So this week is somewhat dedicated to reviewing material.

read the chapters!!!!!!

There is some intuition that helps and some that hinders so when you look at a topic spend a minute trying to understand in a bit more depth.

Example: Strength of  an interaction

  1. How is I characterizeè electric forceè electric charge [+,-] or α (fine structure)
    1. all interactions will have a characteristic strength è TRUE
  2. How do the multiple charges at the same point behave?
    1. add as scalars
  3. The interaction between quarks is characterized by QCD charges called color.
    1. red, blue, green, antired, antiblue,antigreen
  4. How do the multiple charges at the same point behave?
    1. add as scalars: FALSE

 

Taking for granted:

The theory of particle behavior is consistent and without basic problems! : FALSE

·       Point particles introduce infinities that are not really resolved in classical physics.

·       As Einstein pointed out the question of what one experiences when riding on a light beam was not completely clear using Galilean relativity.

 

Back to waves

 

Let’s quickly derive the wave equation in 1-d.

  • Shows that a medium is required
  • Shows that the solution falls under the auspices of classical physics
  • Wave behavior is not fundamental it is a “contrived behavior” for complex systems that is based on Newton’s Laws and particles.

 

 

 

String Wave Equation Development

Analysis of the forces on a segment of stretched string gives two relationships:

 

Combining gives

and relating this to the slopes at the segment ends gives

In the limit Δx→ 0 this becomes

 

One of the classic examples of waves and a classic problem that when viewed IDEALLY present many of the basic notions of QM is the two slit experiment.

 

Firstly we develop the problem using Huygen’s principle:

·       There are two equivalent ways to describe the waves impinging on the slit

o      plane waves

o      sum of circular waves

Why would one use circular waves?

Are there any limitations that one can see based on this model?

·       need a point aperture otherwise one needs to include single slit interference as well

 

 The phase difference between the waves is just due to difference in the path length between the two alternate routes to x.

 

 

Thus as you move up the screen the phase difference changes.  The sum of the two waves (assume  ) for a give angle  and a specific time t will be:

 (Note: The details of the mathematics are not as important as the concept but are included for completeness. The point is that the sum of the waves will lead to a classic interference pattern.)

 

If you find magnitude squared of the sum

 

 

The cosine squared dependence shows that the amplitude will change with α.  Two slit interference of waves is a standard example of how waves interfere. The key elements of importance:

  • Wave intensity is the magnitude squared of the wave amplitude.
  • Wave amplitude may change with position and time.
  • The total wave is the sum of all the waves that arrive at the point of interest.
  • Adding amplitudes and finding magnitude squared generates interference.

Note: There seems to be a problem with this derivation in that the cenral region is not brighter than the edges. Why? (Neglected the single slit part!)

 

Consider the classical particle version:

  • Particles are bullets or chunks they come in discrete units.
  • Particles pass distinctly through pone and only one slit.
  • The outcome of an experiment where one slit is closed for 1/2 of the experiment time interval and then the other slit is closed will be the same as for both slits open simultaneously for 1/2 of the time. (At least if the rate of particles is low.)

 

Now before we consider the quantum nature of the two slit experiment.

Vector spaces:

WKIPEDIA

Let F be a field (such as the real numbers or complex numbers), whose elements will be called scalars. A vector space over the field F is a set V together with two binary operations,

  • vector addition: V × V → V denoted v + w, where v, w ∈ V, and
  • scalar multiplication: F × V → V denoted av, where a ∈ F and v ∈ V,

satisfying the axioms below. Four of the axioms require vectors under addition to form an abelian group, and two are distributive laws.

  1. Vector addition is associative:

For all u, v, w ∈ V, we have u + (v + w) = (u + v) + w.

  1. Vector addition is commutative:

For all v, w ∈ V, we have v + w = w + v.

  1. Vector addition has an identity element:

There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.

  1. Vector addition has inverse elements:

For all v ∈ V, there exists an element w ∈ V, called the additive inverse of v, such that v + w = 0.

  1. Distributivity holds for scalar multiplication over vector addition:

For all a ∈ F and v, w ∈ V, we have a (v + w) = a v + a w.

  1. Distributivity holds for scalar multiplication over field addition:

For all a, b ∈ F and v ∈ V, we have (a + b) v = a v + b v.

  1. Scalar multiplication is compatible with multiplication in the field of scalars:

For all a, b ∈ F and v ∈ V, we have a (b v) = (ab) v.

  1. Scalar multiplication has an identity element:

For all v ∈ V, we have 1 v = v, where 1 denotes the multiplicative identity in F.

 

Most of the intuition that you need about vector spaces can be derived from simple 3-d vector space.

 

Change in notation: