Proving Orthogonal Functions: Integral of $\phi_{m}^* \phi_{n}$

In summary, the conversation discusses proving a mathematical expression and integrating it over x to show that the integral of two functions is equal to 0 if the energies of the functions are not equal. It is also mentioned that the process of integration is simple.
  • #1
stunner5000pt
1,461
2
In the first hald of this question it was proven that

[tex] -\frac{\hbar^2}{2m} \frac{d}{dx} \left[ \phi_{m}^* \frac{d \phi_{n}}{dx} - \phi_{n} \frac{d \phi_{m}^*}{dx}\right] = (E_{m} - E_{n}) \phi_{m}^* \phi_{n} [/tex]

By integrating over x and by assuming taht Phi n and Phi m are zero are x = +/- infinity show that

[tex] \int_{-infty}^{infty} \phi_{m}^*(x) \phi_{n}(x) dx = 0 [/tex] if Em is not En

so for this do i simply integrate that above expression wrt x?? is it really that simple?
 
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  • #2
I'd say so.
 

1. What is the definition of an orthogonal function?

An orthogonal function is a type of mathematical function where the integral of the product of any two different functions is equal to zero. This means that the functions are perpendicular to each other and do not overlap.

2. Why is proving orthogonal functions important?

Proving orthogonal functions is important because it allows us to simplify complex mathematical problems by breaking them down into smaller, perpendicular components. This can make calculations and analyses easier and more efficient.

3. How do you prove that two functions are orthogonal?

To prove that two functions are orthogonal, you need to show that the integral of their product is equal to zero. This can be done using various mathematical techniques, such as integration by parts or substitution.

4. What is the significance of the integral of $\phi_{m}^* \phi_{n}$ in proving orthogonal functions?

The integral of $\phi_{m}^* \phi_{n}$ is significant because it represents the inner product of two functions. If this inner product is equal to zero, then the functions are orthogonal. Therefore, by calculating this integral, we can determine if two functions are orthogonal or not.

5. Can you provide an example of proving orthogonal functions using the integral of $\phi_{m}^* \phi_{n}$?

One example is proving that the functions $\sin{x}$ and $\cos{x}$ are orthogonal on the interval $[0, \pi]$. We can calculate the integral of $\sin{x} \cos{x}$ over this interval, which is equal to $-\frac{1}{2}\cos^2{x} \Big|_0^\pi = -\frac{1}{2}(-1-1) = 0$. This shows that these two functions are orthogonal on this interval.

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