Question

The unit vectors \( \hat{\theta} \) and \( \hat{\phi} \) (where \( \theta \) and \( \phi \) are the polar and azimuthal angles, respectively), in the spherical coordinate system, under the operation of inversion (i.e., reflection through the origin) have:

• A. Even parity and odd parity, respectively
• B. Both odd parity
• C. Odd parity and even parity, respectively
• D. Both even parity

Hint:

Parity transformation (inversion) affects unit vectors differently in **spherical coordinates**. - \( \hat{r} \) and \( \hat{\theta} \) retain **even parity**. - \( \hat{\phi} \) has **odd parity** due to its dependence on \( \phi \).

Answer 1

The Correct Option is A

The behavior of the unit vectors under inversion (reflection through the origin) can be determined as follows:
- In spherical coordinates, the position vector is given by:
\[ \mathbf{r} = r \hat{r} \] - Under inversion \( \mathbf{r} \to -\mathbf{r} \), the radial unit vector transforms as:
\[ \hat{r} \to -\hat{r} \] - The polar unit vector \( \hat{\theta} \) is defined as:
\[ \hat{\theta} = \frac{\partial \hat{r}}{\partial \theta} \] - Since \( \hat{r} \) changes sign under inversion, its derivative with respect to \( \theta \) remains unchanged.
- Therefore, \( \hat{\theta} \) has even parity.
- The azimuthal unit vector \( \hat{\phi} \) is defined as:
\[ \hat{\phi} = \frac{1}{\sin\theta} \frac{\partial \hat{r}}{\partial \phi} \] - Since \( \hat{r} \) changes sign under inversion, its derivative with respect to \( \phi \) also changes sign.
- Hence, \( \hat{\phi} \) has odd parity.