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A property of a physical system, such as the entropy of a gas, that stays approximately constant when changes occur slowly is called an adiabatic invariant. By this it is meant that if a system is varied between two end points, as the time for the variation between the end points is increased to infinity, the variation of an adiabatic invariant between the two end points goes to zero.
In thermodynamics, an adiabatic process is a change that occurs without heat flow; it may be slow or fast. A reversible adiabatic process is an adiabatic process that occurs slowly compared to the time to reach equilibrium. In a reversible adiabatic process, the system is in equilibrium at all stages and the entropy is constant. In the 1st half of the 20th century the scientists that worked in quantum physics used the term "adiabatic" for reversible adiabatic processes and later for any gradually changing conditions which allow the system to adapt its configuration. The quantum mechanical definition is closer to the thermodynamical concept of a quasistatic process and has no direct relation with adiabatic processes in thermodynamics.
In mechanics, an adiabatic change is a slow deformation of the Hamiltonian, where the fractional rate of change of the energy is much slower than the orbital frequency. The area enclosed by the different motions in phase space are the adiabatic invariants.
In quantum mechanics, an adiabatic change is one that occurs at a rate much slower than the difference in frequency between energy eigenstates. In this case, the energy states of the system do not make transitions, so that the quantum number is an adiabatic invariant.
The old quantum theory was formulated by equating the quantum number of a system with its classical adiabatic invariant. This determined the form of the Bohr–Sommerfeld quantization rule: the quantum number is the area in phase space of the classical orbit.