Yes, in quantum mechanics, all stationary states are eigenstates of the Hamiltonian operator. The Hamiltonian operator, often denoted as H, represents the total energy of a quantum system. An eigenstate of the Hamiltonian operator is a state in which the outcome of measuring the energy of the system is certain, yielding a specific eigenvalue.
The stationary states, also known as energy eigenstates or eigenfunctions, are solutions to the time-independent Schrödinger equation. These states have well-defined, quantized energies and do not change over time. Mathematically, they satisfy the equation Hψ = Eψ, where H is the Hamiltonian operator, ψ represents the wavefunction of the system, E is the associated energy eigenvalue, and the equation indicates that the Hamiltonian acting on the eigenstate yields the same state multiplied by a constant.
These energy eigenstates form a complete basis for expanding the wavefunction of a quantum system. Any state, whether stationary or time-dependent, can be expressed as a superposition or linear combination of these eigenstates. The coefficients in the linear combination determine the probability amplitudes associated with each eigenstate, allowing for the calculation of probabilities of different measurement outcomes.
In summary, stationary states in quantum mechanics are eigenstates of the Hamiltonian operator, meaning they have definite energies and satisfy the equation Hψ = Eψ. They play a crucial role in the description and analysis of quantum systems.