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Quantum Mechanics after hydrogen atom
Cards in this deck (5)
- 1Hamiltonian of hydrogen atom\[ H = - \dfrac{\hbar^2}{2\mu} \nabla^2
- \dfrac{e^2}{4\pi\epsilon_0}\dfrac{1}{r} \] - 2Bohr radius\( a = \dfrac{4\pi \epsilon_0 \hbar^2}{\mu e^2} \propto \dfrac{1}{\mu e^2}\)
Tip: Positronium has \( \mu \approx \frac{m_e}{2} \), so its Bohr radius is twice the hydrogen's - 3Ground state Wavefunction of hydrogen\[ \Psi_1(r) \propto e^{-\frac{r}{a}} \]
Tip: Due to electron-proton attraction, \(\Psi=1 \) (largest) near the origin - 4Ground state Energy of hydrogen\[ - E_1 = \dfrac{\hbar^2}{2\mu a^2} = \dfrac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2} \propto \mu (e^2)^2 = 13.6 eV \]
Tip: For nucleus of charge 2e, V increases by factor of 2 and E_1 by factor of 2^2=4. - 5Energies of hydrogen\[ - E_1 = \dfrac{\hbar^2}{2\mu a^2}\dfrac{1}{n^2}, \ n=1, \ 2, \ 3, \ ... \]
Tip: Gives Rydberg formula \( f \propto \dfrac{1}{n^2_f} - \dfrac{1}{n^2_i} \)
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