Verifying Operator Equation: d^2/dx^2 - x^2 -1

[sanitykey]

Homework Statement

Verify the operator equation

(d/dx + x)(d/dx - x) = d^2/dx^2 - x^2 -1

(where d is meant to be the partial derivative symbol)

Homework Equations

None that are obvious to me?

The Attempt at a Solution

The truth is I'm not really sure how i should be going about this i tried expanding the brackets to get:

d^2/dx^2 + d/dx*x - d/dx*x - x^2

I didn't know whether having d/dx*x means i should differentiate or just leave it but i thought they'd cancel either way giving d^2/dx^2 - x^2 so where does the -1 come from?

 

[genneth]

You can't reorder things:

$$\left(\frac{d}{dx} + x\right)\left(\frac{d}{dx} - x\right) = \frac{d^2}{dx^2} + x \frac{d}{dx} - \frac{d}{dx}x - x^2$$

 

[Dick]

Let your operator operate on something, like f(x). That will make what's going on much clearer.

 

[sanitykey]

I think i understand why it's important to keep the order so if i applied the operator to f(x) i think it'd become:

d^2/dx^2(f[x]) + x*d/dx(f[x]) - d/dx(x*f[x]) - x^2 *f[x]

f''(x) + x*f'(x) - {f(x) + x*f'(x)} - x^2 *f(x)

f''(x) + x*f'(x) - x*f'(x) - x^2 *f(x) - f(x)

f''(x) - x^2 *f(x) - f(x)

which is the same as if i'd done this to the function? d^2/dx^2 - x^2 -1

 

[Dick]

I think you are right.

 

[sanitykey]

Thanks for the help :)