# Eigenfunction/Eigenvalue question

• noahcharris
In summary, an eigenfunction/eigenvalue pair is a mathematical concept used in linear algebra and differential equations. Eigenfunctions are functions that remain unchanged when multiplied by a constant, while eigenvalues are the constants by which the eigenfunctions are scaled. They are important because they simplify complex problems and have many applications in science and engineering. To calculate them, an eigenvalue equation must be solved. Eigenfunctions and eigenvalues can have complex values, which are necessary in some cases. Each linear operator has a unique set of eigenfunctions and eigenvalues.
noahcharris
Hey all,

So I was watching an MIT OCW video on intro QM and came across this 'clicker' question. (Shows up at 4:45)

Apparently the correct answer is B, but C looks valid to me as well. Could someone explain why C is incorrect?

Thanks

Work it out with pencil and paper carefully, and you'll see that there are only two combinations that work.

noahcharris
Oh wow, I seriously misread the question... Thank you. This thread should be closed probably.

## 1. What is an eigenfunction/eigenvalue?

An eigenfunction/eigenvalue pair is a mathematical concept used in linear algebra and differential equations. An eigenfunction is a function that, when multiplied by a constant, remains unchanged except for a scaling factor. The corresponding eigenvalue is the constant by which the eigenfunction is scaled.

## 2. What is the importance of eigenfunctions/eigenvalues?

Eigenfunctions/eigenvalues are important because they allow us to simplify complex mathematical problems and systems by breaking them down into smaller, more manageable components. They also have many applications in physics, engineering, and other scientific fields.

## 3. How are eigenfunctions/eigenvalues calculated?

The calculation of eigenfunctions/eigenvalues involves solving an eigenvalue equation, which is a mathematical equation involving a linear operator and an unknown function. The resulting eigenfunctions and eigenvalues can be found by solving the characteristic equation associated with the linear operator.

## 4. Can eigenfunctions/eigenvalues have complex values?

Yes, eigenfunctions/eigenvalues can have complex values. In fact, in some cases, complex eigenvalues and eigenfunctions are necessary to accurately describe certain physical systems and phenomena.

## 5. Are eigenfunctions/eigenvalues unique?

Yes, eigenfunctions/eigenvalues are unique for a given linear operator. This means that different linear operators can have different sets of eigenfunctions/eigenvalues. However, for a single linear operator, the eigenfunctions and eigenvalues are always unique.

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