The nabla symbol is used in maths (and physics of course) to denote a differential operator. It was introduced by Hamilton around 1837. Its name apparently goes back to a joke by Maxwell. According to Wikipedia, W. Thomson wrote in 1884:
"I took the liberty of asking Professor Bell whether he had a name for this symbol and he has mentioned to me nabla, a humorous suggestion of Maxwell's. It is the name of an Egyptian harp, which was of that shape"
I am kind of glad he didn't suggest to use the Greek name 'psaltery' as I admittedly have no idea how to pronounce it. You might be interested to hear though that it makes an appearance in the bible, Psalm 33:2
"Rejoice in the LORD, O ye righteous: for praise is comely for the upright.
Praise the LORD with harp: sing unto him with the psaltery and an instrument of ten strings. "
The word 'operator' is a very sophisticated expression for a thing that assigns things to things. The telephone operator for example, assigns incoming calls to the desired connection. Its correct mathematical notation is
An operator can be almost everything. Your kid who never tidies up is an operator that assigns toys to places in your living room. If you buy tickets for the opera, the online booking system is an operator that assigns seats to the audience.
A differential operators specifically acts on functions by differentiating them. The nabla for example, when applied to a scalar field, gives the gradient of that field. If you think about the scalar field as an altitude in a mountain range, then the gradient points towards the direction where the increase is the steepest.
Operators are the core concept of quantum mechanics. Quantities that in a classical theories are functions, like the position or energy of an object, become operators. To make something useful out of them, they now have to act on a function - that being the purpose of an operator. In quantum mechanics, it is the well-known wave-function that they act on.
But the usefulness of the operator concept is that one can deal with them on their own without applying them all the time. It's a bit like replacing 'classical' money with a credit card. If you want to see something 'real' you have to 'apply' it to an ATM to get cash. Most often the result is quantized, say, you can only get multiples of $10 or so. You also typically have an offset, a smallest possible amount that you can get. But for most cases, you are fine dealing with the card itself. You have to be a bit careful though if you use it together with other cards, say the club card (payback card, member card, VIP card, whatever) from your local groceries. For your total, it matters in which order you present them at the register. We say that the operators don't commute: the result depends on the order of use.
The nabla is essentially the operator that, when acting on the wave-function, gives the momentum. That is, up to a constant - in this case a relevant constant. But this may be subject of another post.
TAGS: PHYSICS, NABLA, MATHEMATICS, OPERATOR