States, operators and matrices

1
States, operators and matrices
Dirac notation
Starting with the most basic form of the
Schrödinger equation, and the wave function (?)
Note to extract the expectation value of a
property, one needs to sum over the expanded
state
The state of a quantum mechanical system is ?.
This state can be expanded to a vector and the
system can be more complicated (e.g. with spin,
etc.)
Combining operators
Operator T, transforms the original state.
Here, a hermitian matrix M was introduced, to
extract properties from the wave function
(hermitian matrices have real eigenvalues).
Hence, to get the expectation value
2
Adding spin (1)

• Projection is known (m quantum number)
• Length of the two spins is known (j1 and j2)
• Several possibilities to construct projection by
  adding the two spins

Example m jmax - 2
j1 - 2
j2
lt j1, j1 - 2 j2, j2 gt
lt j1, m1 j2, m2 gt
jmax - 2
m
3
Adding spin (1)

• Projection is known (m quantum number)
• Length of the two spins is known (j1 and j2)
• Several possibilities to construct projection by
  adding the two spins

Example m jmax - 2
j1 - 1
j2-1
lt j1, j1 - 1 j2, j2 - 1 gt
lt j1, m1 j2, m2 gt
jmax - 2
m
4
Adding spin (1)

• Projection is known (m quantum number)
• Length of the two spins is known (j1 and j2)
• Several possibilities to construct projection by
  adding the two spins

Example m jmax - 2
j1
j2- 2
ltj1, j1 j2, j2 - 2 gt
lt j1, m1 j2, m2 gt
jmax - 2
m
5
Adding spin (2)

• Projection is known (m quantum number)
• Length of the maximum total spin known
• jmax j1 j2
• Several possibilities to construct projection
  from different sizes of total spin j

j1
j2
j1 j2 j j1 j2
j
Note j1, j2 and j can be interchanged. However,
changing the composite state with one of the
constituent states is not trivial and requires
re-weighting of the constituent states
j j1 j2
j j1 j2 -1
j j1 j2 -2

lt j1 j2, j1 j2 - 2 gt
lt j, m gt
jmax - 2
lt j1 j2 - 1, j1 j2 - 2 gt
m j
lt j, m gt
m
lt j, m gt
lt j1 j2 - 2, j1 j2 - 2 gt
6
Adding spin (3)
mjmax m1j1 m2j2 jj1j2 mjmax
mjmax-1 m1j1 m2j2-1 m1j1-1 m2j2 jj1j2 mjmax-1 jj1j2-1 mjmax-1
mjmax-2 m1j1 m2j2-2 m1j1-1 m2j2-1 m1j1-2 m2j2 jj1j2 mjmax-2 jj1j2-1 mjmax-2 jj1j2-2 mjmax-2

• Note
• - j1 ? m1 ? j1 , thus m1 has (2 . j11) possible
  values and m2 (2 . j2 1).
• Each combination shows up exactly ones in the
  second column of the table
• so the total number of states is (2 . j11) (2 .
  j2 1).
• The third column has the same amount of states as
  the second column.
• The quantum number j is a vector addition, thus
  it will never be lower than j1 j2, which is
  called jmin.
• For m lt jmin the amount of states is jmax jmin
  1 (j1j2) j1 j2 1 for each m.
• This situation occurs for - j1 j2 ? m ? j1
  j2 , thus (2 j1 j2 1) times.
• Number of states (2 j1 j2 1) .
  (j1j2)-j1 j2 1
• For m ? -jmin and m ? jmin the amount of states
  is jmax ?? m 1. This results in the following
  sum
• Number of states

7
Adding spin Clebsch Gordan
Example m jmax - 2
e.g.
From symmetry relations and ortho-normality, the
C coefficients can be calculated. The first few
½ x ½ 1, 1 1, 0 0, 0 1, -1
½, ½ 1 0 0 0
½, -½ 0 ½ ½ 0
-½, ½ 0 ½ ½ 0
-½, -½ 0 0 0 1
JM
m1,m2
8
Combining spin and boost

• Lorentz transformations
• For
• Jackson (section 11.7) calculates the
  corresponding operator
• Extracting rotations
• Allowing definition of the canonical state

9
Intermezzo rotation properties

• The total spin commutes with rotation
• However, the projection is affected with a phase.
• Consider the rotation around the quantization
  axis
• Euler rotations, convention z y z
• Advantage quantization axis used twice for
  rotation
• Rotations are unitary operators. The rotation
  around y includes a transformation of the
  previous rotation.
• This results in

i.e. Rotations can all be carried out in same
coordinate system when order is inverted.
10
Intermezzo the rotation matrix

• Summary of the result from the previous page
• Euler z y z convention makes left and right term
  easy
• The expression for djmm is complicated, but is
  used mainly for the deduction of symmetry
  relations

Note 2 Since rotations are hermitic, the
conjugate matrix is Djmm(-?,-?,-?)Djmm(???)
Note 1 Inverse rotation is accomplished by
performing the rotations through negative angles
in opposite order ( Djmm(?, ?, ?) )-1
Djmm(-?,-?,-?)
Note 3 Combination of all this gives
Djmm(???)(-)m-mDj-m,-m(???)
11
Rotations, boosts and spin
Describing the rotation of a canonical state
Remember
i.e. rotation of a canonical state rotates the
boost and affects the spin state in the same way
as it would in the particle at rest state.
12
Helicity
The helicity state is defined with
Note
Compared to the same operations on the spin state
Hence
Note that the helicity state does not change with
rotations
In other words helicity (?) is the spin
component (m) along the direction of the momentum.
Note that the helicity state does not change with
boosts (as long as the direction is not reversed)
13
Discrete symmetries
Parity
Charge conjugation
Dirac
particle
Conjugatingtransposing gives
Commutation relations
C is the matrix doing the transformation
Mirror analogy
Resulting in
Left-handed
Left-handed
antiparticle
e.g.
14
Time reversal
e.g.
Note that time reversal changes t in t and input
states in output states (in other words lt bra
to ket gt ).
Another way to show this
i.e. the transformed state does not obey the
description of motion of the Hamiltonian, it
needs an extra sign.
The solution is to make time reversal
anti-unitary
Note this can also be shown with the commutation
relation
15
Time reversal continued
Next, the the time reversal operator is split in
a unitary part and a complex conjugation. The m
states consist of real numbers, i.e. projections.
Hence a time reversed expectation value can be
described with
Calculating the expectation value of operator Â
And combining with the time reversal operator
Since a second time reversal should restore the
original equation
Hermitic
16
Time reversal and spin
From the commutation relation
Consistent with
And
The equation
holds if
17
Time reversal and spin continued
Time reversal of a canonical state
Time reversal of a helicity state
Hence
Should give
18
Parity and spin
The parity operation on a canonical state
The parity operation on a helicity state
Hence
Should give
but
Note
Since helicity states include rotational
properties
19
Composite states
Bs ground state
b-quark
s-quark
Bs-meson
J/?-meson
(easy to detect)
This ground state can decay to two vector mesons
Isospin 0 Spin 1 Parity - (C-parity -)
c-quark
c-quark
W
Bs-meson
?-meson
Isospin 0 Spin 1 Parity - (C-parity -)
b-quark
s-quark
Isospin 0 Spin 0 Parity -
s-quark
s-quark
20
Two body decay properties
Spin states of Bs, J/? and ?
Not the complete story Consider momentum in two
particle decay (Bs rest frame)

Normalization
Hence
Remember
Still no complete story Consider angular
momentum
With
Total spin
Angular momentum
Missing Formalism that describes angular
momentum and Yml states.
21
Rotation with angular momentum
Split up the Yml state, to match with new
rotation
i.e. the transformation is a product of the
rotation of two rest states
22
Angular momentum and spin
We can also express them as states with a sum of
angular momentum and spin
Note MJ m MS
Note MS m1 m2
Since
(see 1 page back)
And
The state can be rotated with
Note that (as expected) the sum of angular
momentum and spin (J) is not affected by the
rotation, neither are the angular momentum (l)
and the total spin (S). This result is the
equivalent of the non-relativistic L-S coupling.
23
Two body decay and helicity (1)
As with spin, we need to consider momentum in the
helicity state
Angles are zero. Particles are boosted back to
back along the positive and negative Z-axis
The link with previous page is provided via the
relation between canonical and helicity states
Single state normalization. Rest mass
w Momentum p
MS m1 m2
? ?1 - ?2
Why?
24
Two body decay and helicity (2)
Expressing states with the sum of angular
momentum and spin in helicity states
Intermezzo, check
NJ
Normalization (later, easier the other way around)
25
Two body decay and helicity (3)
Check the transformation properties for rotations
Transforms as it should Remember
i.e. Mj transforms, but ?1 and ?2 do not.
26
Canonical versus helicity states
(2 pages back)
(3 pages back)
Combine
(4 pages back)
Normalization
27
Decay amplitudes
Remember
The transition amplitude to a helicity state is
calculated with the matrix element
Momentum of decay products p Resonance rest
mass w
And
Complete set
With
Check
Hence
Helicity amplitude
28
Helicity amplitude
Remember
Switch to canonical states
And
Complete set
Partial wave amplitude als
Canonical states
29
Bs0 ? J/? ?, tree level
J/?
s
Vcs
W-
Bs0
W
W
?
s
c
b
Vcb
Oscillation
Decay
30
The basics
Schrödinger
No mixing, just 2 states
Particle and anti-particle
Eigen states of the Hamiltonian
en
Time evolution of eigen states
31
The basics
Mixing
Hermitic
Note to obtain properties with real values, the
matrix needs to be Hermitic.
32
The basics
The matrix equation is 0 if
Eigen-states
33
The basics
Eigen-states
Note The Hamiltonian describes the quantum
mechanical system. Only for the eigen-states of
the Hamiltonian Mass and Decay time have meaning
Note 1 No particle and anti-particle. Two
different masses and decay times
Note 2 What is the meaning of time (t) at this
point. This is a composite system, one particle
contains two states. The time is calculated in
the rest frame of the particle On the next slide
it becomes obvious.
34
The basics
Note So time (t) is just the decay time of the
measured particle.