Matter Waves

1.28 × 10^-12 m

Convert energy to joules: K = 500 × 1.6 × 10^-19 J = 8.0 × 10^-17 J. Then use λ = h/√(2mK) with proton mass 1.67 × 10^-27 kg. This gives λ ≈ 1.28 × 10^-12 m.

0.274 Å

For electrons, λ = 12.27/√V Å. With V = 2000 V, λ = 12.27/√2000 Å ≈ 12.27/44.72 Å ≈ 0.274 Å.

2.76 × 10^-11 m

For gas molecules, λ = h/√(3mkT). Using the mass of N2 molecule, temperature 300 K, and k = 1.38 × 10^-23 J K^-1 gives λ ≈ 2.76 × 10^-11 m.

1.67 Å

For electrons accelerated through V volts, λ = 12.27/√V Å. For V = 54, λ = 12.27/√54 Å ≈ 1.67 Å.