postulates of qm part 1: comparing classical and quantum mechanics part 2: complications and...

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.

Ex: Barium atoms made in a nuclear reactor and Barium atoms left over from the earliest stars are exactly the same.

Spectral lines, lifetimes




Observation of matter waves: Davison and Germer

: Spectral lines, lifetimes

Davison and Germer shined an electron beam on cleaned crystaline Nickel and found that the electronscame off at the “Bragg” angle, rather as if they werea form of x-rays. This was in accordance with the deBroglie relation,




Observation of matter waves: Davison and Germer

: Spectral lines, lifetimes

Quantities exist without any classical counterpart: Spin

Spin is like an internal angular momentum that particles possess. It is not actually angular momentumin that no spatial version of spin exists as far as we know, whereas angular momentum will be associatedwith spatial properties of the particles' wave.

Comparison of QM and Classical Physics


QM Classical

Condition of system


QM Classical

Hilbert (complex vector) Space

Phase Space

Condition of system “The State”

QM is a temporal succession of vectors in this vector space. Classical is the motion of

the x,p(t) point in

States can be:

|Barium>

1) States representing a single thing: for example a single barium atom.

=

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 numbersNote 2: The fact that a quantum theory must have facility

with superpositions is the reason that that linear algebrastudied in Chapter 1 is going to be so useful.


QM Classical

Condition of system

Observables/measurement

eigenvalues of

, a linear operator


QM Classical

Condition of system

Observables/measurement

eigenvalues of

NOTE: A list...(could be discrete)

Generally, Continuous Functions

, a linear operator


QM Classical

Condition of system

Observables

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!

, a linear operator

eigenvalues of

Measurement and Observables

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.

Throughout the following pages let the be the

orthonormal basis of eigenvectors of a hermitian operator

with

Measurement and Observables

Ex 0: Suppose we start with a pure state w.r.t.

=

Then, the repeated measurement of while in this state will

and the state vectoralways yield the number

will remain

Ex 1: Mixed state case: the non-degenerate case

Suppose :

The a single, isolated measurement of

return just one value of three possible numbers

will

, , 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+

from a single isolated measurement of . Essentially, this is atomism. In words, no matter how the system wasprepared (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.

In quantum mechanics, as we are teaching it here, a single measurement actively places the state in the eigen-basis corresponding to the eigenvalue measured.

...So if it was a mixed state of a Barium and Yterbiumatom...and you measured it to be Barium...then what?

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,

BUT: Then what does it really mean to be in a mixed state?

...The Barium atom would remain the Barium atom....

Probability and Quantum Mechanics

The probability of a particular measurement outcome is proportional to the norm square of the state's overlap with the associated eigenbasis.

“The Copenhagen Interpretation”

=

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.

(Important note: this formula assumes that both and

are normalized.)

'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

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 stateby the activity of measurement...

of that same observable to again give

This is the so-called 'collapse of the state vector'...in the sensethat 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 completelyobliterated 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-





with the same

eigenvalue

NOW... Suppose that we make a measurement of

and find the value . What is the state of the system after the measurement?





with the same

eigenvalue

NOW... Suppose that we make a measurement of

and find the value . What is the state of the system after the measurement?

ANS:

Averages

By the Copenhagen interpretation, the average

value of the measurement of while the system is in state is given by

=

This is called the Expectation Value of in

Ex 1: For

The expectation value is

= ¼ ¼+ + ½

Ex 2: If the state were

What would be?

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) anexpectation 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 deviationin which you use as the distribution.

Uncertainty: an exampleGiven

Compute the uncertainty on of on this state.

Uncertainty: an exampleGiven

Compute the uncertainty on of on this state.

ANS: Recall the goal is:

And recall from the previous page that

= ¼ ¼+ + ½

SO; only need to figure out expectation value

As an operator, since

=

And, applying to this gives,

2=

So we are now ready to assemble these pieces together into a measurement of the uncertainty;

= [ ]22

-2

= -

=]2[

Non-commuting bases of measurement

Take two hermitian operators and .

The commutator

diagonalizes and (the “compatible operators” case)

Iff Then it is possible to find a basis which

Iff is not zero, then in general the operators cannotbe simultaneously diagonal, called then “incompatible operators”.

Non-commuting bases of measurement

Take two hermitian operators and .

The commutator

In Pictures !

Let:

And take as a starting state the vector

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 b2Then we have

Note this is not the same as first measuring to get b2

And then measuring to get a1

And finally,

...

If the observables are compatible...

=0 So they can be simultaneouslydiagonalized...

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 and

Let . Be the respective probabilities

in a single measurement.

Let be the joint probability of measuring

and then immediately measuring

Then: In general note that:

respectively.

(incompatible case)

Note: There is really no way to reduce the probabilisticnature of quantum reality to probability functions (strictly positive, single valued densities) on classical phase space.

iff (compatible case)

Then


QM Classical

Condition of system

Observables

Reality

Determinism

Given the state at time t=0

and the complete Hamiltonian, one can find the state At any subsequent time with no uncertainty.

Quantum Determinism:

Classical Determinism:

Given the position(s) and momenta at time t=0 with complete precision, and the complete Hamilton,

Specified with complete precision

the subsequent position(s) and momenta are then known at any subsequent time with no uncertainty.

Given the state at time t=0

and the complete Hamiltonian, one can find the state At any subsequent time with no uncertainty.

Quantum Determinism:

Classical Determinism:

Given the position(s) and momenta at time t=0 with complete precision, and the complete Hamilton,

Specified with complete precision

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

QM Classical

x

p)

Poisson BracketCanonical Commutation Relation

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 combinationsof 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.....

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?

Operator Ordering


QM Classical

x

p)


In the oft-used position basis above, these are equivalent to:


QM Classical

x

p)


In the oft-used position basis above, these are equivalent to:

But not everyobservable has a classical version...


QM Classical

x

p)


Spin

?

The Schroedinger Equation

Setting up the equation: Find an operator realization

that captures the physical details of thethe

system.

Often this can be done by promoting the co-ordinatesand 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 inproblems involving

(a) Spatial dimension (b) Multi-particle systems.

Ex: Co-ordinates in 3-d commute!

THUS, the eigenvalues of the position operators mustform 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 wenote only that in the co-ordinate basis this differentialoperator 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

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!

Multi-particle case

Ex: to follow later....

The Schroedinger Equation

Objective : Given

Solve

EX: Time Independent

Step1: Find energy eigenbasis:

with

To find:

Step 2: Study the temporal evolution of the energy eigenbasis:

Let :

Then, the Schroedinger equation become an ordinary DEfor 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-dimensionalvector 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


JJ wants to cruise through time with the object of his affection.

Fortunately, he knows the Hamiltonian that evolves itswavefunction so he can track it and get an idea of what he can expect of measuring affection as time passes. The Hamiltonian is

Case 1:

If they commute, SHOW: The |SLM> and |SLMN> must be eig-vects of

and further, let its eigenvalues be

and

(wolog)|SLM> = E1 |SLM>

|SLMN> = E0 |SLMN>


Case 1: (con't)

SO, If at time t=0, JJ measures to find 1. The State of the system at t=0 is then =|SLM>

Then use

But since |SLM> is an energy eigenstate, this is solved by

Which means that JJ will always measure to be 1.

Show: =0 for this state of affairs... BUT...


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


General Case:(con't)

So now, he again measures affection at time t=0 to findone ( so But, converting that to

H-eigenvectors and evolving them gives,

Where are the energies of the states in frequencyunits.

This solution means that after a certain time the two components of so that the state will lie along the |SLMN> direction !

will accumulate enough relative phase

So measuring will give time dependent results!



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?



So

So, the time-averaged value of the measurements is 1/2

It clearly can be 0, in which case one must conclude thatwe are in the |SLMN> state. Operationally, the measurement of this observable oscillates in time...a “quantumbeat” at the difference frequency of the states.


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 measurementof 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

postulates of qm part 1: comparing classical and quantum mechanics part 2: complications and...

Documents

classical counterpart

dofswhy quantum mechanics

lifetimeswhy quantum

quantum theory

quantum mechanicspart

single barium atom

linear combination

spectral lines