Agenda: Quantum Mechanics Postulates & Consequences
• Quantum Postulates (Claims)
Correspondence principle
The quantum probability density flow
Eigen and expectation values
Overlap integrals, Dirac notation
Compatible and incompatible measurements
Time dependence
• Copenhagen interpretation of QM
System-detector interactions
Paradoxes, entanglement
Alternative trial interpretations
• Case study: Exploring electron spin polarization
• Stern-Gerlach polarimeters,
• Electron spinor states
W. Udo Schröder, 2020
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Prob
1
Overlap Integral and Expectation Value
W. Udo Schröder, 2020
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2
Overlap integral of 2 normalized functions measures their similarity (example: spatial symmetry). Maximally dissimilar (orthogonal) functions have zero overlap.
2* 2
2
: ( ...) ( ,...)
1 0
1
: 0 ; and
G definitions for any no
r
rmalized system wfs
x x dx wit
eneral
Ove
n
h
Rang o
rla
g
p
are to eacoe ortho l h thea
−
= = =
=
ˆ ˆ, .
1
:
ˆ ˆ
a
a a a a a a
Exception EigIn general A is dissimilar to A
w
enfunctio
Aith a a
ns
A a
−
= → = =
*
** *
:
ˆ ˆ( ...) ( ,...)
ˆ ˆ ˆ( ...) ( ,...) ( ,...) (
ˆ
...)
Necessary requirements for operator corresponding to observable A
A x real valA x dx
A A x x dx x A x dx
ue
A
−
− −
= =
= =
ExpectationValue
*ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
D
A
irac not o
A A A
a
A A A A
ti n
= → = = = =
Dirac Notation for Overlap Integrals
W. Udo Schröder, 2020
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3
*, ;
.etc
→ = =
+ = +
Per definition (integral):Anti-linear
2
: 0 1; 0 and
G
r
eneral definitions for any normalized system wfs
Ra are orthogonal to each othnge e =
( ) ˆ
1
"
ˆ ˆ
: " ker
a a a
a a
a
a a
aa
a
Eigenfunction for any observabi le operator A
with a a
l
s
A A
e
a
Kronec dSpecia f ature
ex st
elta
=
=
= → =
* †
( ...)
( ...) ( )
wf x state vector in Hilbert space
wf x vector in adjunct conjugate
H;
H
Dirac Notation for usein overlap integrals:
" " , " "bket ra
Expansion in Dirac Notation
W. Udo Schröder, 2020
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4
( )
( )
ˆ ˆ:
ˆ ˆ
Matrix element of A A square matrix A
A A numeric values
= =
a orthonormalized set of wfs =: a a aaSpecial feature =
" " aaKronecker delta
( )
2
: ...
, ( )1
a aa
aa
ac Norm
G a
alizati
eneric w ve p
on for discrete
c
E
k
V
a e
a
t c
c
=
=
Also solution to TDSE:(Superposition Theorem
Expansion Theorem)
( ) ˆ
ˆ ˆ 1a a a a a a
a for any observable operator A
with a a a
Wave functions are solutions to TDSE
E exiig c senfun tions
A A
t
= → = = Because only eigen values are measured
Instant Quiz
W. Udo Schröder, 2020
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Prob
5
ˆˆa a a
a a aa
for operatorConsider eigenfunction A with a
Show t
s A
hat
=
=
ˆ ˆa a a a aUse A Aa and a with a = =
Agenda: Quantum Mechanics Postulates & Consequences
• Quantum Postulates (Claims)
Correspondence principle
The quantum probability density flow
Eigen and expectation values
Overlap integrals, Dirac notation
Compatible and incompatible measurements
Time dependence
• Copenhagen interpretation of QM
System-detector interactions
Paradoxes, entanglement
Alternative trial interpretations
• Case study: Exploring electron spin polarization
• Stern-Gerlach polarimeters,
• Electron spinor states
W. Udo Schröder, 2020
Post
ulat
es
Prob
8
Measurement Process: Reduction of Wavefunction
W. Udo Schröder, 2020
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Prob
9
III. A system can only be observed in one of the eigen-states of the corresponding indication operator  → Âa = a·a. Orthonormal basis set a
ˆ: ˆA H internal energy of atomExample = =
( )in a a in aa
c or c a da
= =
Expand generic system wf = linear combination of eigen functions a = E
( )
2 2
22
, ( ) , 1a aa
aa
Expectation value with P a c c
A a c or A a c a da
= =
= =
In each (i) of N measurements (detecting fraction i of ensemble), the system expresses one possible eigen value ai at random. Summing all Ni
measurements, one obtains the total initial probability distribution P(a).
Observation
Demonstrate this by explicit construction !
complete
Measurement Process: Reduction of Wavefunction
W. Udo Schröder, 2020
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10
III. A system can only be observed in one of the eigen-states of the corresponding indication operator  → Âa = a·a. Orthonormal basis set a
After one measurement, each sub-part (i) of system ensemble populates one distinct eigen state: Probability Pfin(a=ai) = |cai|
2.
2 2ˆ ˆ
" "
a i a i ai i i
fin ai i
A a A a
System prepared as
= =
→
Separation of sub systems according to measured values ai permits repetition of the same measurement on components i.
Result: Selected system remains in the same normalized eigen state: Pfin(ai) = 1.
Observation
in a aa
c =
( )2
in a a fin i aia
c P a c
= →→
Measurement
in collapses into incoherent sum of components !(ongoing research)
Ideal Expectation: System-Apparatus Int.
Debate from the start of QM: Measure dynamical variable W (observable) of system S with an experimental apparatus A.
W. Udo Schröder, 2020
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11
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:
W. Udo Schröder, 2020
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Prob
12
( ) ( ) ( )( ),
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
Repeated Measurements
N repeated measurements of observable A on many identically prepared
microscopic systems, with the same detector. Accumulate events with ai.
W. Udo Schröder, 2020
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13
( )ˆ ˆ: a aObservable values of A eigen values of A A a
Repeated measurements on events with same a return the same a
= =
Correspondence Principle: For any observable A,
there is a quantal operator projecting mean A & variance
( ) ( )
( )
( )( )
*
22 2
21 2
2
2
ˆ ˆ: , ,
ˆ ˆ
( ): 2 exp
2
all space q
A
A
A
A A dq q t A q t and
Variance A A
A AdP Aprobability distribution
dA
−
= =
= −
− → = −
( ) ( )1
( ): # ,iP a of events i with a a a a a dP a da normalizedN
= + →
( ) ( ) ( )0 0 0" " : , , ,aCollapse of wave function q t t q t t q t →
Instant Quiz
Copenhagen interpretation: quantum-statistics (not thermal!)
W. Udo Schröder, 2020
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14
( )
( ) ( )
2
A with a , set of eigen functions ,
, , ;
Probability ( )
a
a a aa
a a a
complete orthonormal x t
system stae te wave function x t c x t c
c P a c
Pur
→ =
= → =
0 1 1 0 1 0 2 0
0E 1E 1E 0E 1E 0E 2E 0E
Preparation of many instances of same quantum state
Measurement
Data Analysis
?? ??Deduce mean energy E and wave function = =
Compatible and Incompatible Measurements: HUR
W. Udo Schröder, 2020
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Prob
17
IV. A series of successive ensemble measurements of observables A, B →
probability distribution P(A), mean and variance are given by
A B
Heisenberg Uncertainty Relation for sequ
Standard uncer
e t
taintie
l
s a
n ia
nd
( )2 2
21
ˆ ˆ ˆ ˆ... ( ) ( )ˆ ˆ ˆˆi i baAB dq q B q BA Aa d BAn
= = −
ˆ
,
( )i
In general wave function is not eigen function of observable A
qA
→ ˆ( ) ( ) (ˆ ˆ )i i iq changes q B qA A → →
ˆ ( )
ˆ ( )
i
i
B q
B q
→ ˆ ˆ( ) ( ) ( )ˆi i iq B change q qAs B → →
:
ˆ ( )
,
iq
In general measuremen e
A
ts of A and B ar incompatible
→
ˆ ˆ ˆˆ ˆˆ 0
0,C
eommutator A
B aB A
m
A and com
t
p t
A and
i
B no co pat l
blB B
eA
ib
= −
=
1 ˆ ˆ, 22
A B A B
Notation
Post
ulat
es
Prob
W. Udo Schröder, 2020
18
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 square well wave functions
x xx c n c n
a a
−
= +
( )
:
ˆ ˆ: , 0
(
ˆ
)
, 0x
General feature for observable A and B conjugat
No separate meas
m
urements o
b
n sy
e
st
t
e
A
m
l
c
s
o
x
p
r
i
c
e
n
s
I omp o
No
e
C
a
l
i
o
l
n
a
i
e
n
o
g
b bs rv
Role of com utat A
p i
e
B
→
=
= −
→
→
=
−
Time Dependence
V. All system wave functions evolve in time according to the t-dependent Schrödinger Equation,
W. Udo Schröder, 2020
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Prob
19
ˆi t H =
Easy to prove for stationary ( ) wave functions : ( )
ˆ ( ) ( )
E
E E
energy eigen x,t
H x,t E x,t
=
ˆ
( ( )
) ( )
: )
( iH t
conservative V V tC formal solution of TDSE
x,t e x, 0
la d
t =
ime
−
=
( ) ( )i E tE Ex,t e x,0 − =
ˆ( 0) ( 0) ( ) ( 0)i H t
E EE
x, c x, x,t e x, = → =
ˆ( ) ( ) ( )E E Ei x,t H x,t E x,tt
= =
TISE
TDSE
Replace op by EH
Easy to prove for any LC(linearity of qm ops)
Time Evolution of Particle Wave Packet
W. Udo Schröder, 2020
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ulat
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Prob
20
( )2 2 2 2
2
:
ˆ0 : ( ) e
ˆ ( ) ( ) ( ) ( ) ( )2 2
ˆ( , ) ( , ) ( , ):
k
ki xk
k k k k
k k k k
Ensemble of particles with mass m
Plane waves at t x A are eigen functions of H
kH x V x x V x E x
m mx
f i x t H x t E x tThere ot
re
= =
− = + = + =
= =
( )
( ) ( )
0
1 42 2
2
0
( , 0) ,
1( ,0) ( )d ,
2
exp2 4
k
Free particle V const wave packet
central momentum k k
x f k x k
a aGaussian profile f k k k
+
−
=
=
=
= − −
( , ) ( , 0)
iE tk
k kx t e x t =
= =
Time Evolution of Particle Wave Packet
W. Udo Schröder, 2020
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ulat
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Prob
21
( ) ( ) ( )
( )
0
1 42 2
2
0
( , 0) ,
1( ,0) ( )d , exp
2 42
1( ,0) ( ,0) d
2
k
k
Free particle V const wave packet central momentum k k
a ax f k x k Gaussian profile f k k k
i x f k i x kt t
+
−
+
−
= =
= = − −
= →
( )1 4 21
4
00 2 0
4 22 2
2 2 2
( )2 4( ) 1 exp
(2 )
ik
i k x tm kx t
x,t t ea m a a
m
te
i m
− −
−
= + − +
( )( )1 2: tan 2Phase factor ma t −=
( )
( )
1 42 2
22
ˆ
0
0
1( ) ( 0) exp
2 42
:
.
2
....
i H t
perfect square
a ax,t x, k k i k x dk
Evaluate integral by converting exponent to for q k k
i
ke i
Gaussian ntegra
tm
l
+
−
−
= = − − + −
=
−
→
( )
2
21
( , ) ( ,0)2
k
ki t
mx t f k e x
−
+
−
=
Linearity of operator
Spreading of Particle Wave Packet
W. Udo Schröder, 2020
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ulat
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Prob
22
( ) ( )21 41 4 2
02 22 2 4 2
2 4( ) 1 exp
(2 )
i i k t gx phx
x,t t e ea m a a i m t
t
−
−
− = + − +
0 0
20
0 0
:
1
2
2
ph g
gk k
Traveling wave packet
Velocity of fundamental waves
Ephase velocity
k p
Velocity of superposition packet
kd d kgroup velocity
dk dk m m
=
= = =
= = =
Numerical solution
Summary: Measurement/Preparation (CI)
W. Udo Schröder, 2020
Post
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Prob
23
Postulate: qm system can only be observed in an eigen-function (state) of the corresponding indication operator  → Âa = a·a. Other states are not realizedOrthonormal “basis” sets {a} allow convenient expansions of wave functions.
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 a), a system can be “prepared” in that state a. Repeat measurements yield same a.
Discontinuous change in t-dependent behavior of wf is not described by Schrödinger Equation.
Some (incompatible) observables cannot be measured simultaneously on systems, uncertainties of corresponding observables are (anti-)correlated.