List of important quantities:

Operators

States

Eigenstates

Examples of operators: H, P, X, R, J

Inner product

Vector Space

Dual Vector (Bra-vectors)

Superposition

Basis state

Completeness

Orthonormal

Spin

Groups

Poincare transformation

·       translations of space and time

·       rotations

·       boosts

·       reflections (parity and time reversal)

 

Fourier transform of sound waves

Fundamental character (What is it?)

Symmetry

Unitary transformations (very simple overview)

Generators as observables (very simple overview)

Irreducible representations (very simple overview, comparing ½, ½ , with 1,0)

 

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A basic review of QM can be found in Feynman volume III.  Also visit this good web site that describes the Feynman approach using spin filters.

 

Another excellent review by Abner Shimony appears in the New Physics. He breaks QM into a set of basic postulates summarized here. He also discusses the problem of entanglement and measurement. A brief summary is provided.

 

1

“Associated with every physical system is a complex linear vector space V, such that each vector of unit length represents a state of the system.

2

“There is a one to one correspondence between the set of eventualities (observables) concerning the system and the set of subspaces of the vector space associate with the system, such that if e is an eventuality (observable) and E Is the subspace that corresponds to it, then e is true in a state |S> if and only if any vector that represents S belong to E; and is false in the state S if and only if any vector that represents S belongs to E(orthogonal) .”   A states described by a vector that has components in both subspaces represents a state with an unspecified value for this observable e.

3

“If |S> is a state and e is an eventuality (observable) corresponding to the subspace E, then the probability that e will turn out to be true if the initially the system is in state |S> and an operation is performed to actualize (measure) it.

v is a unit vector representing S … and PE is the projection.”

 

4

“If 1 and 2 are two physical systems, with which the vector spacesV1 and V2 are associated, then the composite system 1 + 2 consisting of 1 and 2 is associated with the tensor product V1 x V2.”

5

“If a system is in a nonreactive environment between 0 -> t, then there is a linear operator U(t) such that  U(t) |v> represents the state of the system at time t if |v> represents the state of the system at time 0. Furthermore, ||U(t) v||2 = || v||2 for all v in the vector space.”

 

 

 

The author (Shimony) is an expert on quantum measurement, EPR, and Schrodinger cat paradoxes.  He carefully explores the basic structure of quantum theories.

 

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This page is under continuing development.

 

| a,b,c,d  >

symbol for the state of a QM system with parameters that have values of a,b,c,d

 

For a classical system how do we define a particle’s state?

 

Simple system consists of a point mass.

mass

x,y,z

vx, vy, vz

 

                        | mass,x,y,z,vx,vy,vz  >

 

 

 

Lets add charge as well.

 

| mass,x,y,z,vx,vy,vz, Q  >

 

Given a particle in a potential well there is an infinite set of possible orbits. The mass and charge of course pick certain orbits by choosing the strength of the potential and of course the inertial response of the particle. The position and velocity specify which of these classical orbits the particle will follow. Thus given the value of these variables and of course the forces (potential), the orbit is determined and the location and velocity at any time can be found.

 

Parameters are continuous (position and velocity) or discrete (mass and charge).

 

 

For a quantum system we will define a state in terms of a subset of observables as we did classically. However these observables will have peculiar properties. There will be classical observables such as momentum and position that cannot be simultaneously used to specify a system.

 

For every observable we define an operator.

[Operators and states are defined in some sense together. An operator has meaning in the way it changes a state. Rotation can be an operation. We understand the rotation operator by what it does to objects. We can even have a mathematical representation of what we mean by rotating using a column vector and a rotation matrix. What is a rotation? Well if you take this book and ….(you can fill in this answer). Mathematically the column and matrix show all of the features that rotations possess. The abstract nature of a transformation and a mathematical representation that has the same character both have important roles in the way particle states are constructed.  Another question that will be relevant is once I have identified and operation what does it do to all the states of my system.]

 

So, for every observable we define an operator and insist that the operation, state and value for the observable behave according to an eigenvector equation.

 

.

 

is an operator for the observable a.

 

The state  is specified by a value of the observable a.

 

Note not all operators have to correspond to observables. Since observables need to be real only a subset of the operators in the theory will be associated with a measurable quantity.

 

The most general way that an operator can act on a state is to transform it to a new state.

 

.

 

This should not be a surprise.  Using rotation as a model for operators, we expect that in general a vector, such as position, will become a different position after rotation. However some vectors may not be changed. Vectors pointing along the axis of a rotation are not changed.  Thus the rotation about an axis does not change the vectors pointing along the axis and so these vectors would satisfy and eigenequation for this specific transformation or operation.

 

 

Our states will be elements of a vector space.  That means several important things. Most of the properties of vector spaces can be understood by considering the well know space of 3-vectors. 

 

In place of the dot product we introduce the inner product

 

.

 

Some care needs to be taken in how you define this product but we will ignore some of the details for now.  The easiest way to treat the inner product is to first choose a basis set that spans the space.

 

Basis vectors are .

 

Therefore any general state can be written as

 

.

 

  =  is the amplitude and tells us how much  there is in our state .

 

Choosing orthogonal and normal basis vectors means that

 

 

 

An interesting application of this general formalism is to construct your states with respect to eigenstates of position. You find the states that have the character that each state describes a particle at a specific location x.

 

Basis:

 

General state

 

If you examine how various operators behave on the state and express the mathematics in terms of the components . You arrive at a representation of QM called the Schrodinger representation.

 

 

A critical feature of the mathematics of operators is that they may depend on the order of application.  The standard example of this property is the rotation of an object like a blackboard eraser. Try rotating the eraser several times. Vary the order of the rotations. You find the final orientation of the eraser indeed depends on the order in which the rotations are carried out. We express this feature or relationship between operators as the commutator.

 

 

If the commutator is zero then the operations can be performed in any order because

 

Some standard observable and there operators are:

 

 

For now we note  is the commutator for position and momentum. This equation determines the relationship between the observables and makes position and momentum mutually exclusive in the sense that both cannot be simultaneously precisely measured for a quantum system.

 

 

Once the basic structure of QM is laid out we turn our attention to important properties of space-time. We review what we mean by the space-time observables:

 

Sound waves can be used to clarify wave character.

A good example of the wave character of a state is the evaluation of a finger snap. You can describe it as a localized pressure disturbance. You can translate that localized disturbance by moving around and snapping. You can allow the disturbance to evolve (sound propagation). You can Fourier transform the snap. Describe it as a sum of sine waves (tuning fork waves). You can explain that any snap in the room is built from the same set of FT states and so there is a linear operator that allows you to translate the system. You can also see that the evolution is not the same as translation but that it also can be expanded in terms of the FT. Therefore time evolution is also a linear operation on the original.

 

We review how physics experiments change when we transform the coordinate system or the experiment.

 

Without proof we write that the space-time observables are generators of the Poincare group. We review all of the associated transformations.

·       Time and space translations,

·       Rotations

·       Boosts

·       Parity

·       Time reversal.

 

We can examine the a representation of the combined system |1/2, m>1 and |1/2, m>2 , two spin ½ particles. Using the combined basis we get a rotation that is represented by 4x4 matrix operating on column vectors. We then choose to represent the system wrt

 |½, ½ ,1, m, > [m = 3 projections of the total spin coupled to value 1 è m= 1,0,-1],

and

 |½, ½ ,0, 0, > [1 state with spin 0 and projection 0]. 

 

Without writing the matrix down you can show that it must be block diagonal by knowing that a spinning object cannot be transformed into a nonspinning object simply by rotating to a new perspective.  So spin 1 combination cannot mix with a spin 0 combination through a rotation. You can rotate a spinning object so that it has a different z-projection (m in the ket).

 

 

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SYMMETRY

We review symmetry and what it means. Change something based on a transformation (rotation). Can you tell ?

 

One can review what judgments and/or qualities one might use to identify or choose a particle structure for a classical fundamental theory. It’s useful to ask what types of objects you would create as the elementary particles of a classical model for our world.  Normally people choose point particles.  Would you include a fundamental wave-like particle? One often doesn’t but finds that wave nature is related to composite structures. Waves become disturbances in a medium made up of fundamental point particles.  Can a point particle possess intrinsic angular momentum? Usually the answer is no. Can you know exactly where a point particle is and exactly its velocity?  Normally particles do have characteristics like mass and charge that are just assigned.  If you construct such a theory are there any inherent difficulties? How much energy does it take to build an electron of charge Q?

 

For particle physics the goal is to see that elementary particles are perhaps best understood as the irreducible representations of the generators of the transformations that possess symmetry.  In writing down a list of players for the standard model one can associate the particle structure | P, M, I, Q > with the group structure of space-time. One must also introduce intrinsic spaces such as isospin (I) that provide the backdrop for particle properties such as upness and downess. Whether these ideas remain as the best starting point is unclear. However it seems that transformations, symmetry, irreducible representations and groups are a strongly favored starting point for the Standard Model.

 

 

 

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