last lecture - icecube neutrino observatorytmontaruli/phys248/lectures/lecture27.pdflast lecture •...
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Last Lecture• Observed spectra and Bohr’s energy levels
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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
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1λ
= Z 2R 1n f2 −
1ni2
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Ei − E f =hcλ
=k 2me4
2h2Z 2 1
n f2 −
1ni2
⇒
1λ
=k 2me4
4πh3cZ 2 1
n f2 −
1ni2
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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
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−h2
2µ∇2 + (U − E)
Ψ = 0
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∇2 =1r2
∂∂r
r2 ∂∂r
+
1r2 sinθ
∂∂θ
sinθ ∂∂θ
+
1r2 sin2θ
∂ 2
∂ϕ 2
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∇2 =∇ ⋅∇
Separating variables
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Z=1
radial equation colatitude equation
=-m2
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Ylm (θ,ϕ) = Plm (θ)Fm (ϕ)radial equationspherical harmonics
azimuthal equation (depends only on φ)
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sin2θR
ddr
r2 dRdr
+sinθP
ddθ
sinθ dPdθ
+2µr2 sin2θ
h2E −U(r)( ) = −
1Fd2Fdφ 2
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1Rddr
r2 dRdr
+2µr2
h2E −U(r)( ) = −
m2
sin2θ−
1P sinθ
ddθ
sinθ dPdθ
= l(l +1)
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Rnl (r)
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Ψ = R(r)P(θ)F(φ)
The solutions and quantum numbers
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Bound state energy spectrum is quantized,
En = −E0
n2 , n =1,2,3,K
Principal quantum number
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Angular momentum is quantized (depends on n),
L = l(l +1) h2π
, l = 0,1,2,K,n −1( )
Solutions exist when these conditions are satisfied on quantum numbers
Orbital quantum number
Magnetic quantum number
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The direction of angular momentum is also quantized,
Lz = mlh
2π, ml = −l,−l +1,K,−1,0,1,K,l −1,l( )
The z component of L is quantized2l+1states
Another quantum number is needed: the spin!
Quantization of angular momentum
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p
pr
pt
r
α
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L = r ×p⇒ L = rpsinα = rpt
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E =p2
2µ+U(r) =
pr2 + pt
2
2µ+U(r) =
pr2
2µ+
L2
2µr2+U(r) =
pr2
2µ+Ueff (r)
The angular part is:
=-m2
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1Rddr
r2 dRdr
+2µr2
h2E −U(r)( ) = −
m2
sin2θ−
1P sinθ
ddθ
sinθ dPdθ
= l(l +1)
From
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−1
P sinθddθ
sinθ dPdθ
+
1F sin2θ
d2Fdθ 2
= l(l +1)
Defining the operator for L2
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(L2)op = −h21sinθ
ddθ
sinθ ∂∂θ
+
1sin2θ
∂ 2
∂θ 2
This is the angular part of the S. eq.: for a potental U(r) the angular momentum is quantized
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L = l(l +1)h
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l = 0,1,2,...,n −1
O
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(L2)opYlm (θ,φ) = h2l(l +1)Ylm (θ,φ)
Solutions• In order to have well behaved solutions only certain values of the energy
are possible (identical to Bohr’s energy levels!)• Angular wave function
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Plm (θ) = Plm (cosθ)
Fm (ϕ) = Aexp(imφ)
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l = 0,1,2,...,n −1m = 0,±1,...,±l
Associated Legendre Polynomials
• |Ψ|2 = square magnitude of the wavefunction
• P(x)dx = |ψ|2dx = probability to find the particle in a infinitesimal interval dx around x
• It is normalized to 1 because we are sure that the particle must be somewhere along x
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Probability densities
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|ψ |2 dx−∞
+∞
∫ =1
The ground state probability density
• Ground state: lowest energy
• Surface of const probability is surface of a sphere.• Probability decreases exponentially with radius.
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n =1, l = 0, ml = 0
In Bohr’s model the electron is in an orbit of radius a0.The electron can be found at any distance, and the most probable is r=a0
P(r)dr is a prob. per unit volume
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P(r)dr∝ Ψ24πr2dr∝ r2e−2r / a0dr
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Hydrogen atom probabilities
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n = 2, l =1, ml = 0
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n = 2, l =1, ml = ±1
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n = 2, l = 0, ml = 0
2s-state
2p-state 2p-state n=2 Same energy, but different probabilities
http://webphysics.davidson.edu/faculty/dmb/hydrogen/
Energy is lower for 1s than 2s and from radial probability we see that 2s electrons can be on average much farther from the nucleus than 1s electrons
Why do 1s shells fill first before 2s?
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n = 3, l =1, ml = 0
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n = 3, l =1, ml = ±1
3s-state3p-state
3p-state
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n = 3, l = 0, ml = 0
n=3: two s-states, six p-states
3s-state3p-state
3p-state
…ten d-states
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n = 3, l = 2, ml = 03d-state
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n = 3, l = 2, ml = ±1
3d-state
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n = 3, l = 2, ml = ±2
3d-state