Ansatz

An Ansatz is an assumed form for a function or state that is used as the starting point for solving a problem in mathematics and physics. The term comes from German and roughly means an approach or setup. An Ansatz is chosen to reflect known constraints, such as symmetries, boundary conditions, or conservation laws, so that the remaining task reduces to determining a finite set of parameters.

Once proposed, the Ansatz is substituted into the governing equations or variational principle. If it leads to a consistent solution, or if the residual is minimized, it provides a candidate solution. The Ansatz is not guaranteed to be exact; it is a guess that can be exact in special cases or serve as a good approximation. Its success depends on how well the chosen form captures the essential structure of the problem.

Examples include using a polynomial or exponential form to solve differential equations, a Gaussian trial wavefunction in quantum mechanics, or a product form in mean-field theories. The Bethe Ansatz is a notable example in many-body physics that constructs exact eigenstates for certain integrable models. In variational methods, the parameters in the Ansatz are varied to minimize an energy functional, producing an upper bound on the ground-state energy.

Historically, Ansätze have played a central role in developing analytical insights and guiding numerical methods, particularly in systems with symmetry or complex interactions where exact solutions are not readily available.