# Why Does <n',l',m'|\hat{z}|n,l,m> Equal Zero Unless m=m'?

• z2394
In summary, the conversation discusses how to show that <n',l',m'|\hat{z}|n,l,m> = 0 unless m=m' by using the form of spherical harmonics and the equation for spherical harmonics. The individual attempting the solution is unsure where to start since there are no simple eigenvalues for \hat{z}|n,l,m>, but they have a feeling that normalization of spherical harmonics may play a role. Another individual suggests writing down the integral for the inner product to find the answer.
z2394

## Homework Statement

I want to show that <n',l',m'|$\hat{z}$|n,l,m> = 0 unless m=m', using the form of the spherical harmonics.

## Homework Equations

Equations for spherical harmonics

## The Attempt at a Solution

Not sure how to begin here since there aren't any simple eigenvalues for $\hat{z}$|n,l,m>. I have a feeling that it may have something to do with normalization of the spherical harmonics (because they have Legendre polynomials that are P(cosΘ) = P(z) and would also give you a exp(imø)*exp(im'ø) term), but I have no idea how this could actually give you something for $\hat{z}$as an operator, or something you could actually use to figure out $\hat{z}$|n,l,m>.

Any help at all would be appreciated!

You essentially have the answer already. In the coordinate basis, the operator ##\hat{z}## is represented by ##r\cos\theta##. Just write down the integral for the inner product and evaluate it.

1 person
Thanks! I guess I was thinking about it in an operator sense, so it had not occurred to me to do it as an integral instead.

## 1. What is the Z operator and how does it relate to spherical harmonics?

The Z operator is a mathematical operator used in quantum mechanics to describe the angular momentum of a particle. It is closely related to spherical harmonics, which are mathematical functions used to describe the spatial distribution of particles in a spherical system.

## 2. How is the Z operator used in quantum mechanics?

The Z operator is used to determine the angular momentum of a particle in a given direction. It is also used to calculate the position and momentum of a particle in a spherical system.

## 3. What are some properties of the Z operator?

Some properties of the Z operator include being Hermitian, meaning it is equal to its own conjugate transpose, and being an eigenoperator, meaning it has a set of eigenvalues and eigenvectors that can be used to solve quantum mechanical problems.

## 4. How are spherical harmonics used in quantum mechanics?

Spherical harmonics are used to describe the wave function of a particle in a spherical system. They are also used to calculate the probability of finding a particle in a specific location within the system.

## 5. Can the Z operator and spherical harmonics be used in other fields of science?

Yes, the Z operator and spherical harmonics have applications in other fields such as physics, chemistry, and engineering. They can be used to describe the behavior of particles in a variety of systems and are especially useful in understanding the properties of atoms and molecules.

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