# Showing that (x+iy)/r is an eigenfunction of the angular momentum operator

• Edgarngg
In summary, the question is how to show that a function is an eigenfunction of the Lz angular momentum operator. The given function is ψ=(x+iy)/r, and the operator is Lz= (h bar)/i (x \partial/\partialy - y \partial/\partialx). The strategy is to apply the operator to the function and see if it results in the same function multiplied by an eigenvalue. The issue is how to treat the "r" in the function, as it is not clear how to take the derivative with respect to "r". It is suggested to change to polar coordinates or find a different approach to handle the "r" term.
Edgarngg

## Homework Statement

I know that,if (operator)(function)=(value)(samefunction)
that function is said to be eigenfunction of the operator.
in this case i need to show this function to be eigenfunction of the Lz angular momentum:

## Homework Equations

function:
ψ=(x+iy)/r
operator:
Lz= (h bar)/i (x $\partial$/$\partial$y - y $\partial$/$\partial$x)

## The Attempt at a Solution

My question is how do i treat "r", do i have to change to polar coordinates? or is it possible to do it like this.
i know that i have to apply the operator over the function, and that is (h bar/i) (x(partial derivative for y)- y(partial der for x)) and then see if i get the same function multiplied by an eigenvalue.
the problem i have is that i don't know how to treat that function, since i see an "r" there. So when d/dx "r" and "y" would be constant, and that doesn't make sense to me.
Thank you very much

Last edited:
Edgarngg said:

## Homework Statement

I know that,if (operator)(function)=(value)(samefunction)
that function is said to be eigenfunction of the operator.
in this case i need to show this function to be eigenfunction of the Lz angular momentum:

## Homework Equations

function:
ψ=(x+iy)/r
operator:
Lz= (h bar)/i (x $\partial$/$\partial$y - y $\partial$/$\partial$x)

## The Attempt at a Solution

My question is how do i treat "r", do i have to change to polar coordinates? or is it possible to do it like this.
i know that i have to apply the operator over the function, and that is (h bar/i) (x(partial derivative for y)- y(partial der for x)) and then see if i get the same function multiplied by an eigenvalue.
the problem i have is that i don't know how to treat that function, since i see an "r" there. So when d/dx "r" and "y" would be constant, and that doesn't make sense to me.
Thank you very much

r=sqrt(x^2+y^2), isn't it?

## 1. What is an eigenfunction?

An eigenfunction is a mathematical function that, when operated on by a linear operator, results in a scalar multiple of itself.

## 2. What is the angular momentum operator?

The angular momentum operator is a mathematical operator used in quantum mechanics to describe the rotational motion of a particle.

## 3. How is (x+iy)/r derived as an eigenfunction of the angular momentum operator?

This eigenfunction is derived by solving the Schrödinger equation for a particle in a central potential, and then applying the angular momentum operator to the resulting wavefunction.

## 4. What does the eigenvalue of the angular momentum operator represent?

The eigenvalue of the angular momentum operator represents the magnitude of the angular momentum of a particle in a quantum system.

## 5. How is the eigenvalue of the angular momentum operator related to the quantum number?

The eigenvalue of the angular momentum operator is directly related to the quantum number, which describes the energy level and orbital angular momentum of a particle in a quantum system.

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