1. The Origin of Spin Joshua Varner

2. Quantum Mechanics: A History

3. Thomson discovers Electron! “Plum Pudding” model of the 19th century atom. “Plum Pudding” model of the 19th century atom. 1897 J. J. Thomson

4. Zeeman and his effect Applying a Magnetic Field to a charged particle caused its spectrum to split into several components. Applying a Magnetic Field to a charged particle caused its spectrum to split into several components. 1902 Pieter Zeeman B 1897 μ = L e 2mc

5. Quantization of Energy levels follows from Quantization of Angular Momentum. Quantization of Energy levels follows from Quantization of Angular Momentum. Bohr postulates Quantum Theory New Atom Model has electrons circulating around positive nucleus. New Atom Model has electrons circulating around positive nucleus. 1913 Niels Bohr L = nħ 18971902

6. L = ħ l n Lorentz’s new unit of magnetic moment 1913 Hendrik Antoon Lorentz μ = L e 2mc μ B = ±eħ 2mc E B = -μ B ∙B E B = B ±eħ 2mc ⇒ ⇒ The Magnetic Energy either adds or subtracts from the total Energy. The Magnetic Energy either adds or subtracts from the total Energy. Combines the Bohr Model (1) and the Charged particle (2). Combines the Bohr Model (1) and the Charged particle (2). (1) (2) 189719021913

7. Lorentz’s new unit of magnetic moment 1913 Hendrik Antoon Lorentz B E1E1 E2E2 E 1 -E B E 1 +E B E 2 -E B E 2 +E B However, no one questioned the possibility of μ B = 0... Moving on. However, no one questioned the possibility of μ B = 0... Moving on. E B = B ±eħ 2mc 189719021913

8. The Stern-Gerlach Experiment! 1921-1923 Otton Stern Walther Gerlach Arthur Holly Compton Their experiment proved beyond a doubt that space quantization was real. Their experiment proved beyond a doubt that space quantization was real. “[...] the electron itself spinning like a tiny gyroscope, is probably the ultimate magnetic dipole.” 189719021913

9. The Old Quantum Theory Fine Structure of multiplet spectra due to interactions between magnetic moments. μ B = ±eħ 2mc 1922 B E B = B ±eħ 2mc WTF?? Assumed l = 1 does not include 0 L = l(l+1) ħ√ ? 1897190219131921

10. Physicists at the time contrived a “rump” model for the atom to include magnetic moment interactions. Landé introduced a ratio between the atomic angular momentum and the Magnetic “rump” Moment. The gyromagnetic ratio. Rump angular momentum may be a half- integral of ħ. (Questionable) Pauli: “Large atomic nuclei possess angular momentum.” No news about proton though. Physicists at the time contrived a “rump” model for the atom to include magnetic moment interactions. Landé introduced a ratio between the atomic angular momentum and the Magnetic “rump” Moment. The gyromagnetic ratio. Rump angular momentum may be a half- integral of ħ. (Questionable) Pauli: “Large atomic nuclei possess angular momentum.” No news about proton though. Landé, and his pesky g-factor 1923 “rump” g = 2.0023193043617 μ = L ge 2mc l? l? ⇒ ħ2ħ2 18971902191319211922 OQT ?

11. Proposition that the Entity responsible for half-integral Angular momentum, and mysterious g-factor is electron itself! From that discovery came the notion of the electron spin. Their results astounded the scientific community. Proposition that the Entity responsible for half-integral Angular momentum, and mysterious g-factor is electron itself! From that discovery came the notion of the electron spin. Their results astounded the scientific community. The Old Quantum Theory still reigns however. Lorentz makes an objection!!! The Old Quantum Theory still reigns however. Lorentz makes an objection!!! Uhlenbeck & Goudsmit rewrite the rules 1925 George Eugene Uhlenbeck Samuel Goudsmit Hendrik Antoon Lorentz Albert Einstein “rump” SeSe “NO! This is wrong for two reasons.” First, if the electron had a spin and magnetic moment of one magneton, then: First, if the electron had a spin and magnetic moment of one magneton, then: E B = μ 2 /r e 3 E = mc 2 ⇒ r e ≈ 10 -12 cm Second, if the electron had a classical radius and a quantum angular momentum: Second, if the electron had a classical radius and a quantum angular momentum: r = e mc 2 L = ħ/2 ⇒ v s ≈ 10c The inadequacy of the Old Quantum Theory and classical thinking had then become apparent. g 18971902191319211922 OQT 1923 ?

12. The Fantastic Four 1926 Werner Heisenberg Pascual Jordan Erwin Schödinger Max Born iħ ψ = - ∇ 2 ψ + V(ψ) ∂ ∂t ħ 2 2m ΔxΔp ≥ ħ2ħ2 New Matrix Mechanics Zeeman Hamiltonian g 18971902191319211922 OQT 19231925 ?  α > = ( )  β > = ( ) 0101 1010 Ĥ Z = μ (S e ⋅ B eff )

13. Treats Spin Angular Momentum as an independent variable for the first time. Links directly to the two-spin wave functions. Treats Spin Angular Momentum as an independent variable for the first time. Links directly to the two-spin wave functions.  > = ( ) β + _ α “Large atomic nuclei possess angular momentum.” 1923 Pauli joins the effort, reveals spin matrices! 1927 Wolfgang Pauli g 18971902191319211922 OQT 192319251926 ? 0 1 1 0 σ x = 0 -i i 0 σ y = 1 0 0 -1 σ z = 0101 1010

14. Phipps & Taylor get it right! 1927 g 18971902191319211922 OQT 1923192519261927 ? Hydrogen atoms μ B = eħg 2mc Confirmed!! √ 1H1H Spin-quantum number √ l = ½ On the other hand, nothing yet required elementary particles to have an intrinsic Angular Momentum until a year later.

15. Dirac brings it all together The Relativistic Schrödinger Equation. The Relativistic Hamiltonian. The Relativistic Schrödinger Equation. The Relativistic Hamiltonian. 1928 Paul Adrien Maurice Dirac [p 0 - ρ 1 (σ ⋅ p) - ρ 3 mc]ψ = 0 Ĥ rel = [c(α ⋅ p) + ρ 3 mc 2 ]ψ g 18971902191319211922 OQT 1923192519261927 ?

16. The New Quantum Theory 1928 - 1939 [p 0 - ρ 1 (σ ⋅ p) - ρ 3 mc]ψ = 0 Ĥ rel = [c(α ⋅ p) + ρ 3 mc 2 ]ψ SeSe B iħ ψ = - ∇ 2 ψ + V(ψ) ∂ ∂t ħ 2 2m ΔxΔp ≥ ħ2ħ2 g 18971902191319211922 OQT 19231925192619271928 ? 0 1 1 0 σ x = 0 -i i 0 σ y = 1 0 0 -1 σ z = μ B = eħg 2mc Hydrogen atoms  α > = ( )  β > = ( ) 0101 1010 WORLD WAR II

17. The New Quantum Theory 1945 [p 0 - ρ 1 (σ ⋅ p) - ρ 3 mc]ψ = 0 Ĥ rel = [c(α ⋅ p) + ρ 3 mc 2 ]ψ SeSe B iħ ψ = - ∇ 2 ψ + V(ψ) ∂ ∂t ħ 2 2m ΔxΔp ≥ ħ2ħ2 g 18971902191319211922 OQT 19231925192619271928 ? 0 1 1 0 σ x = 0 -i i 0 σ y = 1 0 0 -1 σ z = μ B = eħg 2mc Hydrogen atoms  α > = ( )  β > = ( ) 0101 1010

18. Nuclear Magnetic Resonance is observed! Basis of study derived from the principles of The New Quantum Theory. Basis of study derived from the principles of The New Quantum Theory. 1946 Felix Bloch Edward Mills Purcell g 18971902191319211922 OQT 19231925192619271928 NQT ? 1946

19. Relativistic Quantum Theory: Dirac’s Greatest Gift g 18971902191319211922 OQT 192319251926192719281946 NQT ?

20. Equation of Motion: Criteria DeBroglie and Einstein relations. Classical Mechanics Linear and Homogeneous Differential on the First Order of time. DeBroglie and Einstein relations. Classical Mechanics Linear and Homogeneous Differential on the First Order of time. Paul Adrien Maurice Dirac p = h/λE = hν E = p 2 /(2m) + V Ψ = c 1 ψ 1 + c 2 ψ 2 Ψ(r,t)

21. Equation of Motion iħ ψ = - ∇ 2 ψ + V(ψ) ∂ ∂t ħ 2 2m -iħ ∇ = Ĥ = p 2 1 2m -E = iħ ∂ ∂t Ĥ = ∇ 2 -ħ 2 2m Schrödinger Equation Quantum Mechanics Classical Mechanics p E = Ĥ + V

22. Special Relativity x y z x’ y’ z’ vt S S’ x’ = x - vty’ = y z’ = z x 2 + y 2 + z 2 = c 2 t 2 (x’) 2 + (y’) 2 + (z’) 2 = c 2 (t’) 2 t’ = t Albert Einstein t’ = (1 - ) ½ v2c2v2c2 t - x’ = (1 - ) ½ v2c2v2c2 x - vt vx c 2 dt’ = (1 - ) ½ v2c2v2c2 dt m(v) = (1 - ) ½ v2c2v2c2 m0m0 m 2 c 2 - m 2 v 2 = (m 0 ) 2 c 2 m 2 c 2 - p 2 = (m 0 ) 2 c 2 E = (c 2 p 2 + (m 0 ) 2 c 4 ) ½ Flash Bulb when origins Coincide @ t = t’ = 0 1) 2) Starting point of the Lorentz Transformation Time Dilation and Velocity-dependent mass m 2 c 4 - p 2 c 2 = (m 0 ) 2 c 4 Relativistic Energy Equation

23. E = (c 2 p 2 + (m 0 ) 2 c 4 ) ½ Relativistic Quantum Mechanics Paul Adrien Maurice Dirac “The question remains as to why Nature should have chosen this particular model of an electron spin instead of being satisfied with the point charge.” 0 1 1 0 σ x = 0 -i i 0 σ y = 1 0 0 -1 σ z =

24. Relativistic Quantum Mechanics Paul Adrien Maurice Dirac E = c(p x 2 + p y 2 + p z 2 + (m 0 ) 2 c 2 ) ½ Ĥψ = Eψ = (p x 2 + p y 2 + p z 2 + (m 0 ) 2 c 2 ) ½ ψ iħ ∂ψ ∂t [p 0 2 - p 1 2 - p 2 2 - p 3 2 - (m 0 ) 2 c 2 ]ψ = 0 p 0 2 = ħcħc ∂ 2 ∂t 2 -( ) 2 [p 0 2 - (p 1 2 + p 2 2 + p 3 2 + (m 0 ) 2 c 2 ) ½ ]ψ = 0 E = (c 2 p 2 + (m 0 ) 2 c 4 ) ½ 0 1 1 0 σ x = 0 -i i 0 σ y = 1 0 0 -1 σ z = Shrödinger Equation New indices, all terms are coordinate independent. iħ c ∂ ∂t { = p 0 } Multiply by: {p 0 + (p 1 2 + p 2 2 + p 3 2 + (m 0 ) 2 c 2 ) ½ } c Klein-Gordon Equation

25. Dirac’s Trick Paul Adrien Maurice Dirac (p 0 - α 1 p 1 - α 2 p 2 + α 3 p 3 + β)ψ = 0 [p 0 2 - Σ 123 [(α 1 p 1 ) 2 + (α 1 α 2 + α 2 α 1 )p 1 p 2 + (α 1 β + βα 1 )p 1 ]ψ = 0 [p 0 2 - p 1 2 - p 2 2 - p 3 2 - (m 0 ) 2 c 2 ]ψ = 0 It was necessary to find a Linear Wave equation in the first time derivative. Such an equation required the existence of an intrinsic degree of freedom which turned out to behave like spin. Dirac Started from this simple form: Multiply by: {p 0 - (α 1 p 1 2 + α 2 p 2 2 + α 3 p 3 2 + β} The next task is to obtain values for α 1,2,3 and β.

26. [p 0 2 - Σ 123 [(α 1 p 1 ) 2 + (α 1 α 2 + α 2 α 1 )p 1 p 2 + (α 1 β + βα 1 )p 1 ]ψ = 0 Integrating Dirac’s Trick into the picture Paul Adrien Maurice Dirac α 1 2 = 1 β 2 = α m m 2 c 2 α m 2 = I α 1 α 2 + α 2 α 1 = 0 α 1 β + βα 1 = 0 [p 0 2 - p 1 2 - p 2 2 - p 3 2 - (m 0 ) 2 c 2 ]ψ = 0 Equalizing his trick with the Klein-Gordon Equation, Dirac began the search for the missing terms. From these equalities, the conditions for such operators could then be written in a compact form. α a α b + α b α a = 2δ ab {a, b = 1, 2, 3, m}

27. = [p 0 - ρ 1 (σ ⋅ p) - ρ 3 mc]ψ Spin Matrices Applied Paul Adrien Maurice Dirac α 1 = ρ 1 σ 1 α 2 = ρ 1 σ 2 α 3 = ρ 1 σ 3 α m = ρ 3 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 σ 1 = 0 -i 0 0 i 0 0 0 0 0 0 -i 0 0 i 0 σ 2 = 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1 σ 3 = 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ρ 1 = 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 ρ 3 = Ĥ rel = [c(α ⋅ p) + ρ 3 mc 2 ]ψ α a α b + α b α a = 2δ ab {a, b = 1, 2, 3, m} Indeed, such operators exist as 4x4 versions of the Pauli Matrices. (p 0 - α 1 p 1 - α 2 p 2 + α 3 p 3 + β)ψ = 0 Ĥ = ∇ 2 -ħ 2 2m

28. Ĥ rel = [c(α ⋅ p) + ρ 3 mc 2 ]ψ 0 = [p 0 - ρ 1 (σ ⋅ p) - ρ 3 mc]ψ ConclusionsConclusions Paul Adrien Maurice Dirac σB 0 eħ 2mc The relativistic Schrodinger Equation predicts the existence of Spin. The relativistic Hamiltonian, when applied in a magnetic field B 0, allows the derivation of the Magnetic Moment μ of the electron. μ B = An ab initio derivation of the gyromagnetic ratio of the electron! An unparalleled accomplishment in particle physics! “Thus, it is proven.”

29. ReferencesReferences P. A. M. Dirac, The Principles of Quantum Mechanics, 4th Edition, Oxford University Press, 1959. N. Zumbulyadis, Whence Spin?, Concepts in Magnetic Resonance, 1991, 3, 89-107. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th Edition, Oxford University Press, 1959. N. Zumbulyadis, Whence Spin?, Concepts in Magnetic Resonance, 1991, 3, 89-107.

30. Thank You Questions?Questions?