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Lecture 22

Relativistic Quantum Mechanics

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Background

Why study relativistic quantum mechanics?

1 Many experimental phenomena cannot be understood within purelynon-relativistic domain.

e.g. quantum mechanical spin, emergence of new sub-atomicparticles, etc.

2 New phenomena appear at relativistic velocities.

e.g. particle production, antiparticles, etc.

3 Aesthetically and intellectually it would be profoundly unsatisfactory

if relativity and quantum mechanics could not be united.

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Background

When is a particle relativistic?

1 When velocity approaches speed of lightcor, more intrinsically,when energy is large compared to rest mass energy, mc2.

e.g. protons at CERN are accelerated to energies of ca. 300GeV(1GeV= 109eV) much larger than rest mass energy, 0.94 GeV.

2 Photons have zero rest mass and always travel at the speed of light they are never non-relativistic!

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Background

What new phenomena occur?

1 Particle production

e.g. electron-positron pairs by energetic -rays in matter.

2 Vacuum instability: If binding energy of electron

Ebind=

Z2e4m

22 >2mc

2

a nucleus with initially no electrons is instantly screened by creationof electron/positron pairs from vacuum

3 Spin: emerges naturally from relativistic formulation

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Background

When does relativity intrude on QM?

1 When Ekin mc2, i.e. p mc2 From uncertainty relation, xp>h, this translates to a length

x> h

mc =c

the Compton wavelength.

3 for massless particles,c=

, i.e. relativity always important for,

e.g., photons.

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Relativistic quantum mechanics: outline

1 Special relativity (revision and notation)

2 Klein-Gordon equation

3 Dirac equation

4 Quantum mechanical spin

5 Solutions of the Dirac equation

6 Relativistic quantum field theories

7 Recovery of non-relativistic limit

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Special relativity (revision and notation)

Space-time is specified by a 4-vector

A contravariant 4-vector

x= (x) (x0, x1, x2, x3) (ct, x)

transformed into covariant 4-vector x=gx by Minkowskiimetric

(g) = (g) =

11

11

, gg =g

,

Scalar product: x y=xy =xyg=x

y

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Lorentz group: consists of linear transformations, , preservingxy, i.e. for x x = x =xy

x y =gxy

=g

=g

xy =gxy

e.g. Lorentz transformation along x1

=

v/c

v/c 1 0

0 1

, =

1

(1 v2/c2)1/2

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4-vectors classified as time-like or space-like

x2

= (ct)2

x2

1 forward time-like: x2 >0, x0 >0

2 backward time-like: x2 >0, x0

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Lorentz group splits up into four components:

1 Every LT maps time-like vectors (x2 >0) into time-like vectors

2 Orthochronous transformations 00 >0, preserveforward/backward sign

3 Proper: det = 1 (as opposed to1)4 Group of proper orthochronous transformation: L

+ subgroup

of Lorentz group excludes time-reversal and parity

T =

1

11

1

, P=

11

11

5 Remaining components of group generated by

L =TL+, L

=PL

+, L

+ =TPL

+.

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1 Special relativity requires theories to be invariant under LT or, moregenerally,Poincare transformations: x x +a

2 Generators of proper orthochronous transformations, L+, canbe reached by infinitesimal transformations

=

+

,

1

g

=g++ +O(

2) !=g

i.e. = , has six independent componentsL+ has six independent generators: three rotations and three boosts

3 covariant and contravariant derivative, chosen s.t. x = 1

=

x =1

c

t, , = x = 1c t, .4 dAlembertian operator: 2 =

= 1

c22

t2 2

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3 Dirac equation





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Klein-Gordon equation

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How to make wave equation relativistic?

According to canonical quantization procedure in NRQM:

p= i, E=it, i.e. p (E/c, p) p

transforms as a 4-vector under LT

What if we apply quantization procedure to energy?

p

p = (E/c)2

p2

=m2

c2

, m rest massE(p) = +

m2c4 +p2c2

1/2 it= m2c4 2c221/2 Meaning of square root? Taylor expansion:

it=mc

2

2

2

2m

4(

2)2

8m3c2 + i.e. time-evolution of specified by infinite number of boundaryconditions non-locality, and space/time asymmetry suggeststhat this equation is a poor starting point...

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Alternatively, apply quantization to energy-momentum invariant:

E2 =p2c2 +m2c4, 22t= 2c22 +m2c4Setting kc=

2

c=

mc

, leads to Klein-Gordon equation,

2 +k2c= 0Klein-Gordon equation is local and manifestly Lorentz covariant.

Invariance of under rotations means that, if valid at all,Klein-Gordon equation limited to spinless particles

But can ||2 be interpreted as probability density?

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Klein-Gordon equation: Probabilities

Probabilities? Take lesson from non-relativistic quantum mechanics:

Schrodinger eqn. it+

222m

= 0,

c.c.

it+

222m

= 0

i.e. t||2 i

2m ( ) = 0

cf. continuity relation conservation of probability: t+ j= 0

= ||2, j= i 2m

( )

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Klein-Gordon equation: Probabilities

Applied to KG equation: 1c2

2t 2 +k2c= 02t(

t t) 2c2 ( ) = 0

cf. continuity relation conservation of probability: t+ j= 0.

= i

2mc2(t t) , j= i

2m( )

With 4-currentj = (c,j), continuity relation j = 0.

i.e. Klein-Gordon density is the time-like component of a 4-vector.

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Klein-Gordon equation: viability?

But is Klein-Gordon equation acceptable?

Density= i

2mc2(t t) is not positive definite.

Klein-Gordon equation is not first order in time derivativetherefore we must specify and t everywhere at t= 0.

Klein-Gordon equation has both positive and negative energysolutions.

Could we just reject negative energy solutions? Inconsistent localinteractions can scatter between positive and negative energy states

2 +k2c=F() self interaction(+iqA/c)2 +k2c

= 0 interaction with EM field

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Relativistic quantum mechanics: summary

When the kinetic energy of particles become comparable to restmass energy, p mcparticles enter regime where relativity intrudeson quantum mechanics.

At these energy scales qualitatively new phenomena emerge:

e.g. particle production, existence of antiparticles, etc.

By applying canonical quantization procedure to energy-momentuminvariant, we are led to the Klein-Gordon equation,

(2 +k2c)= 0

where = c

2 =

mcdenotes the Compton wavelength.

However, the Klein-Gordon equation does not lead to a positivedefinite probability density and admits positive and negative energysolutions these features led to it being abandoned as a viablecandidate for a relativistic quantum mechanical theory.

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Lecture 23

Relativistic Quantum Mechanics:

Dirac equation

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3 Dirac equation





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Dirac Equation

Dirac placed emphasis on two constraints:

1 Relativistic equation must be first order in time derivative (andtherefore proportional to = (t/c,)).

2 Elements of wavefunction must obey Klein-Gordon equation.

Diracs approach was to try to factorize Klein-Gordon equation:(2 +m2)= 0 (where henceforth we set = c= 1)

(im)(i m)= 0

i.e. with p=i

(p m)= 0

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Dirac Equation

(p m)= 0

Equation is acceptable if:

1 satisfies Klein-Gordon equation, (2 +m2)= 0;2 there must exist 4-vector current density which is conserved

and whose time-like component is a positive density;3 does not have to satisfy any auxiliary boundary conditions.

From condition (1) we require (assuming [, p] = 0)

0 = (p+m) (p m)= (

(+)/2

pp m2)

= 12{, }pp m2= (gpp m2)= (p2 m2)i.e. obeys Klein-Gordon if{, } + = 2g

, and therefore , can not be scalar.

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Dirac Equation: Hamiltonian formulation

(p m)= 0, {, }= 2g

To bring Dirac equation to the form it= H, consider

0(p m)=0(0p0 pm)= 0

Since (0)2 12{0, 0}= g00 = I,

0(p m)=it 0 p m0= 0

i.e. Dirac equation can be written as it= H with

H= p+m, = 0, =0

Using identity {, }= 2g,

2 = I, {,}= 0, {i,j}= 2ij (exercise)

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Dirac Equation: matrices

it= H, H= p+m, = 0, =0

Hermiticity ofHassured if = , and =, i.e.

(0) 0 =0, and 0 =0

So we obtain the defining properties of Dirac matrices,

=00, {, }= 2g

Since space-time is four-dimensional, must be of dimension at

least 4 4 has at least four components.However, 4-component wavefunction does not transform as4-vector it is known as a spinor (or bispinor).

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Dirac Equation: matrices

=00, {, }= 2g

From the defining properties, there are several possiblerepresentations ofmatrices. In the Dirac/Pauli representation:

0 =

I2 0

0 I2

, =

0 0

Pauli spin matrices

ij=ij+iijkk, i =i

e.g., 1 = 0 11 0 , 2 =

0 ii 0 , 3 =

1 00

1

So, in Dirac/Pauli representation,

= 0=

0 0

, =0 =

I2 0

0 I2

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Dirac Equation: conjugation, density and current

(p m)= 0, =00

Applying complex conjugation to Dirac equation

[(p m)] =i m

= 0,

()

Since (0)2 = I, we can write,

0 =0

(i0 0

m0) =

i+m

0

Introducing Feynman slash notationa a, obtain conjugateform of Dirac equation

(i +m) = 0

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Dirac Equation: conjugation, density and current

(i +m) = 0, =

The, combining the Dirac equation, (im)= 0 with its

conjugate, we have (i +m)= 0 = (im), i.e.

+=(

j

) = 0

We therefore identify j = (,j) = (,) as the 4-current.

So, in contrast to the Klein-Gordon equation, the density

= j0

=

is, as required, positive definite.

Motivated by this triumph(!), let us now consider what constraintsrelativistic covariance imposes and what consequences follow.

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Relativistic covariance

If(x) obeys the Dirac equation its counterpart (x) in a LTframe x = x

, must obey the Dirac equation,i

x m

(x) = 0

If observer can reconstruct (x) from (x) there must exist alocal (linear) transformation,

(x) =S()(x)

where S() is a 4 4 matrix, i.e.

S()ix

x

(1)S()S1()

x

mS()(x) = 0

Compatible with Dirac equation if S()S1() = (1)

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(x) =S()(x), S()S1() = (1)

But how do we determine S()? For an infinitesimal (i.e. properorthochronous) LT

=

+

, (

1)= +

(recall that generators, =, are antisymmetric).

This allows us to form the Taylor expansion ofS():

S() S(I+) =S(I)

I

+

S

i4 +O(2)

where = (follows from antisymmetry of) is a matrix inbispinor space, and is a number.

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S() = I i4

+ , S1() = I+ i

4

+

Requiring that S()S1() = (1), a little bit of algebra

(see problem set/handout) shows that matrices must obey therelation,

[, ] = 2i

This equation is satisfied by (exercise)

= i

2[, ]

In summary, under set of infinitesimal Lorentz transformation,x = x, where = I+, relativistic covariance of Dirac equationdemands that wavefunction transforms as (x) =S() whereS() = I i

4

+O(2) and = i2

[, ].

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S() = I

i

4

+

Finite transformations (i.e. non-infinitesimal) generated by

S() = exp i

4

, = g

1 Transformations involving unitary matricesS(), whereSS= I translate to spatial rotations.

2 Transformations involving Hermitian matricesS(), whereS =Stranslate to Lorentz boosts.

So what?? What are the consequences of relativistic covariance?

A l d i

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Angular momentum and spin

For infinitesimal anticlockwise rotation by angle

around n

x x+n x x, I+ n

i.e. ij= ikjnk, 0i =i0 = 0.

In non-relativistic quantum mechanics:

(x) =(x) =(1x) ((I )x) (x) x (x) + =(x) in x (i)(x) +

= (1 inL)(x

) + U(x

)

cf. generator of rotations: U=einL.

A l d i

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But relativistic covariance of Dirac equation demandsthat (x) =S()(x)

With ij= ijknk, 0i=i0 = 0,

S() I i4

= I i4ijikjnk

In Dirac/Pauli representation k Pauli spin matrices

ij= i

2[i, j] =ijk

k 0

0 k

, i=

0 ii 0

i.e. S() = I inS where

Sk= 14ijkij=

14 ijkijl

jjkl jljk= 2kll I= 1

2

k 00 k

A l d i

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Altogether, combining components of transformation,

(x) =

I

inS

S() (x1x) (I inL)(x)

(I in(S+L))(x)

we obtain

(x) =S()(1x) (1 inJ)(x)

whereJ= L+S represents total angular momentum.

Intrinsic contribution to angular momentum known as spin.

[Si, Sj] =iijkSk, (Si)2 = 1

4 for each i

Dirac equation is relativistic wave equation for spin 1/2 particles.

P i

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Parity

So far we have only dealt with the subgroup of properorthochronous Lorentz transformations, L+.

Taking into account Parity, P =

11

11

relativistic covariance demands S()S1() = (1)

S1(P)0S(P) =0, S1(P)iS(P) = i

achieved ifS(P) =0ei, where denotes arbitrary phase.

But since P2 = I, ei = 1 or1

(x) =S(P)(1x) =0(Px)

where =1 intrinsic parity of the particle

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Lecture 24

Relativistic Quantum Mechanics:

Solutions of the Dirac equation

R l ti i ti t h i tli

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3 Dirac equation

4 Quantum mechanical spin5 Solutions of the Dirac equation



F ti l l ti f Di E ti

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Free particle solutions of Dirac Equation

(p

m)= 0,

p=i

Free particle solution of Dirac equation is a plane wave:

(x) =eipxu(p) =eiEt+ipxu(p)

where u(p) is the bispinor amplitude.

Since components of obey KG equation, (pp m2)= 0,

(p0)2 p2 m2 = 0, E p0=

p2 +m2

So, once again, as with Klein-Gordon equation we encounterpositive and negative energy solutions!!

F ti l l ti f Di E ti

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Free particle solutions of Dirac Equation

(x) =eipxu(p) =eiEt+ipxu(p)

What about bispinor amplitude, u(p)?

In Dirac/Pauli representation,

0 = I2

I2 , =

the components of the bispinor obeys the condition,

(p m)u(p) =

p0 m p p p0 m

u(p) = 0

i.e. bispinor elements:

u(p) =

,

(p0 m)= pp= (p0 +m)

F ee a ticle sol tions of the Di ac E ation

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Free particle solutions of the Dirac Equation

u(p) =

, (p0 m)= pp= (p0 +m)

Consistent when (p0)2 =p2 +m2 and = p

p0 +m

u(r)(p) =N(p) (r)

p

p0+m(r)

where (r) are a pair of orthogonal two-component vectors withindex r= 1, 2, and N(p) is normalization.

Helicity: Eigenvalue of spin projected along direction of motion

12

p|p|

() S p|p|

() =12()

e.g. ifp= pe3, (+) = (1, 0), () = (0, 1)


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So, general positiveenergy plane wave solution written in eigenbasis

of helicity,

()p (x) =N(p)eipx

()

|p|

p0+m()

But how to deal with the problem of negative energy states? Mustwe reject the Dirac as well as the Klein-Gordon equation?

In fact, the existence of negative energy states provided the key thatled to the discovery ofantiparticles.

To understand why, let us consider the problem of scattering from apotential step...

Klein paradox and antiparticles

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Consider plane wave, unit amplitude, energy E, momentum pe3,and spin(= (1, 0)) incident on potential barrier V(x) =V(x3)

in =eip0t+ipx3

(+)

p

p0+m(+)

At barrier, spin is conserved, component r is reflected (E,pe3),

and component t is transmitted (E

=E V, p

e3)From Klein-Gordon condition (energy-momentum invariant):

p20 E2 =p2 +m2 and p02 E2 =p2 +m2


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in =eip0t+ipx3

(+)

pp0+m(+)

Boundary conditions: since Dirac equation is first order, require onlycontinuity of at interface (cf. Schrodinger eqn.)

1

0p/(E+m)

0

+r1

0p/(E+m)

0

=t1

0p/(E +m)

0

(helicity conserved in reflection)

Equating (generically complex) coeffi

cients:

1 +r=t, p

E+m(1 r) = p

E +mt


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1 +r=t (1), p

E+m

(1

r) =

p

E

+m

t (2)

From (2), 1 r=t where

=p

p

(E+m)

(E +m)

Together with (1), (1 +)t= 2

t= 2

1 +,

1 +r

1 r = 1

, r=

1 1 +

Interpret solution by studying vector current: j= =

j3 =3, 3 =03=

3

3


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j3 = 33 (Up to overall normalization) the incident, transmitted and reflectedcurrents given by,

j(i)3 =

1 0 p

E+m 0 0 3

3 0

10p

E+m

0

= 2pE+m ,j(t)3 =

1

E

+m

(p +p)|t|2, j(r)3 =

2p

E+m

|r|2

where we note that, depending on height of the potential, p maybe complex (cf. NRQM).


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=p

p

E+m

E +m

Therefore, ratio of reflected/transmitted to incident currents,

j(r)3

j(i)3=|r|2 =

1 1 +

2

j(t)3

j(i)3

=|t|2(p +p)

2p

E+m

E +m =

4

|1 +|21

2(+ ) =

2(+)

|1 +|2

From which we can confirm current conservation, j(i)3 =j

(r)3 +j

(t)3 :

1 + j(r)3

j(i)3

= |1 +|2 |1 |2

|1 +|2 =

2(+)

|1 +|2 =

j(t)3

j(i)3


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j(r)3

j(i)3=|r|2 =

1 1 +

2

Three distinct regimes in energy:

1 E (E V)> m:i.e. p2 =E2 m2 >0, p >0 (beam propagates to right).

Therefore p

p

E+m

E +m>0 and real; |j

(r)3 |< |j

(i)3 | as expected,

i.e. for E >m, as in non-relativistic quantum mechanics, some ofthe beam is reflected and some transmitted.

2 m> E > m:i.e. p2 =E2 m2

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Physical Interpretation:

Particles from right should be interpreted as antiparticlespropagating to right

i.e. incoming beam stimulates emission of particle/antiparticle pairsat barrier.

Particles combine with reflected to beam to create current to left

that is larger than incident current while antiparticles propagate tothe right in the barrier region.


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Negative energy states

Existence of antiparticles suggests redefinition of plane wave stateswith E2m the potential step is in a precarious situation: Itbecomes energetically favourable to create particle/antiparticle pairs cf. vacuum instability.

Incident beam stimulates excitation of a positive energy particlefrom negative energy sea leaving behind positive energy hole anantiparticle.


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cf. creation of electron-positron pairvacuum due to high energy photon.


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Therefore, for E2m, theory describes creation of particle/antiparticle pairs.

must be viewed as a quantum field capable of describing anindefinite number of particles!!

In fact, Dirac equation must be viewed as field equation, cf. waveequation for harmonic chain. As with chain, quantization of theoryleads to positive energy quantum particles (cf. phonons).

Allows reinstatement of Klein-Gordon theory as a relativistic theoryfor scalar (spin 0 particles)...

Quantization of Klein-Gordon field

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Quantization of Klein Gordon field

Klein-Gordon equation abandoned as candidate for relativistictheory on basis that (i) it admitted negative energy solutions, and

(ii) probability density was not positive definite.

But Klein paradox suggests reinterpretation of Dirac wavefunctionas a quantum field.

If were a classical field, Klein-Gordon equation, (2 m2)= 0would be associated with Lagrangian density,

L= 1

2

12

m22

Defining canonical momentum, (x) =L=(x)

H= L= 12

2 + ()2 +m22

H is +ve definite! i.e. if quantized, only +ve energies appear.

Quantization of Klein-Gordon field

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Quantization of Klein Gordon field

Promoting fields to operators and , with equal timecommutation relations, [(x, t),(x, t)] =i3(x x), (for m= 0,

cf. harmonic chain!)

H=

d3x

1

2

2 + ()2 +m22

Turning to Fourier space (with k0 k =

k2 +m2)

(x) = d3k

(2)32k

a(k)eikx +a(k)eikx

, (x) 0(x)

where

a(k), a(k)

= (2)32k3(k k),

H= d

3k

(2)32k

k a(k)a(k) +1

2Bosonic operators a and a create and annihilate relativistic scalar(bosonic, spin 0) particles

Quantization of Dirac field

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Dirac equation associated with Lagrangian density,

L= (i m), i.e. L= (i m)= 0

With momentum =L= i0 =i, Hamiltonian density

H= L=

i

0

0 L= (i+m)

Once again, we can follow using canonical quantization procedure,promoting fields to operators but, in this case, one must imposeequal time anti-commutation relations,

{(x, t),(x

, t)} (x, t)(x

, t) + (x, t)(x

, t)=i3(x x)


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Turning to Fourier space (with k0 k =

k2 +m2)

(x) =

2r=1

d3k(2)32k

ar(k)u(r)(k)eikx +br(k)v(r)(k)eikxwith equal time anti-commutation relations (hallmark offermions!)

ar(k), as(k

)= br(k), bs(k

)= (2)32krs

3(k k)ar(k), as(k)= br(k), bs(k)= 0which accommdates Pauli exclusion ar(k)

2 = 0(!), obtain

H=2

r=1 d3k

(2)32k

k ar(k)ar(k) +b

r(k)br(k)

Physicallya(k)u(r)(k)eikx annihilates +ve energy fermion particle(helicityr), and b(k)v(r)(k)eikx creates a +ve energy antiparticle.

Low energy limit of the Dirac equation

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Previously, we have explored the relativistic (fine-structure)

corrections to the hydrogen atom. At the time, we alluded to theseas the leading relativistic contributions to the Dirac theory.

In the following section, we will explore how these correctionsemerge from relativistic formulation.

But first, we must consider interaction of charged particle with

electromagnetic field.

As with non-relativistic quantum mechanics, interaction of Diracparticle of charge q(q= e for electron) with EM field defined byminimal substitution, p p qA, where A = (, A), i.e.

(p qAm)= 0


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For particle moving in potential (, A), stationary form of DiracHamiltonian given by H=E where, restoring factors ofand c,

H=c (p qA) +mc2+q=

mc2 +q c (p qA)c (p qA) mc2 +q

To develop non-relativistic limit, consider bispinor T = (a,b),

where the elements obey coupled equations,

(mc2 +q)a+c (p qA)b=Eac (p qA)a (mc2 q)b=Eb

If we define energy shift over rest mass energy, W =E

mc2,

b= 1

2mc2 +W qc (p qA)a


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b= 1

2mc2 +W

q

c (p qA)a

In the non-relativistic limit, W mc2 and we can develop anexpansion in v/c. At leading order, b 12mc2 c (p qA)a.Substituted into first equation, obtain Pauli equationHNRa =Wa where, defining V =q,

HNR= 12m

[ (p qA)]2 +V.

Making use of Pauli matrix identity ij=ij+iijkk,

HNR= 1

2m(p qA)2 q

2m (A) +V

i.e. spin magnetic moment,

S= q

2m=g

q

2mS, with gyromagnetic ratio, g= 2.


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b= 1

2mc2

+W Vc (p

qA)a

Taking into account the leading order (in v/c) correction (withA= 0 for simplicity), we have

b 1

2mc2 1 W V2mc2 c paThen substituted into the second bispinor equation (and taking intoaccount correction from normalization) we find

H p2

2m +V p4

8m3c2 k.e.

+ 12m2c2 S(V) p spinorbit coupling

+

2

8m2c2 (2V) Darwin term

Synopsis: (mostly revision) Lectures 1-4ish

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Sy p ( y )

1 Foundations of quantum physics:

Historical background; wave mechanics to Schrodinger equation.2 Quantum mechanics in one dimension:

Unbound particles: potential step, barriers and tunneling; boundstates: rectangular well, -function well; Kronig-Penney model.

3 Operator methods:

Uncertainty principle; time evolution operator; Ehrenfests theorem;symmetries in quantum mechanics;Heisenberg representation;quantum harmonic oscillator; coherent states.

4 Quantum mechanics in more than one dimension:

Rigid rotor; angular momentum; raising and lowering operators;representations; central potential; atomic hydrogen.

non-examinable *in this course*.

Synopsis: Lectures 5-10

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y p

5 Charged particle in an electromagnetic field:

Classical and quantum mechanics of particle in a field; normal

Zeeman effect; gauge invariance and the Aharonov-Bohm effect;Landau levels, Quantum Hall effect.

6 Spin:

Stern-Gerlach experiment; spinors, spin operators and Paulimatrices; spin precession in a magnetic field; parametric resonance;

addition of angular momenta.

7 Time-independent perturbation theory:

Perturbation series; first and second order expansion; degenerateperturbation theory; Stark effect; nearly free electron model.

8

Variational and WKB method:Variational method: ground state energy and eigenfunctions;application to helium; Semiclassics and the WKB method.



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y p

9 Identical particles:

Particle indistinguishability and quantum statistics; space and spin

wavefunctions; consequences of particle statistics; ideal quantumgases; degeneracy pressure in neutron stars; Bose-Einsteincondensation in ultracold atomic gases.

10 Atomic structure:

Relativistic corrections spin-orbit coupling; Darwin term; Lamb

shift; hyperfine structure. Multi-electron atoms; Helium; Hartreeapproximation and beyond; Hunds rule; periodic table; LS and jjcoupling schemes; atomic spectra; Zeeman effect.

11 Molecular structure:

Born-Oppenheimer approximation; H+2 ion; H2 molecule; ionic and

covalent bonding; LCAO method; from molecules to solids;application of LCAO method to graphene;molecular spectra;rotation and vibrational transitions.



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12 Field theory: from phonons to photons:

From particles to fields: classical field theory of harmonic atomicchain; quantization of atomic chain; phonons; classical theory of theEM field; waveguide; quantization of the EM field and photons.

13 Time-dependent perturbation theory:

Rabi oscillations in two level systems; perturbation series; suddenapproximation; harmonic perturbations and Fermis Golden rule.

14 Radiative transitions:

Light-matter interaction; spontaneous emission; absorption andstimulated emission; Einsteins A and B coefficents; dipole

approximation; selection rules; lasers. non-examinable *in this course*.


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15 Scattering theoryElastic and inelastic scattering; method of particle waves; Bornseries expansion; Born approximation from Fermis Golden rule;scattering of identical particles.

16 Relativistic quantum mechanics:Klein-Gordon equation; Dirac equation; relativistic covariance andspin; free relativistic particles and the Klein paradox; antiparticles;coupling to EM field: minimal coupling and the connection tonon-relativistic quantum mechanics; field quantization.


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