What is the role of linear operators in quantum mechanics?

[Roodles01]

Homework Statement

Just starting third level Uni. stuff & am faced with linear operators from Quantum Mechanics & need a little help.
OK, an operator, Ô, is said to be linear if it satisfies the equation
Ô(α f1 + β f2) = α(Ô f1) + β(Ô f2)
Fine

but I have an equation I can't wrap my head around, maybe just rusty, a hint would be nice, though.

Homework Equations

Ô1 = d/dx;
Ô2 =3 d/dx +3x^2;

　
Find the new functions obtained by acting with each of these operators on
(a) g(x, t) =3 d/dx +3x^2
(b) h(x, t)=α sin(kx − ωt).

The Attempt at a Solution

Now
Ô1 g(x,t) = 6xt^3
But not sure about how to get
Ô2 g(x,t) =

how to get this middle bit, please . . . . .

Answer is 18xt^3 + 9x^4 t^3

 

[dextercioby]

You realize there's no 't' dependence in g(x,t), so that the outcome must be independent of t, right ? Actually the way you wrote it, g(x,t) is Ô₂, not a wavefunction, but also an operator.

 

[Roodles01]

I went back to the question, scanned the problem in & looking again showed a huge error in seeing things.
What a twit.