Waves

Quantum mechanics will develop a new framework for the description of matter and fields.Often the new framework is developed using the familiar concepts of waves and particles of classical physics.Of course, QM being the more fundamental theory, the best way of understanding would be have an intuition at the quantum level as to how matter and energy behave and then extrapolate that to the large scale level.This of course is not really possible because very few people have quantum experience as undergraduates.So we review how classical waves behave. We try to understand in what context they appear classically. The we can use classical wave behavior as a guide but at the same time realize that classical waves and quantum wave-like behavior are not at all the same thing.Classical waves are due to a medium of things that do not behave at all like a wave. Quantum wave-like behavior is just how things behave and all matter will act according to these rules.

 

  • 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.
  • Sound
  • Waves are:
    • Local
    • Spread out
    • Add via interference rules (constructive and destructive)

 

Review methods to solve classical problems.

Review some mathematical relationships.

  • Complex numbers, complex conjugates,
    • x+iy

 

Review of waves:

  • Wave equation is basically Newtonís laws applied to a arrangement of particles.

    • Vector calculus
    • Partial derivatives
  • 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.
  • The relationship in general:

  • This relationship is similar to 3-d vectors:

 

Consider a different problem.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.

 

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.

 

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.

 

Interference is a general property of waves.

 

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