Functional variation

Functional variation refers to the change in a functional’s output as a result of a small change in the functional’s input on a specified domain.



What is a functional?


A functional is a function whose value depends on a function instead of an independent variable. For example, the expectation value of the Hamiltonian is a functional, where .


For a small variation in ,

where is a small arbitrary change in .

Since is small, , and

where .

If , we have or

which means that a small change in the functional’s input yields no change in the functional’s output. This implies a minimum energy according to the variational principle.


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