Orthogonality and orthonormality ?

[new_986]

Orthogonality and orthonormality ??

Hi
What does mean orthogonality and orthonormality physically ? e.g. orthogonal or orthonormal wavefunctions
king regards

Nawzad A.

 

[K^2]

Do you understand what orthogonality and orthonormality means for vectors?

Suppose you have two vectors, a and b. They are orthogonal to each other if:

$$\sum_i a_i b*_i = 0$$

Here, b* denotes complex conjugate of b. If b is real, b*=b.

Vector a is a unit vector if

$$\sum_i a_i a*_i = 1$$

Vectors a and b are orthonormal if a and b are unit vectors that are orthogonal to each other.
This works for finite number of dimensions. A function can be thought of as a vector in infinitely many dimensions. (Hilbert Space is the formal name). Each point in a function is a component. The x coordinate takes place of index i, and the y coordinate is the magnitude of the function.

So the two functions a(x) and b(x) are orthogonal if:

$$\int a(x) b*(x) dx = 0$$

Similarly, a normalized function a(x) is the one that conforms to following condition.

$$\int a(x) a*(x) dx = 1$$

Note that I'm not placing boundaries on the integration, even though these should be definite integrals. The reason is that you may want to define your wave functions over all space, in which case integrals are from -∞ to ∞, or over some interval, in which case the integration is done over that interval.

 

[Meir Achuz]

The previous post is a good description.
I just want to add that most physicists use the word orthogonal with the understanding that they really mean orthonormal.

 

[new_986]

Thank you very much this is the clearest description i think,, i can imagine now