontaneous decay. In these cases, an electron emits photons to drop down to the lower 1s energy level.

This decay happens because of something called vacuum fluctuations. While the approximate math suggests higher states are steady, the true Hamiltonian shows they are not. The 1s ground state is unique because it remains a stationary state according to both the approximate and the true Hamiltonian. This highlights the difference between an idealized mathematical model and the actual physical world. It also shows how even the "emptiness" of a vacuum can influence the stability of a particle's state.

In chemistry, you will often encounter the concept of an orbital. An orbital is very similar to a stationary state, but the context is slightly different. For a molecule that contains only a single electron, such as H2+, the orbital is exactly the same as the total stationary state. However, for molecules with many electrons, an orbital is not the total state of the system. Instead, an orbital describes the stationary state of just one individual electron within that larger group.

To make sense of many-electron systems, scientists use the single-electron approximation. This assumes that we can ignore the instantaneous repulsion between electrons to simplify the math. By using this trick, we can treat the total state of a molecule as a collection of individual electron orbitals. This allows researchers to study complex chemistry using the manageable rules of stationary states. It connects the deep mathematics of quantum mechanics to the practical study of how atoms and molecules behave.