Einstein convention & vector notation: components, column vectors, unit vectors

Einstein convention repeated indices imply a sum.

Vectors

 

Partial derivative

are any three parameters used to locate a point in space. Typically one chooses .Then

Given a function you find the change in the function for a change in a variable with the other variables chosen held constant.Keeping two of the variable constant defines a step in 3-d space that is unique and well defined. The total derivative of change in a function can then be approximated as the sum of three steps each steps.

Which is exact as the step size goes to zero

Vector calculus

Three vectors

Where momentum, force and position are vectors.

Four vectors

 

In three space invariant is the distance between points

In rotated and translated coorinate systems the distance between two point is a scalar. Also if you choose Galilean transformations (non relativistic velocity transformations) the distance between two points is constant.

And momentum conservation is true so long as you donít change your inertial frame.

 

In 4-vector space-time.

 

The distance between two points is space include the time separation and this distance is unchanged no matter what BOOST is performed.Also a particle with 4-monetum in one frame will have the same rest mass in all frames.

 

Consider the following problem Ť How can we describe the sound in a room?In general we would specify the pressure at every point in the room at some time t0.The wave equation with appropriate absorption and reflection at the walls would determine the sound at some later time.

P(x,t=t0)evolve using the wave equation to P(x,t).

 

Consider the sounds you might hear and record with a microphone sitting in a room at location x0.

 

Snap of your finger or the clap of you hands:this would be localized in time. It is a short duration excitation that quickly disappears.

 

SHORT DURATION SOUND

 

 

Whereas if you struck a tuning fork the sound would continue and by listening you could identify the pitch of the tuning fork.

 

TUNING FORK WAVE

 

There is a relationship between duration and frequency.If a sound pulse is of short duration it cannot be characterized as a single frequency. This is fairly obvious once the wave is so short it doesnít complete even one oscillation. As a matter of fact in order for a sound wave to have an exact frequency the tuning fork must play forever.The theory of Fourier series and/or Fourier transform shows us that the short duration pulse can be adequately described as a sum of sine or tuning fork waves of varying pitch. The shorter the duration the more frequencies need to be added to duplicate the sound profile.So short duration sounds are made up of many frequencies but long duration sounds can be characterized by one. This inverse relationship between duration and frequencies required is a property of waves.This relationship will govern certain variables in quantum problems.

 

Linear Algebra

Generalized vector space

In linear algebra you discover that the set of general functions that are mathematically well behaved can be treated as an infinite dimensional vector space.There are sets of functions, such as sines and cosines, that span the space and are orthogonal [an inner product must be defined in order to specify what orthogonal means].Fourier transforms are one example of this general character.These ideas are central to QM and therefore to aspects of particle physics.The three vector description of space is expanded to n-dimensions where n can be infinite.

 

 

For classical systems wave phenomena is not surprising or unexpected. We understand the nature of waves by considering the response of a medium.The medium carries the wave.The underlying parts of the medium (air molecules in a room) all follow all of the classical rules of mechanics.Waves are aggregate phenomena.One interesting aspect of QM is the need to apply the features or properties of waves to the fundamental elements of the system.The wave function is NOT an aspect of a medium but describes the behavior of a fundamental particle.So picturing a sound wave spread throughout a room is straightforward whereas imagining an electron spread out through an area seems untenable.Never the less understanding QM will require us to take some easily understood classical wave ideas and extrapolate them to particle behavior. This extrapolation leads to ideas that are non intuitive.

 

 

 

SHORT DURATION SOUNDŤ To

SNAP

SINEŤf

TUNING FORK WAVE

 

The point is that these two measurable quantities are exclusive.

 

QMŤ A particle of definite momentum cannot have an observable location.

 

This relationship itself isnít counter intuitive but using it to describe classical particle is counter intuitive. CM is wrong.

 

One can in addition easily imagine that the short duration pulses could be combined to build a sine wave and therefore one can think of a sine wave as compilation of snaps.

Experience with Fourier series tells us that the opposite approach is also valid. A short duration pulse can be built form sine waves.

 

We will rely heavily on this idea that some ways of viewing systems are labeled by mutually exclusive observables but the two approaches are related because each view can be built from the other.

 

 

The Fourier series leads to and inverse relationship between the duration of sound and the frequency interval required to describe the sound.This is a general property of waves.

 

What is more fundamental the frequency representation or the localized representation?

 

In the general formalization of quantum mechanics there emerges a very central and critical idea. There are states that represent the possible arrangement or character of a system.

Mechanics

 

Systemís state [ball in a gravitational field] is determined by †††††

 

 

Forces determine the evolution of the state and the initial conditions are sufficient to determine the orbit.

 

Vectors have the generic property:

 

  • The relationship in general:

  • This relationship is similar to 3-d vectors:

 

For sound this vector nature results in a relationship between duration and frequency.The specific problem is usually revealed through the Fourier representation of a wave.The theory of Fourier series and/or Fourier transform shows us that the short duration pulse can be adequately described as a sum of sine or tuning fork waves of varying pitch. The shorter the duration the more frequencies need to be added to duplicate the sound profile. So short duration sounds are made up of many frequencies but long duration sounds can be characterized by one. This inverse relationship between duration and frequencies required is a property of waves.

 

Again it is a generic feature of vector spaces that a particular entity [wave, vector Ö] can be decomposed into a sum of special waves. In this case we choose the special waves to be sines or cosines.Vector spaces have the property that these sums are equivalent. A momentum vector can be expressed in terms of unit vectors wrt any Cartesian coordinate system I choose. As a matter of fact I do not need to choose orthogonal vectors I can combine any number of vectors in any arbitrary direction so long as they sum to the original vector.††

 

Review of waves:

 

  • Classical waves are a manifestation of the behavior of a conglomeration of fundamental objects moving according to fundamental rules.Waves are not fundamental.Wave theory in this regard is like thermodynamics.
  • Waves on a string, sound, light Ö
  • Waves are:
    • Local
    • Spread out
    • Add via interference rules (constructive and destructive)
  • Wave behavior changes to fundamental when one embraces a Quantum viepoint.

 

  • Wave equation is basically Newtonís laws applied to a arrangement of particles.
  • Necessity of a medium
  • Medium is an idealized construct that follows Newtonís laws for particles. You imagine that at some scale the medium consists of masses (point particles) and springs (forces between the particles).Wave mechanics is based on particle behavior and is not a fundamental aspect of nature.
  • Sine, cosine waves are special waves.A tuning fork is a good example of an almost pure sine wave.
  • Impulses or snaps are also acceptable waves.
  • F(x-vt) is a general function moving along a string with no distortion and is also an acceptable wave.
  • Impulses and sine waves are related. See Fourier transforms or Fourier series.
  • Linear equation thus any sum of solutions is a solution.
  • The relationship in general:

  • This relationship is similar to 3-d vectors: