### Step-by-Step Solution
#### (a) State de Broglie Equation and its Significance
1. **De Broglie Equation**: The de Broglie equation is given by the formula:
\[
\lambda = \frac{h}{mv}
\]
where:
- \(\lambda\) is the wavelength associated with a particle,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \(m\) is the mass of the particle, and
- \(v\) is the velocity of the particle.
2. **Significance of the De Broglie Equation**:
- The de Broglie equation signifies that all matter exhibits wave-like properties, not just light. This concept is fundamental in quantum mechanics, indicating that particles such as electrons can behave like waves.
- It is particularly significant for microscopic particles, such as electrons and atoms, where wave-particle duality becomes observable. It helps in understanding phenomena such as electron diffraction and the behavior of particles in quantum mechanics.
#### (b) Finding the Wavelength of Helium Atoms
1. **Given Data**:
- Velocity of helium atoms, \(v = 2.0 \times 10^3 \, \text{ms}^{-1}\)
- Mass of a helium atom, \(m \approx 4.0 \times 10^{-27} \, \text{kg}\) (since the atomic mass of helium is approximately 4 u, and 1 u = \(1.66 \times 10^{-27} \, \text{kg}\))
- Planck's constant, \(h = 6.626 \times 10^{-34} \, \text{Js}\)
2. **Substituting into the De Broglie Equation**:
- Using the de Broglie equation:
\[
\lambda = \frac{h}{mv}
\]
- Substitute the values:
\[
\lambda = \frac{6.626 \times 10^{-34} \, \text{Js}}{(4.0 \times 10^{-27} \, \text{kg})(2.0 \times 10^3 \, \text{ms}^{-1})}
\]
3. **Calculating the Denominator**:
- Calculate \(mv\):
\[
mv = (4.0 \times 10^{-27} \, \text{kg})(2.0 \times 10^3 \, \text{ms}^{-1}) = 8.0 \times 10^{-24} \, \text{kg m/s}
\]
4. **Calculating Wavelength**:
- Now substitute back into the equation:
\[
\lambda = \frac{6.626 \times 10^{-34}}{8.0 \times 10^{-24}} = 8.28375 \times 10^{-11} \, \text{m}
\]
- Rounding off, we get:
\[
\lambda \approx 8.28 \times 10^{-11} \, \text{m} \, \text{or} \, 82.8 \, \text{pm}
\]
### Final Answers
- **(a)** De Broglie Equation: \(\lambda = \frac{h}{mv}\) with significance in wave-particle duality for microscopic particles.
- **(b)** Wavelength of the helium atoms: \(\lambda \approx 8.28 \times 10^{-11} \, \text{m}\).
