Sin(kx) is an eigenfunction of the KE operator?

1. Apr 5, 2006

UrbanXrisis

The function sin(kx) is an eigenfunction of the KE operator?

my work:

$$KE=\frac{- \hbar^2}{2m}\frac{\partial^2}{\partial x^2} sin(kx)$$

$$\frac{\hbar^2}{2m}\frac{sin(kx)}{k^2}$$

i'm not sure how to show that a function is an eigenfuntion. what other work do I need to do?

2. Apr 5, 2006

Physics Monkey

Hi UrbanXrisis,

You've calculated your derivative incorrectly. The definition of an eigenfunction $$f$$ with eigenvalue $$\lambda$$ of the kinetic energy operator is $$-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} f = \lambda f$$. Your job is to see if your function satisfies this equation, and if it does, then you should find the eigenvalue $$\lambda$$.

Last edited: Apr 5, 2006
3. Apr 5, 2006

UrbanXrisis

the eigen value is $$\frac{\hbar^2}{2mk^2}$$ isnt it? So to show that a function is an eigenfunction, it must satisfy $$-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} f = \lambda f$$ with $$\lambda$$ being a repeated term... aka the eigenvalue?

That means the function sin(kx) is an eigenfunction of the KE operator right?

4. Apr 5, 2006

Physics Monkey

Yes, you understand that it's an eigenfunction. However, you still have the eigenvalue wrong because you messed up the derivative.

5. Apr 5, 2006

UrbanXrisis

oh whoops.. i was thinking integral $$\frac{\hbar^2k^2}{2m}$$

so as long as the eigenvalue does not change, then a function is an eigenfunction?

6. Apr 5, 2006

Physics Monkey

Yes, you have eigenvalue right now. Also, the eigenvalue doesn't change, it goes with the eigenvector. The key is that the operator acting on the eigenfunction gives you back the same function simply multiplied by a number, the eigenvalue.

7. Apr 5, 2006

UrbanXrisis

you said that lambda is a repeated term. what do you mean by "repeated term"?

8. Apr 5, 2006

Physics Monkey

Actually, you used the phrase "repeated term", I never mentioned it. $\lambda$ is the eigenvalue, and it appears precisely as I described above.

Hope this helps.

9. Apr 6, 2006

Gokul43201

Staff Emeritus
As a postscript to this discussion :

Urbanxrisis, notice that the eigenvalue must have the dimensions of the operator. So, in this problem, your eigenvalue must have dimensions of energy. That should have alerted you to the error in calculating the derivatives.

10. Apr 7, 2006

dextercioby

Actually, that operator has a purely continuous spectrum. Therefore, terms such as "eigenvalue", "eigenfunction", "eigenvector" are wrongly used in connection to that linear operator.

Daniel.

11. Apr 7, 2006

Physics Monkey

Thanks for the comment, dextercioby, but one must be a little careful here. The spectrum of that operator depends on its domain. For the whole real line, the operator turns out to be essentially self adjoint with continuous spectrum and hence technically no eigenvectors. On the half line, the operator has a deficiency index of (1,1) and thus has self adjoint extensions parameterized by a phase. Again, the spectrum is continuous, but depending on the extension one choooses, a discrete bound state can arise. Finally, on a finite interval, the deficiency index is (2,2) which means self adjoint extensions exist which are parameterized by $U(2)$. In this case, each self adjoint extension has a discrete spectrum with eigenvalues and eigenvectors.

In any case, the use of the eigen- words is commonplace amongst physicists even in situations where mathematicians have defined things differently.