The Origin of SpinThe Origin of SpinThe Origin of SpinThe Origin of Spin

Joshua VarnerJoshua VarnerJoshua VarnerJoshua Varner

Quantum Mechanics:Quantum Mechanics:A HistoryA History

Quantum Mechanics:Quantum Mechanics:A HistoryA History

Thomson discovers Electron!Thomson discovers Electron!Thomson discovers Electron!Thomson discovers Electron!

“Plum Pudding” model of the 19th century atom.

“Plum Pudding” model of the 19th century atom.

1897

J. J. Thomson

Zeeman and his effectZeeman and his effectZeeman and his effectZeeman 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 = Le2mc

Quantization of Energy levels follows from Quantization of Angular Momentum.

Quantization of Energy levels follows from Quantization of Angular Momentum.

Bohr postulates Quantum TheoryBohr postulates Quantum TheoryBohr postulates Quantum TheoryBohr 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

1897 1902

L = ħln

Lorentz’s new unit of magnetic Lorentz’s new unit of magnetic momentmoment

Lorentz’s new unit of magnetic Lorentz’s new unit of magnetic momentmoment1913

Hendrik Antoon Lorentz

μ = Le2mc

μB = ±eħ2mc

EB = -μB∙B

EB = B±eħ2mc

⇒

⇒

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)

1897 1902 1913

Lorentz’s new unit of magnetic Lorentz’s new unit of magnetic momentmoment

Lorentz’s new unit of magnetic Lorentz’s new unit of magnetic momentmoment1913

Hendrik Antoon Lorentz

B

E1 E2

E1-EB

E1+EB

E2-EB

E2+EBHowever, no one questioned the possibility of μB = 0... Moving on.

However, no one questioned the possibility of μB = 0... Moving on.

EB = B±eħ2mc

1897 1902 1913

The Stern-Gerlach Experiment!The Stern-Gerlach Experiment!The Stern-Gerlach Experiment!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 itselfspinning like a tiny gyroscope,

is probably the ultimatemagnetic dipole.”

1897 1902 1913

The Old Quantum TheoryThe Old Quantum TheoryThe Old Quantum TheoryThe Old Quantum Theory

Fine Structure of multiplet spectra due to

interactionsbetween magnetic

moments.

μB = ±eħ2mc

1922

B

EB = B±eħ2mc

WTF??

Assumed l = 1does not include 0

L = l(l+1) ħ√

?

1897 1902 1913 1921

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-factorLandé, and his pesky g-factorLandé, and his pesky g-factorLandé, and his pesky g-factor1923

“rump”

g = 2.0023193043617

μ = Lge2mc

l?

⇒ ħ2

1897 1902 1913 1921 1922

OQT

?

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 Uhlenbeck & Goudsmit rewrite the rulesthe rules

Uhlenbeck & Goudsmit rewrite Uhlenbeck & Goudsmit rewrite the rulesthe rules1925

George Eugene Uhlenbeck

Samuel GoudsmitHendrik Antoon LorentzAlbert Einstein

“rump”

Se

“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:EB = μ2/re

3

E = mc2

⇒ re ≈ 10-12cm

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

mc2

L = ħ/2⇒ vs ≈ 10c

The inadequacy of theOld Quantum Theory and

classical thinking hadthen become apparent.

g1897 1902 1913 1921 1922

OQT

1923

?

The Fantastic FourThe Fantastic FourThe Fantastic FourThe Fantastic Four1926

Werner Heisenberg

Pascual Jordan

Erwin Schödinger

Max Born

iħ ψ = - ∇2ψ + V(ψ)∂∂t

ħ2

2m

ΔxΔp ≥ ħ2

New Matrix Mechanics

Zeeman Hamiltonian

g1897 1902 1913 1921 1922

OQT

1923 1925

?

⎮α> = ( ) ⎮β> = ( )01

10

ĤZ = μ (Se ⋅ Beff)

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.”“Large atomic nuclei possess angular momentum.”

1923

Pauli joins the effort, reveals spin Pauli joins the effort, reveals spin matrices!matrices!

Pauli joins the effort, reveals spin Pauli joins the effort, reveals spin matrices!matrices!1927

Wolfgang Pauli

g1897 1902 1913 1921 1922

OQT

1923 1925 1926

?

0 11 0σx =

0 -i i 0σy =

1 00 -1σz =

01

10

Phipps & Taylor get it right!Phipps & Taylor get it right!Phipps & Taylor get it right!Phipps & Taylor get it right!1927

g1897 1902 1913 1921 1922

OQT

1923 1925 1926 1927

?

Hydrogen atomsμB = eħg2mc

Confirmed!!√

1HSpin-quantum

number√l = ½

On the other hand, nothing yet required elementary particles to have an

intrinsic Angular Momentum until a year later.

Dirac brings it all togetherDirac brings it all togetherDirac brings it all togetherDirac brings it all together

The Relativistic Schrödinger Equation.

The Relativistic Hamiltonian.

The Relativistic Schrödinger Equation.

The Relativistic Hamiltonian.

1928

Paul Adrien Maurice Dirac

[p0 - ρ1(σ ⋅ p) - ρ3mc]ψ = 0

Ĥrel = [c(α ⋅ p) + ρ3mc2]ψ

g1897 1902 1913 1921 1922

OQT

1923 1925 1926 1927

?

The New Quantum TheoryThe New Quantum TheoryThe New Quantum TheoryThe New Quantum Theory1928 - 1939

[p0 - ρ1(σ ⋅ p) - ρ3mc]ψ = 0

Ĥrel = [c(α ⋅ p) + ρ3mc2]ψ

Se

B

iħ ψ = - ∇2ψ + V(ψ)∂∂t

ħ2

2m

ΔxΔp ≥ ħ2

g1897 1902 1913 1921 1922

OQT

1923 1925 1926 1927 1928

?

0 11 0σx =

0 -i i 0σy =

1 00 -1σz =

μB = eħg2mc

Hydrogen atoms

⎮α> = ( ) ⎮β> = ( )01

10

WORLDWAR II

The New Quantum TheoryThe New Quantum TheoryThe New Quantum TheoryThe New Quantum Theory1945

[p0 - ρ1(σ ⋅ p) - ρ3mc]ψ = 0

Ĥrel = [c(α ⋅ p) + ρ3mc2]ψ

Se

B

iħ ψ = - ∇2ψ + V(ψ)∂∂t

ħ2

2m

ΔxΔp ≥ ħ2

g1897 1902 1913 1921 1922

OQT

1923 1925 1926 1927 1928

?

0 11 0σx =

0 -i i 0σy =

1 00 -1σz =

μB = eħg2mc

Hydrogen atoms

⎮α> = ( ) ⎮β> = ( )01

10

Nuclear Magnetic Resonance is Nuclear Magnetic Resonance is observed!observed!

Nuclear Magnetic Resonance is Nuclear Magnetic Resonance is observed!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

g1897 1902 1913 1921 1922

OQT

1923 1925 1926 1927 1928

NQT

?

1946

Relativistic Quantum Relativistic Quantum Theory: Dirac’s Greatest Theory: Dirac’s Greatest

GiftGift

Relativistic Quantum Relativistic Quantum Theory: Dirac’s Greatest Theory: Dirac’s Greatest

GiftGift

g1897 1902 1913 1921 1922

OQT

1923 1925 1926 1927 1928 1946

NQT

?

Equation of Motion: CriteriaEquation of Motion: CriteriaEquation of Motion: CriteriaEquation 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/λ E = hν

E = p2/(2m) + V

Ψ = c1ψ1 + c2ψ2

Ψ(r,t)

Equation of MotionEquation of MotionEquation of MotionEquation of Motion

iħ ψ = - ∇2ψ + V(ψ)∂∂t

ħ2

2m

-iħ∇ = Ĥ = p212m

-E = iħ ∂∂t Ĥ = ∇2-ħ2

2m

Schrödinger Equation

Quantum Mechanics

Classical Mechanicsp

E = Ĥ + V

Special RelativitySpecial RelativitySpecial RelativitySpecial Relativity

x

y

z

x’

y’

z’

vt

S S’

x’ = x - vt y’ = y

z’ = z

x2 + y2 + z2 = c2t2

(x’)2 + (y’)2 + (z’)2 = c2(t’)2

t’ = t

Albert Einstein

t’ =(1 - )½v2

c2

t - x’ =(1 - )½v2

c2

x - vt vx c2

dt’ =(1 - )½v2

c2

dtm(v) =

(1 - )½v2

c2

m0

m2c2 - m2v2 = (m0)2c2

m2c2 - p2 = (m0)2c2

E = (c2p2 + (m0)2c4)½

Flash Bulb when origins Coincide @ t = t’ = 0

1)2)

Starting point of the Lorentz Transformation

Time Dilation and Velocity-dependent mass

m2c4 - p2c2 = (m0)2c4

Relativistic Energy Equation

E = (c2p2 + (m0)2c4)½

Relativistic Quantum MechanicsRelativistic Quantum MechanicsRelativistic Quantum MechanicsRelativistic 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 11 0σx =

0 -i i 0σy =

1 00 -1σz =

Relativistic Quantum MechanicsRelativistic Quantum MechanicsRelativistic Quantum MechanicsRelativistic Quantum Mechanics

Paul Adrien Maurice Dirac

E = c(px2 + py

2 + pz2 + (m0)2c2)½

Ĥψ = Eψ= (px

2 + py2 + pz

2 + (m0)2c2)½ψiħ∂ψ∂t

[p02 - p1

2 - p22 - p3

2 - (m0)2c2]ψ = 0

p02 = ħ

c∂2

∂t2-( )2

[p02 - (p1

2 + p22 + p3

2 + (m0)2c2)½]ψ = 0

E = (c2p2 + (m0)2c4)½0 11 0σx =

0 -i i 0σy =

1 00 -1σz =

Shrödinger Equation

New indices, all terms are coordinate independent. iħ

c∂∂t{ = p0}

Multiply by:

{p0 + (p1

2 + p22 + p3

2 + (m0)2c2)½}

c

Klein-Gordon Equation

Dirac’s TrickDirac’s TrickDirac’s TrickDirac’s Trick

Paul Adrien Maurice Dirac

(p0 - α1p1 - α2p2

+ α3p3 + β)ψ = 0

[p02 - Σ123[(α1p1)2 + (α1α2 + α2α1)p1p2

+ (α1β + βα1)p1]ψ = 0[p0

2 - p12 - p2

2 - p32 - (m0)2c2]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:

{p0 - (α1p1

2 + α2p22 + α3p3

2 + β}

The next task is to obtain values for α1,2,3 and β.

[p02 - Σ123[(α1p1)2 + (α1α2 + α2α1)p1p2

+ (α1β + βα1)p1]ψ = 0

Integrating Dirac’s Trick into the Integrating Dirac’s Trick into the picturepicture

Integrating Dirac’s Trick into the Integrating Dirac’s Trick into the picturepicture

Paul Adrien Maurice Dirac

α12 = 1 β2 = αmm2c2

αm2 = I

α1α2 + α2α1 = 0α1β + βα1 = 0

[p02 - p1

2 - p22 - p3

2 - (m0)2c2]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.

αaαb + αbαa = 2δab

{a, b = 1, 2, 3, m}

= [p0 - ρ1(σ ⋅ p) - ρ3mc]ψ

Spin Matrices AppliedSpin Matrices AppliedSpin Matrices AppliedSpin Matrices Applied

Paul Adrien Maurice Dirac

α1 = ρ1σ1 α2 = ρ1σ2 α3 = ρ1σ3 αm = ρ3

0 1 0 01 0 0 00 0 0 10 0 1 0

σ1 =0 -i 0 0i 0 0 00 0 0 -i0 0 i 0

σ2 =1 0 0 00 -1 0 00 0 1 00 0 0 -1

σ3 =

0 0 1 00 0 0 11 0 0 00 1 0 0

ρ1 =1 0 0 00 1 0 00 0 -1 00 0 0 -1

ρ3 =

Ĥrel = [c(α ⋅ p) + ρ3mc2]ψ

αaαb + αbαa = 2δab

{a, b = 1, 2, 3, m}

Indeed, such operators exist as 4x4 versions of the Pauli Matrices.

(p0 - α1p1 - α2p2

+ α3p3 + β)ψ = 0

Ĥ = ∇2-ħ2

2m

Ĥrel = [c(α ⋅ p) + ρ3mc2]ψ

0 = [p0 - ρ1(σ ⋅ p) - ρ3mc]ψ

ConclusionsConclusionsConclusionsConclusions

Paul Adrien Maurice Dirac

σB0eħ

2mc

The relativistic Schrodinger Equation predicts the existence of Spin.

The relativistic Hamiltonian, when applied in a magnetic field B0, allows the derivation of the Magnetic Moment μ of the electron.μB =An ab initio derivation of the gyromagnetic ratio of the electron! An unparalleled accomplishment in particle physics!

“Thus,it is

proven.”

ReferencesReferencesReferencesReferences

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.

Thank YouThank YouThank YouThank You

Questions?Questions?Questions?Questions?