# Orthonormal Basis: Showing Wave Functions are Orthonormal

• asi123
In summary: The attempt at a solution is to find a function that satisfies the equation A^2=1/2, which can be done by taking the square of both sides.The attempt at a solution is to find a function that satisfies the equation A^2=1/2, which can be done by taking the square of both sides.The mistake is that they are thinking that the function must be a linear function. The answer is that the function can be a logarithmic or exponential function.The mistake is that they are thinking that the function must be a linear function. The answer is that the function can be a logarithmic or exponential function.

## Homework Statement

Hey guys.

http://img39.imageshack.us/img39/2345/27760913.jpg [Broken]

I need to show that these wave functions are orthonormal.
I'm a bit confuse, what's i and what's j?
I mean, do I need to take both of the functions, put them in the integral and to show that the result is the Kronecker delta?
Can I neglect the exponent for this?

Thanks a lot.

## The Attempt at a Solution

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i and j are just arbitrary indices. Yes you have to calculate the integrals for different cases. When i=j the exponential cancels (when you take the complex conjugate it changes sign). When i and j aren't equal you'll get some exponential dependence as well but in that case you should get zero anyway.

phsopher said:
i and j are just arbitrary indices. Yes you have to calculate the integrals for different cases. When i=j the exponential cancels (when you take the complex conjugate it changes sign). When i and j aren't equal you'll get some exponential dependence as well but in that case you should get zero anyway.

Well, where are i and j in my problem?
I mean, this is not a series, it's a function.

asi123 said:
Well, where are i and j in my problem?
I mean, this is not a series, it's a function.

You have two wave functions, $\psi_1$ and $\psi_2$, so the indices $i$ and $j$ can each take on the values $1$ and $2$.

gabbagabbahey said:
You have two wave functions, $\psi_1$ and $\psi_2$, so the indices $i$ and $j$ can each take on the values $1$ and $2$.

Yeah, but I don't have i and j inside the functions so how can I come up with the kronecker delta?

How can I show that if i=j then it's 1 and if i does not equal to j, it's 0 if I don't have i and j?

Thanks.

asi123 said:
Yeah, but I don't have i and j inside the functions so how can I come up with the kronecker delta?

How can I show that if i=j then it's 1 and if i does not equal to j, it's 0 if I don't have i and j?

Thanks.

Showing that

$$\int \psi_i \psi_j dx =\delta_{ij}$$

just means that you need to show:

$$\int \psi_1 \psi_1 dx =\int \psi_2 \psi_2 dx =1$$

and

$$\int \psi_1 \psi_2 dx=\int \psi_2 \psi_1 dx =0$$

gabbagabbahey said:
Showing that

$$\int \psi_i \psi_j dx =\delta_{ij}$$

just means that you need to show:

$$\int \psi_1 \psi_1 dx =\int \psi_2 \psi_2 dx =1$$

and

$$\int \psi_1 \psi_2 dx=\int \psi_2 \psi_1 dx =0$$

Oh, now I get it.

Thanks a lot.

Well, here is the second part of the question

http://img207.imageshack.us/img207/879/95899388.jpg [Broken]

I think they have a mistake, I marked it in the red box.
Shouldn't it be A^2=1/2 ?
Am I missing something?

Thanks a lot.

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## 1. What is an orthonormal basis?

An orthonormal basis is a set of vectors in a vector space that are orthogonal (perpendicular) to each other and have a length of 1. This means that each vector is independent and has the same magnitude, making it a useful tool for representing and analyzing complex systems in mathematics and physics.

## 2. How are wave functions related to orthonormal basis?

Wave functions are mathematical representations of quantum mechanical systems. They can be written as linear combinations of orthonormal basis vectors, which allow us to analyze and manipulate these systems using the principles of linear algebra.

## 3. How can we show that wave functions are orthonormal?

To show that wave functions are orthonormal, we must first prove that they are orthogonal (have a dot product of 0) and that they have a length of 1. This can be done by using the properties of the inner product and the normalization condition of the wave function.

## 4. What is the significance of orthonormal basis in quantum mechanics?

In quantum mechanics, orthonormal basis vectors are used to represent the state of a system, allowing us to make predictions about the behavior of the system. They also play a crucial role in the mathematical framework of quantum mechanics, making it easier to solve complex problems and understand the physical world on a microscopic level.

## 5. Can orthonormal basis vectors be used in other areas of science?

Yes, orthonormal basis vectors are not limited to quantum mechanics and can be applied in various areas of science, such as signal processing, image processing, and data analysis. They provide a useful tool for decomposing complex systems into simpler components, making it easier to study and manipulate them.