IB Physics HL E 2 6 Hl De Broglie Wavelength Questions

Calculate HL de Broglie wavelengths for particles and electrons, relating momentum, kinetic energy, accelerating potential and confinement.

Exam points

apply λ=h/p and, for non-relativistic particles, λ=h/√(2mE_k) to calculate or compare de Broglie wavelengths
predict wavelength changes when kinetic energy, momentum or electron accelerating potential changes, including inverse-square-root relationships
use the electron-in-a-box model and particle mass or energy data to interpret allowed wavelengths and matter-wave scale

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Study E.2.6 (HL)—de Broglie wavelength conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 5]

(a)

The existence of atomic energy levels can be understood by applying the de Broglie hypothesis to an electron confined to move in one dimension in a box of length L.

[ 4 ]

(i)

State the de Broglie hypothesis as it applies to an electron.

[ 1 ]

(ii)

Show that the energy $E_{\mathrm{n}}$ of the electron in the box is given by

E_{\mathrm{n}}=\frac{n^{2} h^{2}}{8 m_{\mathrm{e}} L^{2}}

where $n=1,2,3 \ldots, h$ is the Planck constant and $m_{\mathrm{e}}$ is the mass of the electron.

[ 3 ]

(b)

The magnitude of the least energy that an electron in the hydrogen atom can have is $2.2 \times 10^{-18} \mathrm{~J}$ . Estimate, using the equation in (b)(ii), the radius of the hydrogen atom.

[ 1 ]

$1.7 \times 10^{-10} \mathrm{~m}$ ;

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IB Physics HL E 2 6 Hl De Broglie Wavelength Questions question 1

Question (a)

Question (i)

Question (ii)

Question (b)