Differential operators - the rules

[randybryan]

I always get slightly confused with the rules of differentials.

now $$\frac{d^{2}y}{dx^{2}}$$ is the scond derivative of the function y(x

but rooting this does NOT give the first derivative dy/dx

However, with the operator $$\frac{d^{2}}{dx^{2}}$$, it seems that you can root this and it DOES give the first derivative.

Can someone please explain this to me? I may be wrong, but this seems to be the case in my quantum mechanic notes

 

[tiny-tim]

Hi randybryan!

Yes, d²y/dx² = (d/dx)(d/dx)y, but ≠ [(d/dx)y]².

It's for the same reason that sin(sin(y)) ≠ [sin(y)]²

 

[HallsofIvy]

I always get slightly confused with the rules of differentials.

now $$\frac{d^{2}y}{dx^{2}}$$ is the scond derivative of the function y(x

but rooting this does NOT give the first derivative dy/dx

However, with the operator $$\frac{d^{2}}{dx^{2}}$$, it seems that you can root this and it DOES give the first derivative.

Can someone please explain this to me? I may be wrong, but this seems to be the case in my quantum mechanic notes

I suspect that your "quantum mechanic notes" (don't hold mathematicians responsible for what a physicist says!) are using a special notation in which "square root" has some kind of operational definition. That is, "$f^2(x)$" does not mean f(x) times itself but f(f(x)) and $\sqrt{f}$ is the inverse of that.

 

[randybryan]

Ahh the ongoing feud that is Maths nomenclature vs Physics nomenclature.

 

[blue_raver22]

Differential operators are mathematical tools used in calculus to represent differentiation, or the process of finding the rate of change of a function. They are commonly used in physics and engineering to model physical systems and make predictions about their behavior.

The rules for differential operators are based on the fundamental rules of differentiation, which state that the derivative of a function is equal to the slope of the tangent line to the function's curve at a given point. Specifically, the derivative of a function f(x) is represented by the operator \frac{d}{dx}, and the second derivative is represented by \frac{d^{2}}{dx^{2}}.

When applying these operators to a function, the result is a new function that represents the derivative or second derivative of the original function. However, the order in which these operators are applied matters.

In the case of \frac{d^{2}}{dx^{2}}, this operator represents the second derivative, so applying it once results in the first derivative, \frac{d}{dx}. This is because the second derivative is the rate of change of the first derivative, which is the rate of change of the original function.

On the other hand, taking the square root of \frac{d^{2}}{dx^{2}} does not result in the first derivative, as you mentioned. This is because taking the square root does not change the order of the operators, so the result is still the second derivative.

In quantum mechanics, differential operators are used to represent physical observables, such as position, momentum, and energy. These operators follow the same rules as traditional differential operators, but they may have additional properties and behaviors due to the quantum nature of the systems they describe.

I hope this explanation helps clarify the rules of differential operators for you. It's important to remember that these operators are simply mathematical tools for representing differentiation, and their behavior follows the same rules as traditional derivatives.