Mechanics

System’s state is determined by

Briefly reviewed the quantum outcomes for particles hitting the two slit system:

• Slit blocked è expected single bump
• Both open è wave type interference
• Open and a light source (measure the slit by shining light) è expected two bumps, no interference

• Bullet nature
• averaging over many results
• ensemble will be critical to interpreting QM
• start with many prepared systems all in the exact same quantum state
• ask a question about this state by determining the outcome
• NOT A DEFINITE RESULT IN GENERAL
• Perform the experiment many times and get a distribution that characterizes the behavior of the system

QM is the formalism that allows one to develop this type of system evolution !!!

• Requires a new approach.
• Cannot be the same as CM
• Must somehow reduce to the CM result for macroscopic problems

The first and most important point of view is that every prepared system must have a set of things that you know and a set of things that you can measure.

• QM however in some sense distinguishes these two ideas which are the same in CM
• There are a set of parameters that define the quantum state
• There are a set of parameters that can be measured

Since the Quantum formalism is tied to vectors remember:

Vectors have the generic property:

• The relationship in general:

• This relationship is similar to 3-d vectors:

For sound this vector nature results a relationship between duration and frequency. The specific problem is usually revealed through the Fourier representation of a wave. 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 frequencies required is a property of waves. This relationship will govern certain variables in quantum problems.

Again it is a generic feature of vector spaces that a particular entity [wave, vector …] 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.

The properties of a system characterized by the momentum, position or motion shown as the purple vector must be described completely by considering the momentum, position or motion as a sum. In many instances it is more convenient to analyze a system by breaking, for example, the motion into different directions and analyzing the directions and recombine at the end. If we consider sound to be a sum of amplitudes at various locations or we assume a sound is a sum of waves of definite frequency, our conclusions about the system must be the same. These are two equivalent representations of the system.

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.