Waves, classically waves are not fundamental but they have a mathematical description that could be fundamental.  One typically derives the wave equation for a disturbance on a string by examining infinitesimal mass elements and the forces between them. 

 

see  http://hyperphysics.phy-astr.gsu.edu/hbase/waves/waveq.html

 

 

However if you ignore the source of the mathematics and just address the character of the equations you find that these wave elements have interesting properties.  How does something that satisfies the wave equation behave?  Maxwell’s equations were perhaps the first indication that waves might not require a classical medium.  The ether approach never produced a satisfactory description and claiming that the fields themselves were the medium doesn’t really fix the problem and restore waves to the role of secondary phenomena based on an underlying medium.  The fields are abstract. What is it that exists between the sun and the earth that allows a light wave to propagate?  Quantum mechanics, as we will see, provides a basic element that can describe light and electrons within the same framework. It places these two observable phenomena as manifestations of a single mathematical entity, the quantum state, but with different properties. Feynman at the start of his lectures reports the good news and the bad news.

čgood: light and particles are the same thing

čbad: the underlying theory is confusing to people who live in a classical world

 

It is therefore important to examine certain properties for basic waves.

 

c is the constant wave speed.

 

One approach to solving this problem is to use the separation of variables technique.

Assume

 

 

that the solution is a product of functions that each depend on only one of the variables.

divide by

for simplicity let’s continue in 1-d

If a function of x is equal to a function of t for all values of x,t then that function must be a constant. We pick

Following this method further one arrives at the solution

 

We arrived at some solutions for the wave equation.  This is not the most general treatment but a general solution can be built from these solutions.  This introduces us to basic elements of QM.

 

Lets be sure that the mathematics discussed so far is clear.

 

 

 

Some aspects of sound are reviewed.

 

 

SHORT DURATION SOUND  č To

SNAP

SINE  č  f

TUNING FORK WAVE

 

 

Consider a sound 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.

 

 

 

 

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

 

 

 

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 doesn’t 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 frequency is a property of waves.  This relationship will govern certain variables in quantum problems. 

 

Indeed a general wave could be produced by combining snaps or combining tuning forks. So sound represents a manifestation of how complimentary variables, position in time and frequency, are basic characteristics of the sound but posses an exclusivity.  For sound this not a surprise but a natural relationship.

 

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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