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Course details
Goal is to understand quantum mechanics. Students should use all resources available and may find better sources than provided through course content. I will gladly post additional material on the course website
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- Realignment of view:
Quantum is underlying correct theory.
Classical is the apparent macro scale theory that is completely based on the core or underlying ideas of QM.
While intuition is important and should be developed there needs to be a realization that false intuition needs to be broken down and replaced with quantum intuition.
- Review methods to solve classical problems (based on student questions).
- State of system
- Waves
- Pulse
- Tuning Fork
- Fourier series, Fourier transform
START
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STUDENT APPROACH |
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We need from day one to accept that certain features of our world may be different than what would be predicted based on classical physics. It is certainly surprising that certain relationships and properties of matter are not correctly characterized by classical physics. For example, a perfectly reasonable question in classical physics Particle questions č Where is the ball, atom, electron ? and a reasonable complimentary question č How fast is it going? Wave questions Let us apply similar questions to a sound (tuning fork). č Where is the sound ? and a reasonable complimentary question č What is the frequency? These are well posed classical physics questions that have very different but not surprising answers. Classical particles have the property that they are always localized. So you can explicitly describe the position of a particle to arbitrary precision at any time and simultaneously observe the change and thereby measure a specific precise velocity. Sound however has no such limitation. Sound can be at many locations at the same time. Now wave phenomena is treated in classical physics as a disturbance of a medium. The properties of the medium are the core and a wave is not a basic element. Properties of waves (at least until the advent of light) are understood via the properties of particles. So the first set of questions are directed at fundamental aspects while the second set does not refer to the underlying nature of the theory. Another interesting classical limit enforces forbidden domains. Particles are not allowed to enter a regions based on energy conservation. Thus a barrier can be built to contain a particle [particle in a valley with insufficient energy to climb the walls]. However QM allows for tunneling a process where the particle can temporarily violate energy conservation. In quantum mechanics we no longer have elements in the theory that behave like classical particles. The notion of absolute localization under all situations is not a valid idea. How does one treat a system that behaves in a profoundly different manner? On develops a new paradigm, a new set of rules, and new mathematical description. Student needs to
Student must not
REMBER: QM breaks the rule that elements of the theory, particles, are always localizable. This is completely inconsistent with Classical physics and there is no known way to modify classical physics in some small way and achieve this fundamental breakdown. If you accept that a particle, such as an electron, can have wave-like character and exhibit a spread-out character then you have completely abandoned classical physics. How does the small scale world work when we reject certain classical notions? We need to create a new formalism. QMč NEW FORMALISM THAT IS NOT BASED ON CLASSICAL MECHANICS. If you find yourself saying: “Wait, that doesn’t make sense.” Ask yourself, “Does it make sense for a particle to exist at several locations at once?” You should realize that the second question is the more fundamental and if you accept that premise then you need to find a theory that predicts the answer that particle states are not necessarily localized. Conclusion: vector spaces, operators, commutation relationships, amplitudes… č are the basics and are the necessary and sufficient way to view small scale phenomena. |
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ASIDE: Measuring, seeing, experiencing |
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Seeing [my definition] is a set of observations that lead to a picture or model. Since normal vision is simply the processing of reflected light by the retina and brain there is no special significance afforded this set of measurements as opposed to a microscope view or physics experiment or the data from the LHC. Just as we use our experiences to develop both good and bad intuition as to how systems behave, we can use experimental results and mathematical relationships to guide our ideas and develop pictures for nano-scale phenomena. |
Mechanics
[The book contains a review of basic mechanics. Please read and review these ideas. Questions are welcome as to the underpinnings of classical theory but a formal review of classical physics will not necessarily be part of the lectures. It will be assumed that the student is well versed in these ideas and techniques or will seek help and ask questions.]
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Liboff chapter 1&2 Covered more extensively in class, Students
responsibility |
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Book chapter 1 reviews:
Book Chapter 2 reviews some of the evidence and early ideas
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State of a system
(refining this concept is critical)
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
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 to. 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 
Quantum mechanics will require the refinement of the ideas of what a quantum state will be.
State is the currently active model of a system. It is characterized by some parameters or properties. This is what we know about the system at the present moment. We will expect there to be some set of information that allows us to characterize the state completely. However a complete specification of the state does not mean that all of the possible properties a system can posses are known. It is the old “Can’t have your cake and eat it too” problem. There are mutually exclusive properties. These are well defined characteristics but they cannot simultaneously be well defined for any state. There should be conditions that allow us to determine the state, to prepare a state, and to predict its evolution into other states. QM relies heavily on the notion of states. What a quantum state is and what it means needs to be built from examples, mathematics and probing questions.
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mechanical |
quantum |
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Set of observables: x,p,L,E,V,m |
Set of observables: x,p,L,E,V,m |
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Equation of motion:
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Schrodinger equation: Describes the evolution of your system (time dependence) |
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Solutions to f=ma č orbits |
Sol to schr. Eq č wave function |
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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.
Spring 09 lecture 1 == 5 minutes on waves then done
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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. 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. 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.
It is therefore important to examine certain properties for basic waves.
c is the constant wave speed.
What is the general solution?
Review of waves:
- Wave
equation is basically
- Necessity of a medium
- Medium
is an idealized construct that follows
- 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.
- Linear equation thus any sum of solutions is a solution.
- 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 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.
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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.
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Interference is a
common feature
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Wave speed is a
distinguishing feature.