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Physicists often have a very precise mathematical theory describing how a system will behave. Unfortunately, it is often the case that solving the theory's equations ab-initio in order to produce a useful prediction is not realistic. This is especially true with quantum mechanics, where only a handful of simple models can be solved exactly. This is where the computational physicist steps in.
Challenges in computational physics
Physics problems are in general very difficult to solve exactly. Even apparently simple problems, such as calculating the wavefunction of an electron orbiting an atom in a strong electric field, may require great effort to formulate a practical algorithm (if one can be found).
In addition, the computational cost of solving quantum mechanical problems is generally exponential in the size of the system (see computational complexity theory). Seeing as a typical macroscopic solid has of the order of 1023 constituent particles, it may be somewhat of an understatement to say this is a bit of a problem.
Applications of computational physics
Computational methods are widely used in solid state physics, fluid mechanics and lattice quantum chromodynamics among other areas. Computational physics borrows a number of ideas from computational chemistry - for example, the density functional theory used by computational physicists to calculate properties of solids is basically the same as that used by chemists to calculate the properties of molecules.
Many other more general numerical problems fall loosely under the domain of computational physics, although they could easily be considered pure mathematics or part of any number of applied areas. For example:
- Solving differential equations
- Evaluating integrals
- Stochastic methods, specifically the Monte Carlo method
- Specialised partial differential equation methods, for example the finite difference method and the finite element method
- The matrix eigenvalue problem – i.e. the problem of finding eigenvalues of very large matrices.
- The pseudo-spectral method
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