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Postulates of Quantum Mechanics
1. Normalized ket vector  contains all the information about the state of
a quantum mechanical system.
2. Operator A describes a physical observable and acts on kets.
3. One of the eigenvalues an of A is the only possible result of a measurement.
4. The probability of obtaining the eigenvalue an : P  an
5. State vector collapse :  ' 

Pn 
 Pn 
6. Schrödinger Equation : i d  (t )  H (t )  (t )
dt
Time evolution of a quantum system
2
Sz is the operator that corresponds to the physical observable of spin
along the z-axis.
Write the eigenvalue equation for Sz using the above notation
Write the matrix Sz, and the kets |+> and |-> in matrix notation
In the z-basis, use what we know
about how to write the kets |+>x and
so on (given below) to find the
matrices Sy and Sx
Group work
• Show that the Sx, Sy and Sz matrices can be
written as a linear combination of projection
operators. (Projection operators are outer
products of the eigenvectors with
themselves.) The coefficient of each term is
the eigenvalue associated with the
eigenvector used to make the projection
operator.
Postulates of Quantum Mechanics
1. Normalized ket vector  contains all the information about the state of
a quantum mechanical system.
2. Operator A describes a physical observable and acts on kets.
3. One of the eigenvalues an of A is the only possible result of a measurement.
4. The probability of obtaining the eigenvalue an : P  an
5. State vector collapse :  ' 

Pn 
 Pn 
6. Schrödinger Equation : i d  (t )  H (t )  (t )
dt
Time evolution of a quantum system
2
What do we know so far?
• Atoms (and fundamental particles, i.e., electrons) can
have “intrinsic spin” – can be a 2-level spin system,
where all measurements of the intrinsic angular
momentum yield + or – hbar/2
• Can measure this with S-G device
• Find that by looking at the % of atoms found along
different spin projections (i.e. Sz) we can infer the
initial state
• Find that making a measurement to determine the
state of an atom changes the state -> “collapses” the
atom into the new state – this is related to
“projection”
Group work
• Show that the spin operator matrices: Sx and
Sy can be written as a linear combination of
projection operators, where the projection
operators are outer products of the
eigenvectors with themselves and the
coefficient of each term is the eigenvalue
associated with the eigenvector used to make
the projection operator
• If a spin operator acts on a vector, what
transformation does it correspond to?
Hermetian Operators: At = A
• We normalize the eigenkets!
• ALWAYS have real eigenvalues and orthogonal
eigenfunctions that form a complete basis set
• Linear transformations
– What kind of transformations have you observed with
matrices acting on vectors – what do they do?
– Come up with at least one “physical” example of a linear
transformation (operator) (you don’t need to write the
matrix, just say what it does)
– What did the projection operators do to the vectors?
– What can you say about the operators Sz, Sx, and Sy?
Projection operator
• Operators “embed” the kets and eigenvalues
• The projector operator MODELS
measurements – it tells us what state (ket) the
atom is in after the measurement:
• It tells us about the probability of finding a
particular eigenvalue from a measurement
• P+|ψ> = |+><+| ψ> = ψ+|+> = coefficient of
Psi along +z spin, in the +z spin direction
(this new ket is NOT normalized!)
“Fixing our equation” to make the new ket always normalized
Spin 1 systems:
• Spin measurements can be h, 0 or –h (h =
hbar)
• Write the eigenvalue equations for Sz
• Write the operator Sz in matrix form, and
the kets corresponding to spin-up in Z, spin0 in Z, and spin-down in Z
Spin 1
Small board activity
• As a class you’ll calculate all combinations of Sx,
Sy, and Sz operators acting on all the eigenstates:
|+>, |->, |+>x, |->x, |+>y, |->y, and we’ll try to
determine what this operation “does”
•
•
•
•
•
•
•
There are 18 combinations – each of you will do 2
Count off by 9’s
1’s do Sx on |+> and |->x
2’s do Sx on |-> and |+>y
3’s do Sx on |+>x and |->y
4’s are like 1 with Sy, 5’s like 2 with Sy, 6 like 3 with Sy
7 like 1 with Sz, 8 like 2 with Sz, 9 like 3 with Sz
Your General questions:
• How do we know how many eigenvalues a matrix has?
• What is “COLLAPSING”? (Copenhagen vs. multiverse)
• What “ARE” operators, and what do they do? (better
question: what do Hermitian operators represent?)
• Don’t all measurements “perturb” a system?
• Is this the same phenomena except at very small scales
the perturbation is large?
• Is “quantization” odd?
• What is a basis, and when do we have to be in a
particular basis?
• Should some calculations be basis independent, and does
that mean we don’t have to be careful?
2.2 Measurement
When doing a Quantum measurement, making a measurement changes the state
Is this really so odd?
Two examples: determining the classical trajectory of an object (handout)
Illuminate it to see where it is (optical method)
• Radiation pressure will disturb the motion
• If we want to disturb it less, we have to radiate it less – this means we’ll
know less about the position
• We can reduce the intensity of the light to individual photos to minimize
the disturbance – but then can go no lower
• We can reduce the energy of the photons by increasing the wavelength –
but that reduces our resolution
Mechanical method – use a series of bells
• If the bells are too large there is low resolution
• If the bells are too small there will be too much momentum transfer and
we’ll disturb the projectile
• (This type of measurement is analogous to a tracking chamber in a particle
physics experiment)
Tracking chamber (the inner blue
ring on right – note scale of person)
BOTH classical and quantum measurements perturb a system and have a
fundamental interplay between “resoultion” and “perturbation” (x and p)
What is different between classical and quantum measurement?
•Classical: a single measurement can be compared to theory, multiple
measurements simply reduce the systematic and measurement errors
•Quantum: a single measurement tells us something about the
eigenvalues, but does not tell us about the original state or tell us that a
similar measurement will yield the same result
•MORE and MORE measurements do not minimize uncertainty in
quantum measurements! It’s all probabilistic!
•We will need to apply statistics to understand our measurements!
Example: Single measurement of Sz cannot determine the original state, and can
result in more than one value!

X
x
Prepare atoms in
a particular state
N+
Z
-
N=N++N-
NPerform the same experiment on
each identically prepared atom
Expectation Value (also called the mean or average)
The expected mean value of a large number of experiments
A   A  THIS IS NOT the expected value of any single experiment!!
THIS IS NOT a time average – it is an average over many
identical experiments!!
Example: rolling dice
We expect an average value different from any individual measurement
1
1
1
1
1
1 21
n  1  2   3   4   5   6  
 3.5
6
6
6
6
6
6 6
Sz
In words: the MEAN result of a
 
 
  |    P()   -  P(-) |   measurement will be value x, X% of
 2
 2
the time plus value y, Y% of the time
2
2




  - -     -  - -
2
2
2
2

 

     - - -     S z    S z - - 
2
 2

  S z     - -    S z 
What do we expect to get if we take the expectation value of Sz
for a system prepared in state |+ ?
Do the calculation on your whiteboards!
(1)
Sz


 Only possible result:
this must be the
  Sz        
2
2
2 expectation value.
Or we could do the full calculation: (1 0)  1 0
2 0 -1
1
0
What do we expect to get if we take the expectation value of Sz
for a system prepared in state |+x ? Let’s calculate it!
(2) S z  x  S z 
x
1
1
  1 0  1 1 

   1 1   0
1 1 

2  0 - 1 2 1 4
2
 - 1
The two possible results   each have 50% possibility,
2 so we’d find  (.5) + -  (.5) = 0
2
2
Standard Deviation: spread of measurements about the mean
General Formula written in QM notation:
A 
A -
A

2
•What does the thing under the square root mean?
•Why do we square then take the square root?
•(What would we get if we didn’t take the square?)
-> Let’s simplify this expression by expanding it out
Let’s find this for the two previous examples:
What do you expect for the standard deviation of the
measurement for Sz in state |+? What do we find?
What do you expect for the standard deviation of the
measurement for Sz in state |+x? What do we find?
2
(1)
S z2

2
  Sz      
2
2


     
2
2
 S z 
S
2
z
2
- Sz
2
2
 
   -   0
2 2
2

2
S z  0
Sz 
N
-

2


2
Sz
(2)
S x  S 
2
z
2
z
x
1
  1 0    1 0  1 1
 

 
1 1 

2  0 - 1 2  0 - 1 2 1
2
2
2
1
0
1
1






1
1

     1 1    
   1 1
22
 0 - 1 - 1 2  2 
1  2 
2
2
 S z 
S z2 - S z
2


   -0 
2
2
N
Sz  0
S z 
-

2


2

2
Sz
In quantum mechanics, the uncertainty is fundamental, meaning that you cannot
design the experiment any better to improve the result.
2.3 Commuting Observables
Incompatible observables: we already saw and discussed this in
the context of our experiments
•Measuring x altered our state and when we re-measured z we
no longer knew anything about the initial state!
•However, if we measured z in the middle, we did not alter
our state – so some sequential measurements can be done
without loss of information about the initial state

Z
100
X
x
50

25
Z
-x
-
100
-
50
25
Commuting property in mathematics: if A and B commute, then AB = BA
Commutator: A, B  AB - BA
Operator A and B commute if [A, B]=0.  AB-BA=0  AB=BA
Let |a be an eigenstate of the operator A with eigenvalue a: A a  a a
If A and B
Commute:
We get another
Eigenvalue equation
With eigenvector B|a>
an eigenstate of A with
the same eigenvalue a
BA a  Ba a
AB a  aB a
We can move out
our constant a
AB a   aB a 
B a b a
So B|a> must be a scalar
multiple of |a> !
IF Operator A and B commute THEN A and B have common
(or simultaneous) sets of eigenstates.
If operator A and B commute:
 The second measurement does not change the eigenstate |a.
A
a1
a2
a3
a1
B
b1
b2
b3
a1
a1
A
a1
a2
a3
100
a2 0
a3
0
We say that these two observables can be measured simultaneously.
 We can measure one observable without erasing our knowledge of the
previous results of the other observable.
 Observable A and B are said to be compatible.
If operator A and B do not commute:
 A and B are incompatible operators.
 A and B cannot be measured simultaneously.
Based on what we have seen thus far, what can we conclude about the
commutation relationship between Sz and Sz? What about Sz and Sx?
-> Let’s check them!!!
Example:
 1 0   0 1  0 1  1 0 
 
 - 
 

S z , S x   
2  0 - 1 2  1 0  2  1 0  2  0 - 1

  
2
 iS y
2
 0 1   0 - 1    2  0 2 
 - 
    


 - 1 0   1 0   2   - 2 0 
0
These two operators do not commute.
Time permitting – let’s check another one (this problem was
removed from your homework assignment – let’s make sure we
can all do this calculation)
S , S   iS
S , S   iS
x
y
y
z
z
x
S z , S x   iS y
None of the spin projection observables
commute each other – DO THESE IN
CLASS!!!.
2.4 Uncertainty Principle
1
AB  A, B 
2


1

 iS z 
Sz
2
2
2

  
If the initial state is |+, S S 
S

  0
x
y
z
2
2 2 2
 S x  0, S y  0 Recall ΔSz is zero for this case!
• If we knowzspin along
 one direction, we don’t know it along the other directions:
we can never know two non-commuting observables simultaneously!!
S z zˆ
S
Example:
1
S x S y 
Sx , S y
2
 point in a specific direction. Our classical image of spin does not
•Spin does NOT
work! We can2 only know a projection of it onto one axis at a time!
y
• “Spin up” means that spin projection along that axis is up.
x
(no time left for this – see book for
derivation) Group white boards
• Write a general unit vector in spherical
coordinates that can point anywhere in 3-d
space (this vector is usually called n )
• If a total spin vector S can be written as
Sxi + Syj + Szk , write S (dot) n
• What operator might this correspond to?
• Check what it might correspond to by
putting in specific values for theta and phi
(pointing n in specific known directions)
WITHIN the 2-d spin vector space –
i.e., how to write ANY spin vector
No matter which way the spin vector “points” – what will we measure??
2.5 S2 Operator:
As with any vector, if we square it we have information about the
magnitude but we loose information about the direction!

 2
2
2
2
2




S
S x S y S z   s x  s y2  s z2
2
Find S2 in matrix form
2

What operators will S2 commute with?
What can we conclude about simultaneously knowing the value
of S2 and the projection of S along x, y or z?
What are the eigenstates of the operator S2? What are the
eigenvalues?
What is the expectation value of S2? What can you deduce
about the “length” of the spin vector?
3
S 2   2 
4
For any | in the spin-1/2 system
NOT just our eigenstates of Sx, Sy and Sz z
The spin vector is longer than any of the
measured projections (ħ/2): the full spin vector
can never be fully aligned along any axis!!

S

S z zˆ

2
x
This vector describes the "direction" in which the spin is pointing,
corresponding to the classical concept of the axis of rotation. The
spin vector does have a well-defined experimental meaning: It
specifies the direction in ordinary space in which a subsequent
detector must be oriented in order to achieve the maximum possible
probability (100%) of detecting every particle in the collection. For
spin-1/2 particles, this maximum probability drops off smoothly as
the angle between the spin vector and the detector increases, until at
an angle of 180 degrees — that is, for detectors oriented in the
opposite direction to the spin vector—the expectation of detecting
particles from the collection reaches a minimum of 0%.
y
YOUR awesome words:
• As was so clearly stated by your comments
in class:
• IF spin were in 3-d physical space, then we
could in principle align our S-G device
along S and measure the full value of
root(3)/2 hbar – but we can NEVER
measure that value – we can only measure
the projections hbar/2 at any one time!
Generalize to other spin systems:
•s = spin quantum number- the “spin” of the system, such
as ½ in what we’ve done so far
•m = the projection of the spin (magnetic quantum
number) – it can take on values from –s to s in integer
steps
S 2 s, m  s ( s  1) 2 s, m
S z s, m  m s, m
Our spin-1/2 system
in this notation:
1 3
s  0, ,1, , 
2 2
m  - s, - s  1,, 0, , s - 1, s
1
1
s  ,m 
 ,
2
2
1
1
s  ,m  -  2
2
Write down what some states can look like for a spin 3/2 system