B the size required to emit light with energy δ e is

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b) The size required to emit light with energy Δ E is a = h 2 π 2 2 m Δ E ! 1 / 2 The sizes for the given energies work out to red (1.8 eV): a = 0 . 79 nm green (2.4 eV): a = 0 . 69 nm blue (3.5 eV): a = 0 . 57 nm 2. Electron’s distance from nucleus The probability of finding the electron of a hydrogen-like atom between r and r + dr from the nucleus is P ( r ) dr = r 2 | R n,l ( r ) | 2
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where R n,l ( r ) is the radial part of the wavefunction. P ( r ) is known as the radial distribution function. If an experimenter were to make many measurements of the distance from the nucleus to the electron, a histogram of the data would follow P ( r ). Two typical distances can be calculated. The most likely distance occurs where P ( r ) is maximum. The most likely distance has a simple result when the l quantum number has its maximal value n - 1. r Pmax = n 2 Z a 0 for l = n - 1 where a 0 is Bohr’s radius. The average value or statistical mean is computed by integrating rP ( r ) over all r . r mean = Z 0 r 2 P ( r ) dr = n 2 Z a 0 3 2 - l ( l + 1) 2 n 2 ! Calculate r Pmax and r mean for a hydrogen atom with the electron in the 1s, 2p, and 3d states. Answer: Using a 0 5 . 29 × 10 - 11 m, we get 1s: r Pmax = 5 . 29 × 10 - 11 m and r mean = 7 . 94 × 10 - 11 m. 2p: r Pmax = 2 . 12 × 10 - 10 m and r mean = 2 . 65 × 10 - 10 m. 3d: r Pmax = 4 . 76 × 10 - 10 m and r mean = 5 . 55 × 10 - 10 m. 3. Ground state of He We can estimate the ground state energy of helium by using hydrogen-like wavefunctions and then adding in the repulsion energy between the electrons.
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