- #1

Kashmir

- 465

- 74

One of the component of angular momentum operator is ##\hat{L}_{x}=\hat{y} \hat{P}_{z}-\hat{z} \hat{P}_{y}##

I want it's position representation.

My attempt :

I'll find the representation of the first term ##\hat{y} \hat{P}_{z}##. The total representation is the sum of two terms.

The action of ##\hat{y} \hat{P}_{z}## on a ket is :##\left(1 \otimes \hat{y} \otimes \hat{p}_{z}\right) \iiint\left|x^{\prime}, y^{\prime}, z^{\prime}\right\rangle \cdot \psi\left(x^{\prime}, y^{\prime}, z^{\prime}\right) \cdot d x^{\prime} d y^{\prime} d z'##

The position representation is found by acting a bra on it, thus :

##\left\langle x| \otimes\left\langle y|\otimes\langle z|)\left(1 \otimes \hat{y} \otimes \hat{p_{z}}\right) \iiint \mid x^{\prime} y \hat{z}\right\rangle \psi\left(x' y', z^{\prime}\right) d x' dy'dz'\right.##

Which gives ##\int\left\langle x \mid x^{\prime}\right\rangle y^{\prime}\left\langle y \mid y^{\prime}\right\rangle\left\langle z\left|\hat{p}_{z}\right| z^{\prime}\right\rangle \psi\left(x^{\prime}, y^{\prime},z'\right) d x' d y' z^{\prime}## Which is

##y \int\left\langle z\left|\hat{p}_{z}\right| z\right\rangle \psi(x y z) d z=-i \hbar y \frac{\partial}{\partial z} \psi(x ,y ,z)## .

Can anyone please tell Is this correct?

Thank you

I want it's position representation.

My attempt :

I'll find the representation of the first term ##\hat{y} \hat{P}_{z}##. The total representation is the sum of two terms.

The action of ##\hat{y} \hat{P}_{z}## on a ket is :##\left(1 \otimes \hat{y} \otimes \hat{p}_{z}\right) \iiint\left|x^{\prime}, y^{\prime}, z^{\prime}\right\rangle \cdot \psi\left(x^{\prime}, y^{\prime}, z^{\prime}\right) \cdot d x^{\prime} d y^{\prime} d z'##

The position representation is found by acting a bra on it, thus :

##\left\langle x| \otimes\left\langle y|\otimes\langle z|)\left(1 \otimes \hat{y} \otimes \hat{p_{z}}\right) \iiint \mid x^{\prime} y \hat{z}\right\rangle \psi\left(x' y', z^{\prime}\right) d x' dy'dz'\right.##

Which gives ##\int\left\langle x \mid x^{\prime}\right\rangle y^{\prime}\left\langle y \mid y^{\prime}\right\rangle\left\langle z\left|\hat{p}_{z}\right| z^{\prime}\right\rangle \psi\left(x^{\prime}, y^{\prime},z'\right) d x' d y' z^{\prime}## Which is

##y \int\left\langle z\left|\hat{p}_{z}\right| z\right\rangle \psi(x y z) d z=-i \hbar y \frac{\partial}{\partial z} \psi(x ,y ,z)## .

Can anyone please tell Is this correct?

Thank you

Last edited: