Physical Chemistry - Electron Spin and Pauli Spin Matrices_lecture23-24
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Course
MIT Course
Institution
MIT Course
This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...
Compton Scattering (1921): AH Compton suggested that “the electron is
probably the ultimate magnetic particle.”
SternGerlach Experiment (1922): Passed a beam of silver atoms (4d105s1)
through an inhomogeneous magnetic field and observed that they split into two
beams of space quantized components.
Uhlenbeck and Goudsmit (1925) showed that these were two angular
momentum states – the electron has intrinsic angular momentum – "SPIN"
angular momentum
Pauli Exclusion Principle (1925): no more than 2 electrons per orbital, or,
no two electrons with all the same quantum numbers. Additional quantum
number, now called ms, was postulated.
Postulate 6: All electronic wavefunctions must be
antisymmetric under the exchange of any two electrons.
Theoretical Justification
Dirac (1928) developed relativistic quantum theory & derived electron
spin angular momentum
Orbital Angular Momentum
L = orbital angular momentum
L = � l (l + 1 )
l = orbital angular momentum quantum number
l ≤ n −1
Lz = m �
m = 0, ±1, ±2, …, ±l
, 5.61 Physical Chemistry 23-Electron Spin page 2
Spin Angular Momentum
S ≡ spin angular momentum
S = � s (s + 1 ) = � 3 2
s = spin angular momentum quantum number
s =1 2
Sz = ms �
ms = ± 1 2
Define spin angular momentum operators analogous to orbital angular momentum
operators
L2Yl m (θ , φ ) = l ( l + 1) � 2Yl m (θ , φ ) l = 0,1, 2,...n for H atom
LzYl m (θ , φ ) = m�Yl m (θ , φ ) m = 0, ±1, ±2,... ± n for H atom
1
Ŝ 2α = s ( s + 1) � 2α Ŝ 2 β = s ( s + 1) � 2 β s= always
2
1 1
Ŝ zα = ms �α mαs = Ŝ z β = ms �β msβ = −
2 2
Spin eigenfunctions α and β are not functions of spatial coordinates so the
equations are somewhat simpler!
α ≡ "spin up" β ≡ "spin down"
Spin eigenfunctions are orthonormal:
∫ α α dσ = ∫ β β d σ = 1
* *
σ ≡ spin variable
∫ α β dσ = ∫ β α d σ = 0
* *
Spin variable has no classical analog. Nevertheless, the angular momentum of
the electron spin leads to a magnetic moment, similar to orbital angular
momentum.
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