System Measurement and Preparation
Transcript of System Measurement and Preparation
System Measurement and Preparation
Quantum Postulates (Claims)
The quantum probability density
probability flow
Uncertainty relation
System-detector interaction
Copenhagen interpretation of QM
Paradoxes, entanglement
Alternative interpretations
Case study: Exploring the electron spin
Stern-Gerlach polarimeters,
Electron spinor states
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Quantum Postulates for Microscopic Systems
Copenhagen (orthodox) interpretation of QMSystem: Ensemble of microscopic particles and force (el-weak, nuclear) fields.
Relativistic or classical, specific: QED, QCD (not gravitation yet!)
I. A system ensemble is completely (!?) described by a smooth wave function y ({qi},t) . Independent coordinates qi:
II. Any observable A corresponds to quantal operator which extracts the
associated information from the wave function (representing vector in H).
III. A system can only be observed in an eigen-state of the corresponding indication operator Â, → Âya = a·ya. Orthonormal set = basis
IV. A series of ensemble measurements of observables A, B →
probability distribution P(A), mean and variance are given by
V. System wave functions evolve in time according to the t-dependent Schrödinger Equation,
( ) ( )
22 2
1 2ˆ ... ( , ) ( , )
ˆˆ ˆˆ ˆ ˆ, : , 0
ˆ ˆ, 0.
ˆ
2
ˆ ˆ
,
a
A
i
B
iA dq dq q t q t and
Incompatible observables A B AB BA A B
ComHeisenbe lc patibt
A A
a
A
i en At Brg Un er y Relation
y y
y y
= = −
→
=
→
y y = ˆi t H
(( q )) qi ii iy =
†A A=
ˆ ˆ, 0If A H A const = → =
ay
Quantum Probability Density
Non-relativistic differential equation for proper wave function (generally t-dependent Hamiltonian):
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( ) ( ) ( )
( ) ( ) ( )
( ) ( )
22 2 2
ˆ ˆ( , ) ( ) 0 ;
( , ) ( ,0) ( ,0) .
:
ˆFor stationary solution ( , ) ( , ), ;
( , ) ( ,0
(
con
)
st
, ) ( , 0)
y y y
y y
y y
y y
−
−
−
→ → = → =
→
=
=
= = =
=
=
=
i S
i S t
x t
i E t
x
For V x t V x H t H t S real
probability density t x t e x
l
x t e
x
Examp e
H t x t t
x
x t e x tE x
22with ( , )( , ) ( , ) ( , ) ; 1 y y
+
−
= = = x t ddP x t x t dx x t xdx
Max Born(188-1970)
( )2 2
2ˆ( , ) ( , ) ( , ) ( , )
2y y y
− = = +
i x t H t x t V x t x tt m x
Interpretation of wave function y by Max Born →
Probability to find (measure) system (particles) at time t in volume element dx. → No statement about reality of y !
Inherent statistics of quantum theory: Probability defined by ensemble average over many similarly prepared systems. → No prediction for any single particle!
Structure of set of all such square-integrable wave functions → L2 vector space
Probability Flow
Consider arbitrary finite domains, where wfs can be normalized.
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( ) ( )
( )*
2 2
2
*
2
**
*
*
ˆ ˆ
)2
,
)
2
( , y y y y
y y
y
y y y y
+ +
− −
+
−
+ +
− −
= − =
− = − =
= − + = −
L L
L L
L
L
L L
L L
x
H H
x x
jx x x x
x ti
dx dxt
idx
m
idx x t d
m
22; w h( , ( i ,) , ( ) 1) t yy
+
− = = L
L
x t dx x t d t xx x d
** * *( , ) y y y y y y y y
+ + +
− − −
= + = −
L L L
L L L
x xi
dx d dxt t t t
t i it
( )( , ) , 0
+ =
Continuity Equ n
x t j xx
a i
tt
t o
* *
2y y y y
−
=
Probability curr t
x xm
en
ij
ሻ𝜌(𝑥, 𝑡inj outj
Probability is Conserved
→ →Use Schrödinger Equation to transpose t x
Microscopic System Ensembles
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( ),P x y
xy
( / ):
, ,...
large ensemble of independent objects particleM s quantoni s
All degrees
t
of
c
fr
s
e
i
e
cros op c ys em
dom x y
= =
( ), ,...; nProbability densit d fu oe rx oy bp sa et rvy m lit atio
( ) ( )
( )
( )
2.
,y,.
,...; ,.. ;
. ( ): ,y,...
..:
;,
; 1
P
t
robability density dP x x dV
Partial prob d
Normalized
d
x
t t
x t
d dy dP x
P dy P x
t
=
=
2
2
A 0ˆ ( , ) ; ,ˆ ( , ) ( , )
( , ) ( , ) 1: , ) (
Aa
a a a aa a
a aOperator A has Eigen Functions x t with
x t form
A x t a x t
orts x t c xhonormal basis t
a
t wi h c
y =
=
=
=
=
( )
*
0
22
*
0
ˆ
ˆ ˆ ...
)
,
( , ,...; ...) ( , ,...; ...)
( , ,...; ...) ( , ,...ˆ ...
ˆ
; .
ˆ
..)
(A
Any observable operator projects a info out of
Mean x y x y dx
x y x y dx
Variance
a
i
A
A
define a Gaussian probab lity P a
a A dy
A dy
A A
y y
y y
y
= = =
=
= −
Any measurement of observable A returns one possible value a, with probability P(a)=|ca|2
“Collapse” of wave function (stays in a) → Measurement in QM (?, still debated).
“Fourier” expansion
have quantum-statistical uncertainty
and
Math Example: Prob. Moments from Generating Functions
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( )
( ) ( )
( ) ( )
−
=
=
(
:
)
: s x
ik x
s dx e P x Laplace transformation of
Character
P
k dx e
istic functio
P x Fou
ns of P
rier t
x same info o
ransformat
n system
ion of P
x=x 0
( ) ( ) ( ) ( )2*
1 ,
1 . . .
: 1 : particExample
spatia
simple dimensional system
dP x dxl d
lem
o f x
a
x x
s m
x
s−
= = →
( ) ( ) ( ) ( )= =
− = = → = − 0 0
: 1n
nn
ns s
d ds dx x P x x similar x s
ds ds
( ) ( ) ( ) ( )
( ) ( )
*
But must be Her
: ( )
1( )
: ( , ) mitian
ik xIf P x x x dk e P k known Fourier transform
P x k P k k mean wave number momentumi x
Quantal operators work on amplitudes x ty
= =
= → =
→
(Set x=0)
( ) ( ) ( ) ( )
( ) ( )
− − = = −
= →
0 0
:( :
: [ ]
)n n
ns x n s x
n n
nn n
n
d ds dx e P x dx x e P x
ds ds
ds Laplace transform of
Derivatives of s
x P x xds
(Set s=0)
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Precision Limits: Uncertainty Relation (Incomp. Observables)
( )2
2 3 22 2
p xxn = −
Heisenberg Uncertainty
Relation
Heisenberg, 1924
a=10 fm a=30 fm
Probability distributions for position x and momentum px are anti-correlated, minimum widths. → {x},{px} = conjugate spaces, like n and t in classical Fourier analysis
Task: measure position of (catch) particle in an ideal 1D box and measure its speed (momentum px=conjugate to x).
position x
momentum px
pro
bability d
ensity
( ) cos sinn an bn
Particle in a Box wave functions
x xx c n c n
a ay
−
= +
:
ˆ ˆ: , 0
( '
, 0
)
ˆx
General feature for observable A and conju
No Cloning
u
a
Theor
Incomp tib xle observables
Role of comm tato
e
r
i
ga
Can t mak system cop es
te
e
p
B
m
i
B
A
→
=
= −
→
→
−
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Classical Wave UR: Fourier Transforms
T0 2=
Fourier Transforms
Uncertainty of time vs. frequency : t ~ 1/f → t · f > 0
time → time →
frequency → frequency →
inte
nsity →
inte
nsity →
inte
nsity →
inte
nsity →
System Measurement and Preparation
Quantum Postulates (Claims)
The quantum probability density
probability flow
Uncertainty relation
System-detector interaction
Copenhagen interpretation of QM
Paradoxes, entanglement
Alternative interpretations
Case study: Exploring the electron spin
Stern-Gerlach polarimeters,
Electron spinor states
W. Udo Schröder, 2019
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Expectation: Measurement/Preparation (CI)
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Postulate: qm system can only be observed in an eigen-function (state) of the corresponding indication operator  → Âya = a·ya. Other states are not realized ! Orthonormal set {ya} forms “basis.”
QM makes no specific predictions for any single measurement of observable A, except: In any measurement, system will be found in one of the possible eigen states of  with probability amplitude existing at time of measurement.
→Many independent measurements are distributed “statistically.”
BUT: Immediate repeat measurement of A on the same system give identical results, not a distribution!
→Wave function “collapses” (is suddenly reduced) to one component and frozen in that state.
→By measurement of A and selection of eigen value a (→state ya), a system can be “prepared” in that state ya. Repeat measurements yield same a.
Discontinuous change in t-dependent behavior of wf is not described by Schrödinger Equation.
Ideal Expectation: System-Apparatus Int.
Debate from the start of QM: Measure dynamical variable W (observable) of system S with an experimental apparatus A.
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S
A
Info in state vector/wave function S
Projected with some W sensitivity projector P
Argument assumes discrete eigen spectra
) )
ˆ ˆSystem eigen states of : ;n ;n
ˆ ˆApparatus : eigen states of A : A ;m ;m
indicator variable(e.g. position of needle)
m,n auxiliary qu.#s
W W =
=
=
)
)
; ;
( ) ; ;
S A states LinComb of n m
Assume pure initialsimple n m
=
( ) ( ) ( )
( ) ) )
,..., ,...
, ,
ˆ ˆ ˆ:
" "
,,ˆ ; ; ; ;
,,n m
Time t after measurement unitary operator t p t t
entanglement because interaction S with A mixes states
nnt n m p n m
mm
P = P P
P = →
correlation → , to make A proper indicator
Measurement : System-Apparatus
Transition amplitude p. Coherent superposition of A states, emphasis on eigen value (by construction of apparatus).
Macroscopic state of system at time t, after measurement:
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( ) ( ) ( )( ),
ˆ ; , , ;t
n
S t n p n n n
= P = →
S
A
) ( ) )
), ,
ˆ ;
,,;
,,
t
n m
t m
nnp m
m
A
m
= P
= →
Macroscopic state of measurement apparatus at time t, after measurement:
Macroscopic state of system: different from initial state, no longer Weigenstate, but a coherent superposition → subsequent measurements of the same observable on the same event would produce different outcomes →
Not consistent with postulated “collapse of the wave function” attributed to measurement process. → future t solutions??
Density matrix approach more appropriate?QM prep and measurement
System Measurement and Preparation
Quantum Postulates (Claims)
The quantum probability density
probability flow
Uncertainty relation
System-detector interaction
Copenhagen interpretation of QM
Paradoxes, entanglement
Alternative interpretations
Case study: Exploring the electron spin
Stern-Gerlach polarimeters,
Electron spinor states
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Quantum Measurement
W. U
do
Sch
röder,
20
1914
QM States: Philosophical Access
Positivism: Nature revealed through perception, useless to search for
reality that cannot be sensed empirically (e.g., absolute space, time,..), Scientific theory: understand relationships via logical deduction verifiable through experiment, (Ernst Mach, Schlick, and “Vienna Circle”)
Realism: There is an observer-independent reality (unprovable !),
(Schrödinger), local reality.
Objectivity: Observations are independent of observer, verifiable
by others in the same fashion; e.g., Einstein’s special relativity shows that there are no absolute reference frames.
Locality: Physical processes occurring at one place should have no
immediate (faster than light) effect on the elements of reality at another (distant) location (no “spooky action” at a distance).
CI → Paradox Schrödinger’s Cat
Cat in a box connected to a poisoned gas volume. Gas is released upon decay of radioactive nucleus and kills cat.
While the box is closed: It is not known if decay has occurred → nucleus is in superposition of original and decayed state →Cat is both dead and alive at the same time → contradicts classical science/experience
Electronic release triggered by rad. (statistical) decay
Cat in a superposition of basis states
1 2 1 2: , , ; ( ,, ) nucl cat u dG aener ia al ve del w df y y y y = + = =
11 2
2
, , ,atom cat u d
entangled state
either
or
y y y
y
= + →
Measurement (observation):
Reduction (“Collapse”) of wave function. → How and when?
After: Different system, different Schrödinger Equ.
Does the state exist as objective reality?cat
Poison gas
Cat hidden in box
Quantum Measurement
W. U
do
Sch
röder,
20
1916
CI → EPR Paradox: Decay Experiment Violates HUR
Radioactive Decay
Nucleus → collinear correlated products A & B.
Q=xA-xB and P=pA+pB→ HUR [Q,P] ≠ 0 !
1) Measure pA -pB of A with DpA=0 (2 slits)2) Measure xB -xA of B with DxB=0 (1 slit)→ Q(t=0), P(t=0) with DQ=DP=0 Contradictionunless 1) causes immediately DxB→ ∞
→Requires instantaneous (spooky) action at a distance (xA →∞) between 2 local realities, forbidden by Special Relativity.….OR
→ Quantum Mechanics is not a complete theory → Hidden variables to make a system
an objective reality (deBroglie/Bohm). ….OR
→ Microscopic systems are non-local. ….OR…?
→No evidence (yet) for hidden local variables.
QM: Entangled state, incommensurable relative position and relativemomentum → − − 2p p x xA B A B
Einstein, Podolski, Rosen: PR 47, 777(1935)
A B
pBpA
xxBxA
P=(pA+pB)=0
Spin-Spin Correlations Test Local Realism
Experimental setup: Two polarimeters (I,II) in orientations a and b, perform dichotomic measurements of linear polarization of photons n1 and n2. Polarimeters can rotate independently about the axis of the incident photon beam. Electronics counts singles and the coincidences. (after A. Aspect et al., PRL 49, 91 (1982))
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a b
J=0
J=0
J=1
40Ca atomic E1-cascade
n1
n2
Glass cube polarimeters split beams into 2 each, linearly polarized components →
detected simultaneously with Det 1-Det 4
( )I a ( )II bDet1 Det2
Det3
Det4
40Ca atomic E1-cascade emits two (spin-1) photons with anticorrelated polarizations, coupling to J=0.
Spin-Spin Correlations
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( ) ( ),
.
Photons emitted in opposite directions measure spin S correlations with
Oriented polarimeters I a and II b
y y y y y y= =
=
= = + = −
=
12 12 12 12 12 12
2 1,.
" " 1) :
..,
ˆ ˆ1 1abn n
ba
correlated spin orientatio
Expected for local reality for parallel
ns in each of n N events
S S S Sa
orientation a b
b
a
a
b
b
ab
ab
Co-planar Detector setup
( )( )
= →
−1
, : n nn
Spin spin correlation C a b a bN
( ) ( )
−
→
=− →
2) , :
, co, s
1
,, . .
n n
Arbitrary orientation a b
Speci
anti correlation in each event
QM predictions
depend on a b g
fic
C a beometry
a
e ag
b
b
“Entangled” wave functions
( ) ( )y y+ − − + + − − += + = +12 12
1 1/
2 2a a a a and or b b b bQM:
bosons
|S=0,Sz=0>= (qu#1 qu#2) (qu#1 qu#2)
I II
Spin-Spin Correlations Reject Local Realism
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( ) ( )
= = + + − →
→ = =
→ = → →
1: ;
1
2 2
n n n n n n n n n nn
n n n n n n n n n n
n
for spin orientations
C g g a b a b a b a bN
all a b and a b a b a b a b
Assuming local reali
g C
ty
( ) ( ) ( ) ( ) + + − = , , , , 2C a b C a b C a b C a b C
( )!!
cos c: os: abSpecific setup QM C =→ += − 1 2 2
QM Prediction
Expt. vs. Prediction
A. Aspect et al., PRL 49, 91 (1982)
( ) ( ) ( ) ( )
3) , , , :
, , , , , , , ?" "
Consider a a b b
Do genera realisti
pairs of polarimeter or
l constraints exist on C a b C a b C a b
ientatio
a
n
C
s
bc
QM spin formalism agrees with many experiments but violates Bell-type inequalities (constraints).→ Initial spin correlations are not defined in reality
(→ not conserved).
→ Spin correlations arise at time of distant (non-local) measurements, according to QM rules (collapse of entangled wf).
Recent EPR Experiments
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( ) =
→
>
22 2
2
(0.12 0.01)
( .
2
2:
)
r pB rB
HUR for uncorr particles should be
cf paper
Lixiang Chen et al., Ph. Rev. Letters 123, 060403 (2019)
Produce correlated photons by simultaneous parametric down conversion of light (example: light splitting @ Calcite).
→ Entangled (correlated) photons
Measure correlations & uncertainties in radial position and radial momentum.
Quantum Measurement
W. U
do
Sch
röder,
20
192
2
Possibility (?): Many Worlds Scenario
Conjecture: All possible qm outcomes for measurement of single system are realized (somewhere).
Multiverse postulate: System evolves always smoothly, no collapse. At t of measurement, system is in one of the many possible eigen states of  in our Universe.
Other eigen states will appear in other universes →Multi-Universes (Multiverses). → Many paradoxes (e.g.,
time travel, causality)! Scientific viability?
Reduction of to one of its components of Universeat t0 = time of measurement Different
Universes
( ) ( )
( )
( )
( ) ( )
( )
1
2
00 0
3
, ..........
, .........., ( ) ,
, , ....
, ..........
n
n
n nn n
n
x t
x tx t t c t x t
x t t x t
x t
→
→ = →
= →
→...
Our Universe
System Measurement and Preparation
Quantum Postulates (Claims)
The quantum probability density
probability flow
Uncertainty relation
System-detector interaction
Copenhagen interpretation of QM
Paradoxes, entanglement
Alternative interpretations
Case study: Exploring the electron spin
Stern-Gerlach polarimeters,
Electron spinor states
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Peculiarities of QM Measurement
Heisenberg-type uncertainty has classical analog for waves and certain pairs of observables (frequency and time, wavelength and trajectory,…). But not for massive particles!
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Demonstrate quantum properties with Stern-Gerlach experiments on spin-½ electron. To what extent can a quantum state be controlled (“prepared”) for further use, e.g., fixate spin-up or spin-down (qubits in quantum computer) ?
S
BNon-classical (?) particle phenomenon: Spin- ½ electron (etc.) magnetic moment, spatial orientation (polarization)? (→ Compare polarized elm. waves, cf. later).
Spin: Associated with mechanical angular momentum (Einstein-de Haas experiment on magnetized Fe bar.)
Follow e.g. Townsend’s Ch. 4, etc.
Spin and Magnetic Moment of Ag Atom
Stern and Gerlach (1922): Splitting of Ag beam in inhomogeneous B. B field exerts force on e- magnetic dipole
Ag atom: outer unpaired s1/2 electron.
Experiment: Only 2 orientations of e- spin/dipole → 2S+1=2
→ S = 1/2.
= − = = +
( )z zz s s B s
B BEF g m
z z zOtto Stern 1888-1969
Walther Gerlach1889-1979
Prof. Gerlach’s postcard with the announcement to Bohr in Copenhagen.
Bx
y
z = 410zB z T m
Stern-Gerlach Spin Polarimeter
Magnetic field B-spin s=1/2 interaction → Hamiltonian
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int
0 0
ˆ 0
0z
outside magnet t
H c z inside t T
outside magnet T t
= −
=
= − = − = − int
ˆ /2
z z s z z
z z zS Pauli spin operator matr
H B S B
ix
B g
0
:
(
/
()
0
: )z x
Inhomogeneous magnetic field B Maxwell Gauss
Example and B x x BB z B z B
=
== + −
+ −
−
+ −
= + =
= + → =
=
= =0 0
1( , , , ) ; ,
2
1( , , , ) 2 0
2
0 0y
y y
ip y
p
i i i ic z T p y c z T p
p z
x
y
zx y z r p z z with r p e with
x y z e e z e e z component
t
T s pt
p p
Block path of one trajectory by absorbing “slit” → S-G Spin PolarimeterRotate ( /2) to deflect in x direction.→ Preparation of spin state:
+
pr zp
Stern-Gerlach Thought Experiments
Classical: z longitudinal (spin) polarization → no component perpendicular to z(e.g., Townsend, Modern Approach to QM, Ch. 1)
Beam of N0 Ag atoms: S-G setup filters (projects) Ag with in positive magnetic-field direction → Spin Polarimeter
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( ) ( )y y
+ −
+ −= + → =
2
0
: 2
1 2 1
Initial Ag beam is unpolarized states z and z
z z initial intensity N
:
1
0
Orthonormalized basis
z z z z and
z z z z
+ + − −
+ − − +
= =
= =
1st S-G filter polarizes in +z direction (inhibiting z-). 2nd S-G tests polarization in x direction.
AgN0
z+ N0/2 x+ N0/4
x- N0/4ˆ
zP z z
+ + += ˆ
xP x x
+ + +=
: ˆz
Projector on z P z z+ + +
=
1 2
Classically unexpected results of measurement: All N0/2 events entering S-G 2 produce a result at the exit of S-G 2. 50% of them show spin in +x direction, 50% in –x.On average no net spin polarization in x direction (undetermined):
02 2
xMean values x x S x x
+ − + += → = − =
Result of many Sz-measurements
Stern-Gerlach Thought Experiments
Classical: z longitudinal (spin) polarization → no perpendicular component (e.g., Townsend, Modern Approach to QM, Ch. 1)
Beam of N0 Ag atoms: S-G setup filters (projects) Ag with in positive magnetic-field direction → Spin Polarimeter
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( ) ( )y y
+ −
+ −= + → =
2
0
: 2
1 2 1
Initial Ag beam is unpolarized states z and z
z z initial intensity N
:
1
0
Orthonormalized basis
z z z z and
z z z z
+ + − −
+ − − +
= =
= =
AgN0
z+ N0/2 x+ N0/4
x- N0/4ˆ
zP z z
+ + += ˆ
xP x x
+ + +=
( )
20 0
2
ˆ ˆ ˆ ˆ2 4
11 ( )
2
x z x z
i
N NP P x x z with P P
z x with Phase factor e
y y
+ + + + + + +
+ +
= =
→ = + =
+ + +=: ˆ
xPProjector o xx xn
1 2
Repeat experiment: blocking the spin-up component→ same (random) result →once z (Sz) is measured, x (Sx) is randomized. Incompatible measurements
( ) ( ) + + − −
=→ = = ==+ +2 2
,1
11
2 2phase factx z oz sz x rx
1st S-G filter polarizes in +z direction (inhibiting z-). 2nd S-G tests polarization in x direction.→ No net, but equal
Stern-Gerlach (Thought) Experiments
Classical: z longitudinal (spin) polarization → no perpendicular component (e.g., Townsend, Modern Approach to QM, Ch. 1)
Beam of N0 Ag atoms: S-G setup filters (projects) Ag with in positive magnetic-field direction → Spin Polarimeter
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Results of measurement after 2 filters: again equal (random) #s with up or down. Appropriate reduction in total intensity of transmitted component.
( )
y y
+ + ++ + ++ + + + + + + +
+ −+ +
= = =
=
→ → = + = =
220 0 0
22 2
ˆ ˆ ˆ ˆ2 2
ˆ
1... 11 2 ( )
2
ˆz zx x zz
N N NP z x z z with P P P
etc x z z with
P z z
Ph
P x
x z ase
x
: ˆz
Projector on z P z z+ + +
=
AgN0
z+ N0/2 x+ N0/4 z+ N0/8
z- N0/8
ˆz
P z z+ + +
=ˆz
P z z+ + +
= ˆx
P x x+ + +
=
1 2 3 ≠0!! Random results , also with opposite z,xfilters
≠ 0: Strange basis? Doesn’t behave like vector in 3D space.
Stern-Gerlach (Thought) Experiments
Classical: z longitudinal (spin) polarization → no perpendicular component (e.g., Townsend, Modern Approach to QM, Ch. 1)
Beam of N0 Ag atoms: S-G setup filters (projects) Ag with in positive magnetic-field direction → Spin Polarimeter
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: ˆz
Projector on z P z z+ + +
=
Arbitrary magnet settings 1, 2,…N; splitting and recombining trajectories for different spin orientations. only last (S-G) performs measurement w/r z direction.No events get lost, mean polarization in any direction is zero.
Equal #s spin-up and spin-down, total # N0 conserved.
z+ N0/2
z- N0/2
Magnet 1 Magnet 2S-G
z axisSource Ag
N0
( ) ( ) ( )y
+ − + − + −
+ +
−
= + + +
= = →
→
2 2
, :
" " 1 2;
(
1
)
; 2 ,
?
z x y
i
Found in S G experiments that
z z x x y y
Up to overall phase factors relative phase factors
Strange overlaps x
What can be learned ab
z y x
out properties of s
no
pin states ket
x yt p rp
s
ze
Conclusions From S-G Experiments
Found in S-G experiments: Any (2D-binary ) polarization amplitude
can be expressed in terms of LC of perpendicular components.
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( ) ( )( )
( ) ( )( ) ( )( )
+
− ++ −
+ +
− + − ++ −
−
+ + − + −
−− − −
+ + − + − + + −
= + = +
= + = + = +
1
2 2
1:
2 2 2
iii i
i ii ii i
ex e z e z z e z
e ey e z e z z e z Hence y z e z
( ) ( )
( ) ( )( )
( )( )
+ ++ +
+ + − + − +
−−D − D− D D
+ + + − + −
= → D = − D = −
= + + = +
2
1 2 ( : )
122 2
ii iii i
y x abbreviate and
e e ey x z e z z e z e
( ) ( )2
1
2
1dx zaz i znz y
+ −− += =
→ Must have space of vectors with complex components !
( )( ) ( )( ) ( )( )
( )
D − D − D − D
+ +
D = →
= + + = + D − D
→ D − D = → D − D =
=
=
→
D
2 1 1 11 1 2 2cos
2 4 4
cos 0 2. ( . .
1 2
) 0 2
i i
Choose
y ex e
e g
Constructing the Spin Space
Electron spin is an angular-momentum like observable with exactly 2 components w/r to any direction → one Hermitian operator with a
component for each spatial dimension i = (x, y, z). Each has 2 eigen states/kets and eigen values /2.
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2
" ,
ˆ: ,
"
1 1ˆ
1
22 2
:
0
0 1
z z z
Consider z as quantization axis spin operator and eigen kets
S z z and S z z
Choose these kets as members of ortho normalized basis
Represent by column ve t
z S S
c ors z and z
+ + − −
+ −
→ = = −
−
= =
( ) ( )1 2 1 21 1
2 21 2 2
Represent S G spinors in basis kets
x z z and y z i zi
+ − + −
−
= =
2 2 2
ˆ ˆ ˆ: , ,
ˆ ˆ ˆ ˆ ˆ ˆ, 0; , 0; , 0
ˆ ˆ ˆ ˆ ˆ ˆ, , 0; , 0; , 0
x y z
x y x z y z
x y z
S G Expts Incompatible successive measurements in S S S
S S S S S S
However S S S S S S
−
→
= = =
Sˆ
iS ˆ
iS
2 2 2 2ˆ ˆ ˆ ˆ ˆ:x y z
Length or square of S S S S S= + +
2 2ˆ
1ˆ
3
2
4
zS z
z
z
S z
=
=
Polarization Operator & Pauli Spin Matrices
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= → =
−
→ =
− → =
1 01 1 1ˆ ˆ:2 2 20 1
1 2 0 11 1ˆ:2 21 01 2
1 2 01 1ˆ:2 202
ˆ ˆ,
z z z
x
y
y
x
y
x
S z z corresponding operator S
From x corresponding operator S
iFrom y corresponding operator S
ii
S S =
−( 0,ˆ )z
as requiri S and cy ed bycl Si Gc
( )
= = + =
−
− =
c
: ,
os sinˆˆ ˆ ˆsi
,
n2 sin cos
,0
n x
y
z
xS G example unpol Ag beam in y direction n
S S n S S c
n
s
n
o
= ˆˆ
2n n
S S n
z x y
nx y z
n n in
n in n
− = − −
Spin op
Matrix represent.
=
.x
y
z
n
Spin quantization in arbitrary dir n n
n = = + + :n x x y y z z
n n n n
+ + + −= + → = =
:
ˆ ( ) ( )2
.;n n n
eigen value eq
To predict measurement
Solve S n n n up n downu
Wolfgang Pauli1900 - 1958
Experimental Spin Preparation and Measurement
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Preparation of State |z+>
Measurement of State |n+>
n
z
n z
y
Task: Measure the spin polarization in n-direction for a beam that is 100% polarized in z-direction. Remember: An observable like Sn is accessible only via the eigen states of the associated operator and their eigen values.
Spin Operator For Arbitrary Direction
Electronic spin can be polarized in a direction by specific SG setup,
Subsequent measurement confirm polarization → prepared.
Subsequent measurements of components in directions destroy prior measurement (polarization).
But: Subsequent polarization measurements in direction produces
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z
z+
or yx
n
, 2,n z
Check cases 0
=
ˆ ˆ2 2 ,2
n nspin orientations relative n S eigen states n n with S n n
Express in above coordinate system
+ − → − =
( )
ˆ ˆ ˆ: cos sin cos sin
cos 0 0 sin cos sinˆ :2 2 20 cos sin 0 sin cos
n z y z y
n
Spin measurements perpendicular to beam axis
n x z plane
S S S
S
− →
= + +
= + =
− −
( ) ( )ˆ. : 2 2 cosn n n z
Exp value n z S n z
= → = =
Determining Polarization Eigen States
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ˆ, ..
)cos sin
sin cos
. : ; :
ˆ ( 1
n
n
In basis z z componentsof S eigen states n n
Solve EV equation S n expect
−
+ −
+ − + −
+
→ − = =
=
→
−
= =
( )
( )
cos sin
sin cos
cos sin 1 cos sin
sin cos 1 cos sin
+ = →
−
+ = → − =
− = → + =
2 2 2 2 2 2
cos sin
sin cos
cos sin cos sin 1 . . .
0
det 0
0
Non trivial solutions for secular determinant
q e d
→
− =
→ =
→ =
−
− −
+ − → =− + =
( )( ) ( )
( )( )( )
( ) ( )
2
2 2
2sin 2 cos 2 cos 2sin
1 cos sin 22sin 2
cos 2 sin 2 1
applying normalization
+
+
= = = →−
+ =
Rotations in 3D and Spinor Spaces
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:
cos si
, ... ; :
( 1)n
sin cos
In basis z z components n n
Solve EV equation expected
+ −
+ − +
−
+ −
→ =
=
= →
−
=
( )
( )
( )
( )
+
−
+ −
−= =
⎯⎯⎯⎯⎯→
cos 2 sin 2: :
sin 2 cos 2
Rotation n nz
zand
Rotation by an angle of of spin quantization axis in 3D vector space corresponds to rotation of spin-1/2 spinor kets in spinor space by /2.
Rotations of objects in normal 3D space have SU(3) symmetry, rotations in spinor space are SU(2) symmetric.
( ) ( ) ( ) ( )sincosP n
If system
n z
prepared
P
in z
and n n z + + + − − +
+
== =
→
=22 2 22 2
Conclusions From S-G Experiments
Electronic spin can be polarized in a direction by specific SG setup,
Subsequent measurement confirm polarization → prepared.
Subsequent measurements of components in directions destroy prior measurement (polarization).
But: Subsequent polarization measurements in direction produces
→ Measurements of any 2 components of electronic spin are incompatible: Fix one → both others are completely uncertain.
Results of 2 subsequent measurements depend on order, corresponding operators do not commute. S-operators change states!
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z
z+
or yx
ˆ ˆ ˆ ˆ ˆ ˆ ˆ, 0,..... .z y z y y z x
S S S S S S i S etc = − = −
( #) 3 .
,
Spin space is different from regular real D vector space
Complex overlap amplitudes important relative phases→
→ Heisenberg Uncertainty Relations for incompatible observables.
n
( ) ( )ˆ. : 2 2 cosn n n z
Exp value n z S n z +
= → = =
Task of Quantum Models
For theoretical modeling of bound microscopic system, assume (conservative) forces (bond potentials) and inertias (m). Use TISE to predict discrete energy spectrum of internal states.
→ Check by experimental absorption/emission spectroscopy
For theoretical modeling of unbound microscopic system of several complex particles, assume interaction forces (interaction potentials) and inertias (m1, m2,..). Use TDSE to predict scattering probabilities as functions of relative velocities.
→ Check by measuring experimental scattering and internal particle excitation
probabilities as functions of angle, relative velocities.
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Claim:All accessible pertinent information about an ensemble of microscopic systems is contained in the qm (well-defined, smooth, square-integrable) wave function y. → Measurement process
Task: Theory must provide immanent answers to questions:→ How to make and interpret measurements? → For which conditions do qm and classical pictures coincide (or mimic e.o.)?