Calculate HL de Broglie wavelengths for particles and electrons, relating momentum, kinetic energy, accelerating potential and confinement.
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.
State the de Broglie hypothesis as it applies to an electron.
The electron has an associated wavelength lambda given by lambda = h/p, where p is its momentum.
Show that the energy En of the electron in the box is given by
where n=1,2,3…,h is the Planck constant and me is the mass of the electron.
Allowed wavelengths of the electron: lambda_n = 2L/n.
E_n = p^2/(2m_e) = h^2/(2 lambda_n^2 m_e).
Substitute explicitly for lambda_n to get E_n = n^2 h^2/(8 m_e L^2).
The magnitude of the least energy that an electron in the hydrogen atom can have is 2.2×10−18 J. Estimate, using the equation in (b)(ii), the radius of the hydrogen atom.
1.7×10−10 m;