Is \( e^{i\theta} \) an Eigenfunction of the Given Operators?

[QuantumMech]

For the following:

$$\begin{gather*} \hat{\Omega} = \frac{d}{d\theta}sin \theta \frac{d}{d\theta}\\function = e^{i\theta} \end{gather*}$$

Use the operator on the function and is it an eigenfunction of $\hat{\Omega}$?

Thanks. I don't think it is.

There is also another problem with $\hat{\Omega} = \frac{d^2}{dx^2} - 4x^2$. I don't think this is an eigenfunction either.

 

[dextercioby]

It isn't an eigenfunction for any of he 2 op-s.

Daniel.

 

[thepatientmental]

No, you are not missing anything. In order for a function to be an eigenfunction of an operator, it must satisfy the eigenvalue equation \hat{\Omega}f = \lambda f, where \lambda is a constant. In this case, the given function e^{i\theta} does not satisfy this equation, therefore it is not an eigenfunction of \hat{\Omega}. Similarly, the second problem with \hat{\Omega} = \frac{d^2}{dx^2} - 4x^2 also does not satisfy the eigenvalue equation and is not an eigenfunction. It is important to carefully check the requirements for a function to be an eigenfunction before determining its status.