What is an operator in quantum mechanics?

Answer 1

In the dirac view of quantum mechanics, operators are the center of analysis. An operator is some mathematical operation that acts on the wavefunction (psi) which returns an observable. Lets look at some examples:

say psi=exp(ik(dot)r)*exp(iomega*t) (which is the case for a free particle)

the momentum operator is the -ihbar gradiant applying this to our psi- we get hbar k. This is called the observable.

perhaps more familiar the energy operator which would likewise return hbar omega. Now doesn't that look familiar!

Interestingly enough, these two examples point out that the conservation of momentum and energy stem from the laws of physics being invariant, regardless of position and time.

Answer 2

This is the "field" referred to in Quantum Field Theory (qft). The "operator-valued field" is another way of describing what is sometimes called "second quantization", but which might be better described as "second" or "further" superposition. The idea of "further superposition" is this: In so-called "quantum mechanics", the unstated assumption is that there is such a thing as a distinguishable "particle" that is "there", but its exact position and momentum are indefinite. Therefore, "it" can be thought of as in a state which is a superposition of a number of different position "waveform possibilities" and momentum "waveform possibilities". These two sets of possibilities are not independent, but in a sense co-constitute each other (because position can be relatively definite only to the degree that momentum is relatively indefinite, and vice versa), so that the associated operators "do not commute", and are related by "commutation relations", embodied usually in the "Heisenberg Indefiniteness Principle".

Beginning with the Dirac equation and experiments that showed that "particles" could be created and destroyed, it was realized that one could not assume that a distinguishable "particle" was already "there" at some relatively definite position with some relatively definite momentum. Rather, what was "there" at a particular "spacetime point" was only the a superposition of possibilities for their being one or more (or none) "particles" of one or more types. Until creation and destruction operators (now a new kind of operator at the "further superposition" level) were applied, you couldn't even say that a distinguishable quantum (the word "particle" seems to make less and less sense) was "there". This leads to the idea that what's "there" in the most fundamental sense is a set of quantum creation and destruction processes that are mathematically represented by creation and destruction operators. It would be more correct to say that this deeper level type of field is a "creation/destruction process field", rather than an "operator valued field".

There is the further important point that at this deeper level of field manifesting a deeper level of superposition of waveform possibilities, there are also complementary "commutation relations" between certain pairs of observables, one of the most notable being between number and phase. This relation illuminates the fact that from the deeper perspective we can not assume that there is a definite number of a particular type of quantum present: To measure a relatively definite number, we have to assume a relatively indefinite phase of the associated waveform field, and vice versa.

--Samatman

Answer 3

An operator has a dual meaning in the world of quantum mechanics. The more formal explanation is that an operator is a mathematical operation done to an equation. For example, the H operator in quantum mechanics simply means, "Whatever is in front of me, multiply by (-h^2/2m)D^2+V(r,t), where D is the gradient. So if you see Hx, you would multiply x by everything that is in H.

So H simply represents an equation. In this case, it is the Hamiltonian Operator.

A simpler example would be the parity operator, p. The p operator means "multiply everything by -1". So if you have px, you would end up with -x.

Remember that in quantum mechanics, operators may be commutative, or they may not.

Answer 4

In quantum mechanics, an operator is a mathematical entity that acts on a quantum state to produce another quantum state. Operators represent physical quantities, such as position, momentum, energy, and observables. The measurement of an operator in quantum mechanics corresponds to finding the allowed values of the observable it represents.