Last Lecture • Observed spectra and Bohr’s energy levels 15 E = p 2 2 m − kZe 2 r ⇒ E = − 1 2 k 2 Z 2 me 4 n 2 h 2 = −13.6eV Z 2 n 2 For Rydberg atoms the effect of the screening of other electrons can be accounted for by substituting (Z-1) 2 to Z 2 1 λ = Z 2 R 1 n f 2 − 1 n i 2 E i − E f = hc λ = k 2 me 4 2 h 2 Z 2 1 n f 2 − 1 n i 2 ⇒ 1 λ = k 2 me 4 4 πh 3 c Z 2 1 n f 2 − 1 n i 2
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Last Lecture - IceCube Neutrino Observatorytmontaruli/Phys248/lectures/lecture27.pdfLast Lecture • Observed spectra and Bohr’s energy levels 15 € E= p 2 2m − kZe r ⇒E=−
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Last Lecture• Observed spectra and Bohr’s energy levels
15
€
E =p2
2m−kZe2
r⇒ E = −
12k 2Z 2me4
n2h2= −13.6eV Z 2
n2
For Rydberg atoms the effect of the screening of other electrons can be accounted for by substituting (Z-1)2 to Z2
€
1λ
= Z 2R 1n f2 −
1ni2
€
Ei − E f =hcλ
=k 2me4
2h2Z 2 1
n f2 −
1ni2
⇒
1λ
=k 2me4
4πh3cZ 2 1
n f2 −
1ni2
16
Hydrogen atom wavefunction• We have to extend the wavefunction concept from 1D to 3D
• Simplification: Hydrogen has spherical symmetry
• Ψ(x, y, z) is converted to Ψ(r,θ,ϕ)
• We need three quantum numbers to represent 3D states – Radial distance from nucleus– Azimuthal angle around nucleus– Polar angle around nucleus
• Quantum numbers are integers (n, l, ml), and the spin quantum number completes the picture
• We can’t say exactly where the particle is, but we can tell you how likely is to find the particle in a particular location.
The Schroedinger equation for H atom
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Z=1
Spherical coordinates
Laplace operator or Nabla square in spherical coordinates