What is an operator? What is its purpose? Is the Hamiltonian operator one of many operators? What makes it different?

I know of a momentum operator and an energy operator from my physics class. Does anyone else know of more?

 

edit: Ok, looks to be a position operator as well.

there are loads of operators. you can have raising and lowering operators in atoms for example.

Originally posted by MajinVegeta

What is an operator? What is its purpose?

 

An operator is any mathematical object that acts on any other mathematical object to return a third mathematical object.

 

Operators arose in QM because of the discrete spectra of radiators. Heisenberg first noted that the experimental results could be reproduced if one were to regard the state of an atom as a vector and represent the physical observable as a matrix. Schrodinger developed his version of QM (the one we still use today) not in terms of matrix operators, but in terms of differential operators. It turns out that the two approaches of QM are equivalent. Indeed, it is known from mathematics that matrix operators and differential operators are equivalent.

 

Is the Hamiltonian operator one of many operators? What makes it different?

 

All observables in QM (momentum, energy, position, angular momentum,...) are represented by an operator. Also, many non-observables are represented by operators, for convenience. Examples are the ladder operators the Radical Edward mentioned.

 

Tom