Review methods to solve classical problems.

Review some mathematical relationships.

• Complex numbers, complex conjugates,
• x+iy
• 

Review some basic physics.

· Angular momentum: This will play a central role in understanding quantum systems.

· Amplitudes: There will be an important development of the quantum amplitude which is related to amplitudes for waves and E&M fields.

· Vectors: Model the more abstract relationship between quantum states.

o Vectors are written as a sum over a basis.

Review some of the phenomena that showed the need to develop QM. This material should have been covered in previous courses. Understanding these phenomena will not be essential for the course.

• Black body radiation
• Photo electric effect
• Atomic spectra

Review some early ideas and introduce some new ideas

• Probability and waves
• Uncertainty principle
• Some two “slit” experiment: There are two idealized examples of quantum effects that will be used:
• Two slit experiment: This is covered extensively in Feynman and in many places on the web and in the text book (e.g. section 2.5)
• Stern Gerlach filters: Idealized systems that split a beam into two (or more components) and allow for recombination or blocking.
• Optical systems that divide split and recombine.

COMMENTS

Mechanics

Consider a simple mechanics problem. Let a mass M₁ be in the vicinity of a stationary mass M₂. The sun-earth system might be such a system. To simplify the problem we consider M₂ >> M₁ so that M2 remains stationary and M₁ will orbit M₂. To solve this problem we can calculate the force

use Newton’s law

and solve for the orbits or the possible trajectories. There are an infinite number of ways the earth can orbit the sun but given the location of the earth and its velocity you restrict the possibilities down to one. Given position and velocity you specify the orbit that the earth will follow. Thus you can find the location of the earth at any time later. In this way position and velocity specify the state or orbit or trajectory of our system.

You might note that rather than using the force I could specify the gravitational potential. I could then find the orbits using a Lagrangian formulation or a Hamiltonian formulation. The Hamiltonian formulation is important because it provided the framework for developing the equations that govern quantum systems. A review can be found in the text book. For classical systems the techniques are equivalent. The results are the same: a set of orbits where the exact orbit is determined if you know position and velocity at some time t_o. The various methods offer alternative approaches that sometimes lead to easier solutions or perhaps provide insight into the way classical mechanical systems behave.

You might further note that velocity could be replaced by momentum.

System’s state is determined by

Qunatum mechanics will require the refinement of the ideas of what a quantum state will be.

mechanical	quantum
Set of observables: x,p,L,E,V,m	Set of observables: x,p,L,E,V,m
Equation of motion: Newton’s Laws in conjunction with a specified force. Describes time evolution.	Schrodinger equation: Describes the evolution of your system (time dependence)
Solutions to f=ma è orbits	Sol to schr. Eq è wave function
Set of observables that pick the solution or determine the initial conditions so that the state is determined (which orbit)	Set of observables HOWEVER quantum systems have restrictions on the measurability of two observables at the same time.

For a classical system you simply choose a host of things to measure and once you have done enough measurements you can calculate all the other interesting observables.

For quantum we find that indeed a particle can have specific location.

Particle at x is in the state or a particle can have a definite momentum p and therefore be in the state.

These two states are acceptable and they are related but cannot be the same state. This is a new aspect of quantum. In classical mechanics a state of definite position is also characterized by a definite momentum. The notion that somehow these are different requires a new formulation of the laws of physics, quantum mechanics.

#!################################################################

Review of waves:

• Wave equation is basically Newton’s laws applied to a arrangement of particles.
• Necessity of a medium
• Medium is an idealized construct that follows Newton’s 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 x_0.

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 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.

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.

#**********************************************************

One difference between the equations of motion for quantum systems as compared to classical waves is the resulting speed of the wave. The wave equation predicts that all waves travel at the same speed. If sound or light propagates in a medium where the speed depends on frequency then we characterize this as a dispersive medium. The wave equation is modified to reflect this type of medium. Solutions to the Schrödinger equation, the equation governing the motion of small systems, are dispersive. Particles can of course travel at different velocities and the equations of motion will maintain this feature at the subatomic level. All photons, light particles, on the contrary will have a constant speed c even in QM. So the equations of motion for light and electrons will differ. This difference will be linked with the particle’s mass.

Schrödinger equation is not the classical wave equation. Some features of the solutions will be similar to the wave equation but some features will be different.

· Interference is a common feature

· Wave speed is a distinguishing feature.