Is this correct? (eigenfunctions)

[hoch449]

Homework Statement

a) Show that the functions $$f=sin(ax)$$ and $$g=cos(ax)$$ are eigenfunctions of the operator $$\hat{A}=\frac{d^2}{dx^2}$$.

b) What are their corresponding eigenvalues?

c)For what values of $$a$$ are these two eigenfunctions orthogonal?

d) For $$a=\frac{1}{3}$$ construct a linear operator of $$f$$ and $$g$$ which is orthogonal to $$f$$

The Attempt at a Solution

a) $$\hat{A}f=\frac{d^2}{dx^2}sin(ax)=-a^2sin(ax)$$
$$\hat{A}g=\frac{d^2}{dx^2}cos(ax)=-a^2cos(ax)$$

b) the eigenvalues are $$-a^2$$

c)orthogonality condition is: $$\int f^*gdx=0$$

so to satisfy the above condition $$a$$ would have to be $$\pm\frac{n\pi}{2} \pm\n\pi$$ where $$n=\pm1,\pm2,\pm3...$$

d) I have no clue how to do this one.. Any help?

 

[HallsofIvy]

Homework Statement

a) Show that the functions $$f=sin(ax)$$ and $$g=cos(ax)$$ are eigenfunctions of the operator $$\hat{A}=\frac{d^2}{dx^2}$$.

b) What are their corresponding eigenvalues?

c)For what values of $$a$$ are these two eigenfunctions orthogonal?

d) For $$a=\frac{1}{3}$$ construct a linear operator of $$f$$ and $$g$$ which is orthogonal to $$f$$

The Attempt at a Solution

a) $$\hat{A}f=\frac{d^2}{dx^2}sin(ax)=-a^2sin(ax)$$

$$\hat{A}g=\frac{d^2}{dx^2}cos(ax)=-a^2cos(ax)$$

Okay.

b) the eigenvalues are $$-a^2$$

Okay.

c)orthogonality condition is: $$\int f^*gdx=0$$

Integrated over what interval?

so to satisfy the above condition $$a$$ would have to be $$\pm\frac{n\pi}{2} \pm\n\pi$$ where $$n=\pm1,\pm2,\pm3...$$[/quote]
How did you get that?

d) I have no clue how to do this one.. Any help?

What does it mean for a linear functional be be orthogonal to a (function). For that matter why are they asking for a linear operator of f and g? Why not just a linear operator on the set of function f and g belong to?

 

[hoch449]

For part c) the integral is performed under all space.

When you write out the integral, the integrand is $$[sin(\frac{x}{3})]^*cos(\frac{x}{3})$$ with a constant on the outside of the integral. Now for this to be 0. X would have to be those values that I wrote in my previous post. Does this seem right?

and for part d) I believe what they are asking is to make a new function (the linear combination of {f and g}) that will then be orthogonal to f. Using the above orthogonality relation. I am not to sure how to construct the linear combination though..