Question

Question: the operator for angular momentum about the z-axis in spherical polar coordinate is $-i\hslash \left(\frac{\partial }{\partial \varphi }\right)$.find the function $f\left(\varphi \right)$ that would have a well-defined z-component of angular momentum.

Short answer

Answer

The z-component of the general function's angular momentum is well defined is $\exp i\left(\frac{2\lambda c}{\hslash }\right)\varphi$.

Step-by-step solution

Identification of given data

The given data can be listed below,

The spherical polar coordinate of the momentum is,$\hslash \left(\frac{\partial }{\partial \varphi }\right)$

 Step 2: Concept/Significance of angular momentum

An eigenvector is a vector to which a linear operator applies, and the particular multiple is known as the eigenvalue. Everything else that is referred to be an Eigen is one of its eigenvalue or eigenvector, depending on the linear operator.

Determination of the function having well-defined  -component of angular momentum.

If the operator acting on the function is a constant multiplied by the same value. As an Eigen function of the operator is given by,

$\hat{Q}f\left(x\right)=\lambda f\left(x\right)$

Here, $\hat{Q}$ is the operator for momentum, $\lambda$ is the Eigen value of the operator.

Substitute all the values in the above,

$$\begin{gathered} -i\left(\frac{\hslash }{2\lambda }\right)\frac{\partial }{\partial \varphi }f\left(\varphi \right)=cf\left(\varphi \right) \\ \left(\frac{\partial }{\partial \varphi }\right)f\left(\varphi \right)=\frac{2\lambda c}{-i\hslash }f\left(\varphi \right) \\ f\left(\varphi \right)=\exp i\left(\frac{2\lambda c}{\hslash }\right)\varphi \end{gathered}$$

Thus, the z-component of the general function's angular momentum is well defined is $\exp i\left(\frac{2\lambda c}{\hslash }\right)\varphi$.