How do you solve numerical problems on de Broglie wavelength for electrons and other particles?

In one line

Pick the right form of \(\lambda = h/p\) for the given data (speed, kinetic energy or accelerating voltage), substitute constants in SI units, and convert the answer to Å or nm.

Explained simply

Most MU numericals give one of three things: the particle's speed, its kinetic energy, or the voltage it was accelerated through. Each has its own version of the same formula. Choosing the right version and keeping units in SI is 90% of the job.

How it works

1. Speed given: \(\lambda = h/(mv)\).
2. Kinetic energy E given: \(\lambda = h/\sqrt{2mE}\); convert eV to J by multiplying by \(1.6\times10^{-19}\).
3. Voltage V given: \(\lambda = h/\sqrt{2mqV}\); for electrons \(\lambda = 12.27/\sqrt{V}\) Å.
4. Reverse problems: to find energy from \(\lambda\), use \(E = h^2/(2m\lambda^2)\).
5. Comparing particles at the same voltage: \(\lambda \propto 1/\sqrt{mq}\); at the same kinetic energy: \(\lambda \propto 1/\sqrt{m}\).
6. Constants: \(h = 6.63\times10^{-34}\) J s, \(m_e = 9.11\times10^{-31}\) kg, \(m_p = 1.67\times10^{-27}\) kg, \(e = 1.6\times10^{-19}\) C.

Key formula

\(\lambda = \dfrac{h}{mv} = \dfrac{h}{\sqrt{2mE}} = \dfrac{h}{\sqrt{2mqV}}\); \(E = \dfrac{h^2}{2m\lambda^2}\). \(E\) in joules, \(V\) in volts, \(\lambda\) in metres.

Quick example

(a) Electron with kinetic energy 10 eV: \(\lambda = 12.27/\sqrt{10} = 3.88\) Å.
(b) Energy of an electron with \(\lambda = 1\) Å: \(E = (6.63\times10^{-34})^2/(2\times9.11\times10^{-31}\times10^{-20}) = 2.41\times10^{-17}\) J \(\approx 151\) eV.
(c) Electron and proton accelerated through the same voltage: \(\lambda_e/\lambda_p = \sqrt{m_p/m_e} = \sqrt{1836} \approx 42.8\), so the electron's wavelength is about 43 times longer.

Common mistakes

• Leaving energy in eV inside \(\sqrt{2mE}\) without converting to joules.
• Using the 12.27/\(\sqrt{V}\) shortcut for protons or neutrons.
• Squaring wrongly when finding \(E\) from \(\lambda\) (forgetting \(\lambda^2\)).

How MU asks it

Numericals of 3 to 5 marks are very common. For full marks: write the formula, list given data in SI units, substitute with powers of ten visible, and give the final answer with units (Å, nm or eV). Show intermediate steps rather than only the final number.