In quantum mechanics, stationary states are eigenstates of the Hamiltonian operator, which means that their energy is well-defined and does not change over time. However, it's important to clarify the terminology you used.
A determinate state refers to a state with a definite value of a particular observable. In quantum mechanics, the act of measuring an observable generally involves a collapse of the wavefunction into one of the eigenstates of that observable, thereby yielding a determinate value. However, prior to measurement, the system can exist in a superposition of eigenstates, and the outcome of the measurement is probabilistic.
In the case of stationary states, they are determinate states with respect to the energy observable. They have a definite energy value associated with them. If a system is in a stationary state, a measurement of energy will always yield the same value corresponding to the energy eigenvalue of that state.
However, other observables, such as position or momentum, may not have determinate values in a stationary state. In general, stationary states are not eigenstates of position or momentum operators, and measuring these observables will yield a range of possible values with corresponding probabilities.
To summarize, while stationary states are determinate states with respect to the energy observable, they do not necessarily have determinate values for other observables.