Operator boundedness and domain in quantum mechanics ↩

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created: 2021-12-11 00:36:35
modified: 2022-01-10 04:13:03

In quantum mechanics, we use unbounded operators to describe physical quantities, but these operators cannot be self-adjoint. This is because if

is a linear operator defined on all of

Hilbert space and having the property that

for all

, then

is automatically bounded.

For various reasons (both physical and mathematical), we want our operators of quantum mechanics to be self-adjoint though, so we cannot define the unbounded self-adjoint operators on the entire Hilbert space. Thus, to deal with the unbounded operators of quantum mechanics, we must deal with operators that are defined only on a subspace of the relevant Hilbert space, called the domain of the operator.