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Postulates of QM
Part 1: Comparing Classical and Quantum
Mechanics
Part 2: Complications and Uncertainty
Part 3: Heisenberg Picture and DOF's
Why Quantum Mechanics?
Catastrophic Failures of Classical Physics
Radical sameness of atom, etc.
Spectral lines, lifetimes
Ex: Barium atoms made in a nuclear reactor and
Barium atoms left over from the earliest stars are
exactly the same.
Why Quantum Mechanics?
Catastrophic Failures of Classical Physics
Radical sameness of atom, etc.
: Spectral lines, lifetimes
Observation of matter waves: Davison and Germer
Davison and Germer shined an electron beam on
cleaned crystaline Nickel and found that the electrons
came off at the “Bragg” angle, rather as if they were
a form of x-rays. This was in accordance with the
deBroglie relation,
Why Quantum Mechanics?
Catastrophic Failures of Classical Physics
Radical sameness of atom, etc.
: Spectral lines, lifetimes
Observation of matter waves: Davison and Germer
Quantities exist without any classical counterpart: Spin
Spin is like an internal angular momentum that
particles possess. It is not actually angular momentum
in that no spatial version of spin exists as far as we
know, whereas angular momentum will be associated
with spatial properties of the particles' wave.
Comparison of QM and Classical Physics
Comparison of QM and Classical Physics
Classical
QM
Condition of system
Comparison of QM and Classical Physics
Classical
QM
Condition of system
“The State”
Hilbert (complex vector) Space
QM is a temporal succession of
vectors in this vector space.
Phase Space
Classical is the motion of
the x,p(t) point in
States can be:
1) States representing a single thing: for example
a single barium atom.
=
|Barium>
2) Or, any superposition (that is, a linear
combination) of other states;
= a|Barium> + b|Ytterbium>
Note 1: 'a' and 'b' are in general complex numbers
Note 2: The fact that a quantum theory must have facility
with superpositions is the reason that that linear algebra
studied in Chapter 1 is going to be so useful.
Comparison of QM and Classical Physics
Classical
QM
Condition of system
Observables/measurement
eigenvalues of
, a linear operator
Comparison of QM and Classical Physics
Classical
QM
Condition of system
Observables/measurement
eigenvalues of
, a linear operator
NOTE: A list...(could be discrete)
Generally, Continuous Functions
Comparison of QM and Classical Physics
Classical
QM
Condition of system
eigenvalues of
Observables
, a linear operator
Reality
guarantor of real eigenvalues...
i.e. Hermiticity of the operators associated with measurable
quantities is necessary because we only measure real
quantities...no gauges or meters in the lab give complex
numbers directly!
Measurement and Observables
Throughout the following pages let the
be the
orthonormal basis of eigenvectors of a hermitian operator
with
A measurement in quantum mechanics is both
(1) (An Activity: ) a projection into the eigenspace of the
corresponding eigenvalue measured.
(2) (Probabilistic:) the frequency of measuring a particular
eigenvalue is proportional to the square of the overlap of
the given state with the corresponding eigenspace.
Measurement and Observables
Ex 0: Suppose we start with a pure state w.r.t.
=
Then, the repeated measurement of
always yield the number
will remain
while in this state will
and the state vector
Ex 1: Mixed state case: the non-degenerate case
Suppose :
The a single, isolated measurement of
will
return just one value of three possible numbers
,
, or
ONLY !
But you can't know which value until you “measure” it.
We re-iterate, that you will never find an intermediate
value of, say,
(
+
)/2
.Essentially,
from a single isolated measurement of
this is atomism. In words, no matter how the system was
prepared (how mixed), when you perform a measurement
you will always measure a discrete value that is an
eigenvalue of the observable. You can have one Barium
atom. Or one Yterbium atom. Your state can be an
admixture of the two, but it is not real to find for
a single measurement an atom that is some combination
of the two.
...So if it was a mixed state of a Barium and Yterbium
atom...and you measured it to be Barium...then what?
In quantum mechanics, as we are teaching it here,
a single measurement actively places the state in the eigenbasis corresponding to the eigenvalue measured.
For our example, suppose we start with
We perform a single measurement and find
Then, were we to immediately repeat this measurement,
each subsequent measurement would yield the same value,
...The Barium atom would remain the Barium atom....
BUT: Then what does it really mean to be in a mixed state?
Probability and Quantum Mechanics
“The Copenhagen Interpretation”
The probability of a particular measurement outcome
is proportional to the norm square of the state's
overlap with the associated eigenbasis.
=
and
(Important note: this formula assumes that both
are normalized.)
So, the admixture coefficients reflect the likelihood of the
outcomes of any particular measurement...said another way,
the frequency of a particular measurement outcome from many
independent measurements on the (each time identically
prepared) state.
Ex 2:
= ¼ , ¼ , ½ for the outcomes
,
respectively.
,
'Collapse of the State Vector'
But back to that atomisim...if we make a
measurement on an (arbitrary) state vector and find a value
for example, we expect each immediate re-measurement
of that same observable to again give
But this means that the subsequent probability of measuring
the observable and finding
is one. That in turn by the
Copenhagen interpretation means that as a result of the
measurement process itself there can no longer be any
superposition of states with different eigenvalues...for our
example, the state would have to be a pure state.
=
The formerly mixed state has been projected onto a pure state
by the activity of measurement...
This is the so-called 'collapse of the state vector'...in the sense
that the initial mixed state has 'collapsed' onto an eigenstate.
Thus all the information about the mixture of states in the
state vector before the measurement has been completely
obliterated by the measurement process (so defined).
...whether the atom was produced in a nuclear reactor a
microsecond ago or was left over from the earliest stars,
if we measure them both to be Barium atoms they are
radically identical.
When identical measurements could lead to different states
“The Degenerate Case”
Suppose the state vector was
Where, we have two different states
with the same
,
eigenvalue
“Degenerate States”
Assume that they are orthonormal
-WOLOG-
When identical measurements could lead to different states
“The Degenerate Case”
Suppose the state vector was
Where, we have two different states
with the same
,
eigenvalue
NOW... Suppose that we make a measurement of
and find the value
the measurement?
. What is the state of the system after
When identical measurements could lead to different states
“The Degenerate Case”
Suppose the state vector was
Where, we have two different states
with the same
,
eigenvalue
NOW... Suppose that we make a measurement of
and find the value
the measurement?
ANS:
. What is the state of the system after
Averages
By the Copenhagen interpretation, the average
value of the measurement of
state
is given by
while the system is in
=
This is called the Expectation Value of
Ex 1: For
The expectation value is
=¼
+¼
+ ½
in
Ex 2: If the state were
be?
What would
ANS:
= ¼
+3/4
NOTE 1: These expectation values are average values,
and as such can take on continuous sets of values, unlike
individual quantum measurements.
NOTE 2: Physical examples of inescapably average values in
quantum mechanics are things like the lifetime of an excited
state. Lifetime has no real meaning as an individual
measurement, but it does as a (generalization of) an
expectation value.
Uncertainty
Uncertainty denotes a measure of the spread of the
individual measurement of an observable. It is therefore a
state-dependent notion.
A useful mathematical definition is :
NOTE: This uncertainty is positive, real.
NOTE: It is the same as the notion of standard deviation
in which you use
as the distribution.
Uncertainty: an example
Given
Compute the uncertainty on of
on this state.
Uncertainty: an example
Given
Compute the uncertainty on of
on this state.
ANS: Recall the goal is:
And recall from the previous page that
+¼
=¼
SO; only need to figure out
As an operator, since
=
+ ½
expectation value
And, applying
2
to this gives,
=
So we are now ready to assemble these pieces together
into a measurement of the uncertainty;
]2 =
[
=
[
]2=
2
2
-
-
Non-commuting bases of measurement
Take two hermitian operators
.
The commutator
Then it is possible to find a basis which
Iff
diagonalizes
Iff
and
and
(the “compatible operators” case)
is not zero, then in general the operators cannot
be simultaneously diagonal, called then
“incompatible operators”.
Non-commuting bases of measurement
Take two hermitian operators
and
Let:
And take as a starting state the vector
In Pictures !
.
The commutator
Suppose we measure
To find value a1
The process of measurement has projected our system into
state |t2>. We now measure
This is the case
where
is not zero...
Then, suppose one measures b2
Then we have
Note this is not the same as first measuring
And then measuring
to get a1
to get b2
And finally,
...
If the observables are compatible...
=0
So they can be simultaneously
diagonalized...
Now perform measurements on |t1>
After this, one will never get b2...only b1
Summary about measurement in QM
a) Non-commutative joint probability: Let
and
have eigenvalues
respectively.
Let
Let
and
Be the respective probabilities
in a single measurement.
.
be the joint probability of measuring
and then immediately measuring
Then: In general note that:
(incompatible case)
iff
(compatible case)
Then
Note:
There is really no way to reduce the probabilistic
nature of quantum reality to probability functions
(strictly positive, single valued densities) on classical
phase space.
Comparison of QM and Classical Physics
Classical
QM
Condition of system
Observables
Reality
Determinism
Quantum Determinism:
at time t=0
Given the state
Specified with complete precision
one can find the state
and the complete Hamiltonian,
At any subsequent time with no uncertainty.
Classical Determinism:
Given the position(s) and momenta at time t=0
with complete precision, and the complete Hamilton,
the subsequent position(s) and momenta are
then known at any subsequent time with no uncertainty.
Quantum Determinism:
at time t=0
Given the state
Specified with complete precision
one can find the state
and the complete Hamiltonian,
At any subsequent time with no uncertainty.
Classical Determinism:
Given the position(s) and momenta at time t=0
with complete precision, and the complete Hamilton,
the subsequent position(s) and momenta are
then known at any subsequent time with no uncertainty.
UPSHOT: Both QM and Classical are causal theories. All
the 'probability/uncertainty' in QM comes from
the measurement 'process'.
Heisenberg Uncertainty Principle
(
(See Shankar, Chapter 9)
Example Translation from QM to Classical
Classical
QM
x
p
)
Canonical Commutation Relation
Poisson Bracket
Complication: Operator Ordering...
Differential relationship
x and p are just numbers....
Operator Ordering
So, the Universe is bumping and grinding away,
rotating its state vector...and we want to relate combinations
of operations on that state vector as things that make classical
sense to us, for example, angular momemtum or some
kind of perturbation. How do we correspond
(combinations of) operators acting on the state vector with
classical notions? There is in general no unique way to
translate backwards from a classical notion to a quantum
notion! But we can try...
Example:
Kinda like angular
momentum....
This operator is fine in the quantum theory. It does not however
represent the classical quantity 'xp'. For one thing, the classical
quantity is always real, whereas this quantity is not Hermitean
and so does not always have real eigenvalues.....
Operator Ordering
To make
Hermitean and thus have real eigenvalues
one can try the following combinations of related operators
1
2
Q: Are both Hermitean?
Q: Why is only one of these choices related to the
classical observable xp? Which one?
Example Translation from QM to Classical
Classical
QM
x
)
Canonical Commutation Relation
p
Poisson Bracket
In the oft-used position basis above, these are equivalent to:
Example Translation from QM to Classical
Classical
QM
x
)
Canonical Commutation Relation
p
Poisson Bracket
In the oft-used position basis above, these are equivalent to:
But not every
observable has a
classical version...
Example Translation from QM to Classical
Classical
QM
x
)
Canonical Commutation Relation
p
Poisson Bracket
Spin
?
The Schroedinger Equation
Setting up the equation: Find an operator realization
the
that captures the physical details of the
system.
Often this can be done by promoting the co-ordinates
and momenta to operators as we described earlier in
this talk.
The multi-dimensional case:
The maximal subalgebra of all operators that among themselves
commute with each other is called the “Maximal Set of
Commuting Observables” or also, the Cartan subalgebra (CSA).
Since they commute with each other they are compatible. That
in turn means that we can classify all states of the system in
terms of eigenvalues with respect to each operator in the CSA
An example of this is related to the classification of states in
problems involving
(a) Spatial dimension
(b) Multi-particle systems.
Ex:
Co-ordinates in
3-d commute!
THUS, the eigenvalues of the position operators must
form a good basis for the Hilbert space. Now these
operators, being positions, have a continious spectrum.
Thus that Hilbert space describing a single particle in
3-d is simply the space of all function depending on three
(position eigenvalue) co-ordinates.
Ex: ->
The Harmonic Oscillator in 3-d (isotropic case)
Classical Hamiltonian
=
Descends rather simply from replacing the operators
in this Hamiltonian below with their classical counterparts.
=
We'll study the Scroedinger equation that is associated with
this Hamiltonian in somewhat more detail later...for now we
note only that in the co-ordinate basis this differential
operator can be written in different co-ordinate frames.
To do that, go to the co-ordinate basis of the operators;
And so the Hamiltonian becomes;
=
Note the appearance of the laplacian operator,
Which you have already studied in your E&M class.
You know how to transform it to other co-ordinate
systems, so for example, in spherical co-ordinates, one
has;
=
+r2
Multi-particle case
Q: How do we use Quantum Mechanics to talk about
multiparticle systems?
A: The operators associated with the positions of all the
particles are expected to commute with each other; Think
of the physical meaning of the position eigenbasis...
So, in the position eigenbasis, our Hilbert space is the
space of all functions of N variables...where N is the
number of particles times the spatial dimension!
Ex: to follow later....
The Schroedinger Equation
Objective : Given
Solve
To find:
EX: Time Independent
Step1: Find energy eigenbasis:
with
Step 2: Study the temporal evolution of the energy eigenbasis:
Let :
Then, the Schroedinger equation become an ordinary DE
for the coefficients:
Which can be solved as
So that the energy eigenbasis in time is
=
Step 3: Expand the solution in terms of the energy eigenbasis
..and these coefficients must solve the same equation as
before ! Thus,
Where now, evaluating both sides at t=0 gives,
That's it ! Pretty simple! SO the general solution is;
Example of using this solution method: Quantum Beats
Physicist JJ studies the wavevector of the object of
his affection, called
which lives in a two-dimensional
vector space. The affection operator
is an observable
which has two eigenvectors, called |SheLovesMe> (or |SLM>
for short) and |SheLovesMeNot> (|SLMN> for short), of
eigenvalues +1 and 0 respectively.
|SLM> = |SLM>
|SLMN> = 0
Show: if an operator has eigenvalues +1 or 0 only it must
be a projection.
Note:
=
so
is a projection
Example of using this solution method: Quantum Beats
JJ wants to cruise through time with the object of his affection.
Fortunately, he knows the Hamiltonian that evolves its
wavefunction
so he can track it and get an idea of
what he can expect of measuring affection as time passes.
The Hamiltonian is
and further, let its eigenvalues be
and
Case 1:
If they commute, SHOW: The |SLM> and
|SLMN> must be eig-vects of
(wolog)
|SLM> = E1 |SLM>
|SLMN> = E0 |SLMN>
Example of using this solution method: Quantum Beats
Case 1: (con't)
SO, If at time t=0, JJ measures
State of the system at t=0 is then
to find 1. The
=|SLM>
Then use
But since |SLM> is an energy eigenstate, this is solved by
Which means that JJ will always measure
Show:
=0 for this state of affairs...
to be 1.
BUT...
Example of using this solution method: Quantum Beats
BUT: actually time and affection are not so kind to JJ...
General Case:
Meaning that the eigenbasis of
say, |1> and |0>,
are different than the eignebasis of
For definitness, take the bases to be 450 apart, so that
Example of using this solution method: Quantum Beats
General Case:(con't)
So now, he again measures affection at time t=0 to find
one ( so
But, converting that to
H-eigenvectors and evolving them gives,
are the energies of the states in frequency
units.
This solution means that after a certain time the two
will accumulate enough relative phase
components of
so that the state will lie along the |SLMN> direction !
Where
So measuring
will give time dependent results!
Example of using this solution method: Quantum Beats
General Case:(con't)
So, as one measure of the change, lets compute the
expectation value of
in time.
In steps, first note:
So that we get:
Stop & Think: What happens to this in the w1-w0 limit and why?
Example of using this solution method: Quantum Beats
General Case:(con't)
So
So, the time-averaged value of the
measurements is 1/2
It clearly can be 0, in which case one must conclude that
we are in the |SLMN> state. Operationally, the
measurement of this observable oscillates in time...a “quantum
beat” at the difference frequency of the states.
Example of using this solution method: Quantum Beats
General Case:(Class Discussion)
Are there measurement protocols that guarantee happier
outcomes for JJ? How can he pick the petals for more 1's?
Uncertainty of
Let us now compute the uncertainty in the measurement
of this observable. Since L is a projection, we simplify our
uncertainty formula to
which becomes,
So that the time average of the uncertainty is
...
General formula for the solution in terms of a
time-ordered product: The Propagator
=
=
U is called the propagator, since it is the matrix that
gives the state at time t given just the state at time 0
Note that it is a unitary matrix:
In the general case the Hamiltonian at one time does not
commute with the Hamiltonian at another time! We write
the rather formal expression for the propagator ;
'T” means the time ordered product, by which we mean;
=
Where