Question

what a different beetwen eigen function and eigen value?

Answer 1

How i use the eigen function and eigen value,for example this:

Answer 2

I don't have any experience with this application of eigenfunctions and eigenvalues. Sorry. Try doing a google search.

Answer 3

ok i will try..

Answer 4

If you can take a photo of the question and paste that I'll have a look. (I can't read the .docx right now.)

Answer 5

ok james.i will take....in there

Answer 6

Ok. In general, eigenfunctions/values/vectors are associated with linear maps between vector spaces.

For the functions you have given, the vector space in question is the set of infinitely differential functions over the real numbers \( C^\infty(\mathbb{R}) \), and the linear maps are linear combinations of the differential operator \( D \).

For example ...

Answer 7

...for example, \( Df = \lambda f \), which in ODE notation is \( df/dx = \lambda f \). \( D \) is a linear map
\[ D : C^\infty \rightarrow C^\infty \]
and for every real number is \( \lambda \) an eigenvalue because for every real number is there a solution to
\[ Df = \lambda f \ \ --- (*)\]

For a given eigenvalue \( lambda \), the corresponding eigenfunction \( f \) is the solution of the differentiable equation (*); namely,
\[ f(x) = e^{\lambda x} \]

Another linear operator is \( D^2 = d^2/dx^2 \). The solutions of
\[ D^2f = \lambda f \]
are for positive \( \lambda \) the hyperbolic functions sinh and cosh, and for negative \( \lambda \), sin and cos

Now, given all that, can you clarify the wording of your question. Are you looking for the differential operator, i.e., the maps, which would have as eigenfunctions those listed? Or is it something else?

Answer 8

For example, let's look at the first function, \( f(x) = e^{\alpha x} \). One might ask: what linear operator and eigenvalue would correspond to this function f as an eigenfunction.

In the case, such an operator would be just \( D \) itself an the corresponding eigenvalue is \( \alpha \). That is, the function f is the solution of
\[ Df = \alpha f \]

Answer 9

For the second function, \( f(x) = e^{ibx} \), if we ask the same questions:
- what operator and eigenvalue would correspond to this function f as an eigenfunction?

Then \( D^2 \) does the trick with eigenvalue \( -b^2 \). That is, f is a solution of
\[ D^2f = -b^2 f \]
because \( f(x) = e^{ibx} \) is a solution to
\[ \frac{d^2f}{dx^2} + b^2 f = 0 \]

Making sense?

Answer 10

yes.....
but what is ODE?

Answer 11

ODE = ordinary differential equation

Answer 12

I take your silence to mean you're having a hard time following what I've written. But I'm also not absolutely certain of your question. So talk to me. Tell me what you're thinking.

Answer 13

just for what we must find eigen function or eigen value?

Answer 14

I'm sorry, I don't understand that.

Answer 15

I imagine you've asked this question in physics because applying this theory is fundamental in quantum mechanics. If that's right, then you need to make sure you have a strong grasp of linear algebra and differential equations.

If you're uncertain of the concept of eigenvalues/vectors in general, you should go back and review the basic material. For example, begin here with this video and the ones that follow:
http://www.khanacademy.org/video/linear-algebra--introduction-to-eigenvalues-and-eigenvectors?playlist=Linear+Algebra

The application of this concept to functions is more advanced and I don't think it is in Khan Academy material. But I hope you do have a text book that explains it.

Answer 16

please gave me suggest text book which explain about aplication eigen value or quantum physic

Answer 17

Are you taking a course? Do you not have a text book for that?

Answer 18

just from griffit mybook