ADVANCED QUANTUM MECHANICS _ (11 lectures) The Nonrelativistic case – the Schrodinger equation – 6 lectures

• Introduction to Scattering Theory Scattering by a potential The differential cross section Stationary states and the scattering amplitude Calculat ion of the cross section using probability currents

• Integral Scattering Equation Definition of the Green Function The Lipmann Schwinger equation Determination of the Green function ( r and k space) The Born Series Calculation of the Born approximation for a Yukawa potential

• The operator formulation of the Lippmann -Schwinger equation Introduction to the operator formalism The determination of the Green function (regularisation in the complex plane) The Born Series in operator notation

• Scattering by a cent ral potential : the method of partial waves Angular momentum stationary states Expansion of a plane wave in terms of free spherical waves Partial waves in a central potential The definition of a phase shift Expression of the cross section in terms of the phase shifts Unitarity and the optical theorem

Relativistic quantum field theory

Fundamental division of physicist’s world :

slow fast

large

small

Classical Newton

Classical relativity

Classical Quantum mechanics

Quantum Field theory

speed

Ac

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( amplitude )Si

QM e∝

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ADVANCED QUANTUM MECHANICS _ (11 lectures) The Nonrelativistic case – the Schrodinger equation – 6 lectures

• Introduction to Scattering Theory Scattering by a potential The differential cross section Stationary states and the scattering amplitude Calculat ion of the cross section using probability currents

• Integral Scattering Equation Definition of the Green Function The Lipmann Schwinger equation Determination of the Green function ( r and k space) The Born Series Calculation of the Born approximation for a Yukawa potential

• The operator formulation of the Lippmann -Schwinger equation Introduction to the operator formalism The determination of the Green function (regularisation in the complex plane) The Born Series in operator notation

• Scattering by a cent ral potential : the method of partial waves Angular momentum stationary states Expansion of a plane wave in terms of free spherical waves Partial waves in a central potential The definition of a phase shift Expression of the cross section in terms of the phase shifts Unitarity and the optical theorem

The relativistic case - The Klein Gordon equation – 2 lectures

• The construction of a relativistic wave equation Identification of probability density 4-vector formulation Historical perspective –the problem of the negative energy and probability states, the Pauli Weisskopf reinterpretation of the probability density and the Feynman Stuckelberg interpretation of the negative energy states

• The relativistic treatment of scattering The Lorentz invariant form of the electromagnetic potential The scattering am plitude and the current density The propagator of the Klein Gordon equation The determination of the scattering amplitude The relativistic case – The Dirac equation – 3 lectures

• The Dirac equation Dirac matrices and the relativistic generalisation of the Schrodinger equation Dirac’s derivation – the “square root” of the Schrodinger equation Hole theory interpretation Free particle solutions

• The non -relativistic correspondence The introduction of electromagnetism The coupled equations for the u pper and l ower spinor components Non-relativistic limit The gyromagnetic ratio

• Symmetries Angular momentum, spin and helicity Parity

http://www.physics.ox.ac.uk/users/ross

SCATTERING THEORY AND THE DIRAC EQUATION

• Modern Quantum Mechanics, J.J.Sakurai, Addison -Wesley. Contains an excellent of the scattering theory topics of the course. • Quantum Mechanics, C.Cohen -Tanoudgi, B. Diu and F. Laloe, John Wiley & Sons. Vol II A useful introduction to nonrelativistic scattering theory. • Quantum Mechanics, L. Schiff, McGraw -Hill. • Relativistic Q uantum Mechanics, I.J.R.Aitchison, Macmillan. An excellent introduction to the relativistic aspects of the course. • Relativistic Quantum Mechanics, Bjorken and Drell. McGraw -Hill

A “classic text”.

Fundamental experimental objects

1 2 1 2( ... )na a b b bσ → Cross section (Dimension L2=M-2)

dn = Fi

dσdΩ

(θ ,φ) dΩ (1.1)

=

Differential Cross section ≡dσdΩ

=1Fi

dndΩ

≡ σ θ,φ( )( )

Schrodinger equation : 2

2uV u Eu i

m t⎛ ⎞ ∂− ∇ + = =⎜ ⎟ ∂⎝ ⎠

K.E.+ P.E.

Nonrelativistic wave equation :

1 2( )V r r−

1r 2r

Potential scattering (Finite extent assumed here)

2 21 2 1 2 2 2 1 2

1 2

( , , )2 2 T

uV x x y y z z u E u im m t

⎛ ⎞ ∂− ∇ − ∇ + − − − = =⎜ ⎟ ∂⎝ ⎠

Interaction of two particles described by 1 2( )V r r−

2 22 2 , ,( )

2 2X x TV x y z u E uM µ

⎛ ⎞⇒ − ∇ − ∇ + =⎜ ⎟

⎝ ⎠

1 2

1 1 1m mµ

= + Reduced mass

( ) /( , , , , , ) ( , , ) ( , , ) CMi E E tu x y z X Y Z x y z U X Y Z eψ − +=

22

2 X CMU E UM

⇒ − ∇ =2

2 V( , , )2 x x y z Eψ ψ ψµ

− ∇ + =

1 2 1 1 2 2,... M =m xXx xx m x= − + 1 2M m m= + Schiff p89

−

2

2µ∇x

2ψ + V(x, y, z)ψ = Eψ

“Stationary states” – definite energy E i ∂ψ

∂t

−

2

2µ∇x

2ψ + V(x, y, z)ψ = Eψ ≡

2k 2

2µψ (1.2)

2 2 ( ) ( ) 0 (1.3)k U r rψ⎡ ⎤∇ + − =⎣ ⎦

“Stationary states” – definite energy E

Asymptotic form (V=0 region) of stationary scattering states : scattering amplitude

vk

diffractive (r) Aeikz + fk (θ ,φ)eikr

r(1.5)

Incident amplitude Scattering amplitude – want to find this given V

r→∞

2

2( ) ( )V r U rµ=

write E =

2k2

2µ

∂∂t

P(V∫ r ,t)d 3r =

i2µ

∇. ψ *∇ψ − ∇ψ *( )ψ( )V∫ d 3r

J

(r ) =−i2µ

ψ *∇ψ − ∇ψ *( )ψ( ) (1.6)

−

2

2µ∇x

2ψ + V(x, y, z)ψ = i ∂ψ∂t

(1.2)

Probability density *( , ) ( , ) ( , )P r t r t r tψ ψ=

Probability current density

"Probability current density"J

3 3( , ) . .t

( , ) . 0t

V V A

P r

P r t d r J

J

d r dA

t

J∂ = − ∇ = −

∂ + =

∂

∇∂

⇒

∫ ∫ ∫

Ji =kµ

using ψ Aeikz + fk (θ ,φ)eikr

rc. f .(1.5)

Jd =

kµ

1r 2 |fk (θ ,φ)|2

r→∞

( , ) (1.1)idn F dσ θ φ= Ω

i ikF Jµ

= =

2 2k. ( ) |f ( , )| (1.8)d d r

kdn J dS J r d dθ φµ

= = Ω = Ω

σ (θ ,φ) ≡

dndΩ

= |fk (θ ,φ)|2 (1.9)

(1.6) ( )( )* *( )2iJ r ψ ψ ψ ψµ

−= ∇ − ∇

Calculation of the cross section

Units 8

34 2

3.10 m/sec10 kg m /sec

c−

==

Natural Units Length : L Time : T

Energy : E or Mass : m

} Choose units such that :

2

1 /1 . ( . / )

c L TE T M L T

== ≡

1 unit left : choose 9 -101 ( 10 electron volts = 1.6 10 J)E GeV= =