6.4: Particle-Wave Duality: Prelude to Quantum Mechanics

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de Broglie Wavelength

Exercise \(\PageIndex{1}\)

For a proton (mass = 1.673 × 10^–27 kg) moving with a velocity of 8.63 × 10⁴ m/s, what is the de Broglie wavelength of the proton (in pm)?

Answer

4.59 pm

\[E = \frac{1}{2}mv^{2}\]

\[E = \frac{1}{2}(1.673*10^{-27})(8.63*10^{4})^{2} = 6.229*10^{-18}\;J\]

\[\lambda = \frac{h}{\sqrt{2Em}}\]

\[\lambda = \frac{6.626*10^{-34}}{\sqrt{2*(6.229*10^{-18})*(1.673*10^{-27})}}\]
\[\lambda = 4.588*10^{-12}\;m=4.588\;pm\]

Exercise \(\PageIndex{1}\)

What is the de Broglie wavelength of an electron traveling at 15.9% of the speed of light? (c = 3.00 × 10⁸ m/s, h = 6.63 × 10^–34 J·s, m_e = 9.109 × 10^–31 J)

Answer: 1.52 × 10^–11 m

Exercise \(\PageIndex{1}\)

If the de Broglie wavelength of an electron is 90.2 nm, what is its velocity? (The mass of an electron is 9.11 × 10^–31 kg.)

Answer: 8.06 × 10³ m/s

Exercise \(\PageIndex{1}\)

What is the de Broglie wavelength of a 250-g baseball traveling at 85 mph? (1 mi = 1.609 km and h = 6.63 × 10^–34 J·s)

Answer: 7.0 × 10^-35 m

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