Spin angular momentum for electron is:

Question

Accepted Answer

The spin angular momentum for an electron is a fundamental concept in quantum mechanics. It is described by the formula for the magnitude of the spin angular momentum vector, which is given by:

$$ S = \sqrt{s(s+1)}\frac{h}{2\pi} $$

where:
- $ s $ is the spin quantum number for the electron.
- $ h $ is Planck's constant.

For an electron, the spin quantum number $ s $ is $ \frac{1}{2} $. Substituting this value into the formula, we get:

$$ S = \sqrt{\frac{1}{2}(\frac{1}{2} + 1)}\frac{h}{2\pi} = \sqrt{\frac{1}{2} \times \frac{3}{2}}\frac{h}{2\pi} = \sqrt{\frac{3}{4}}\frac{h}{2\pi} = \frac{\sqrt{3}}{2}\frac{h}{2\pi} $$

This matches the form given in Option 1, which is the correct expression for the spin angular momentum of an electron.

Now, let's examine why the other options are incorrect:

2. Option 2: $\sqrt{2s(s+1)}\frac{h}{2\pi}$
   - This formula incorrectly includes an extra factor of 2 inside the square root, which does not align with the standard quantum mechanical expression for spin angular momentum.

3. Option 3: $\sqrt{s(s+2)}\frac{h}{2\pi}$
   - This formula incorrectly uses $ s(s+2) $ instead of $ s(s+1) $. The correct expression for spin angular momentum involves $ s(s+1) $, as derived from quantum mechanics principles.

4. Option 4: $\sqrt{2(s+1)}\frac{h}{2\pi}$
   - This formula incorrectly modifies the expression by using $ 2(s+1) $ inside the square root, which is not consistent with the established formula for spin angular momentum.

In conclusion, Option 1 is the correct expression for the spin angular momentum of an electron, as it accurately reflects the quantum mechanical principles governing electron spin.