Quantum One: Lecture 1a

Entitled

So what is quantum mechanics,anyway?

So what is quantum mechanics?

So what is quantum mechanics?

Evidently, it is some form of mechanics….

So what is quantum mechanics?

Evidently, it is some form of mechanics.

You are probably familiar with, or have heard of other forms of mechanics….

So what is quantum mechanics?

Evidently, it is some form of mechanics.

You are probably familiar with, or have heard of other forms of mechanics….

There is, of course, the original, which for a long time was simply referred to as

So what is quantum mechanics?

Evidently, it is some form of mechanics.

You are probably familiar with, or have heard of other forms of mechanics….

There is, of course, the original, which for a long time was simply referred to as

Mechanics

So what is quantum mechanics?

Evidently, it is some form of mechanics.

You are probably familiar with, or have heard of other forms of mechanics….

There is, of course, the original, which for a long time was simply referred to as

Mechanics

but which these days is usually referred to as

So what is quantum mechanics?

Evidently, it is some form of mechanics.

You are probably familiar with, or have heard of other forms of mechanics….

There is, of course, the original, which for a long time was simply referred to as

Mechanics

but which these days is usually referred to as

Classical Mechanics

So what is quantum mechanics?

Evidently, it is some form of mechanics.

You are probably familiar with, or have heard of other forms of mechanics….

There is, of course, the original, which for a long time was simply referred to as

Mechanics

but which these days is usually referred to as

Classical Mechanics

and which seems to come in several different versions:

Newtonian Mechanics

Newtonian Mechanics

Lagrangian Mechanics

Newtonian Mechanics

Lagrangian Mechanics

Hamiltonian Mechanics

Newtonian Mechanics

Lagrangian Mechanics

Hamiltonian Mechanics

You probably have had a course in, or at least heard of, some other forms of mechanics:

Statistical Mechanics

Statistical Mechanics

Celestial Mechanics

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

So, what is mechanics?

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

In each of these different disciplines, the practitioners involved are interested in two primary goals: 1) providing a statement of the rules governing the behavior of a particular class

of dynamical systems, (which we shall define in a moment) and

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

In each of these different disciplines, the practitioners involved are interested in two primary goals: 1) providing a statement of the rules governing the behavior of a particular class

of dynamical systems, (which we shall define in a moment) and

2) Exploring the predictions that arise from applying those rules to particular examples of each class (i.e., solving problems).

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

We will be interested in both of these primary goals of mechanics, as they apply to Quantum Mechanics, which seems to govern (with incredible accuracy) the interaction of matter and radiation at very small energy and length scales.

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

We will be interested in both of these primary goals of mechanics, as they apply to Quantum Mechanics, which seems to govern (with incredible accuracy) the interaction of matter and radiation at very small energy and length scales.

As you perhaps also know, of the various forms of mechanics, Quantum Mechanics is considered a fundamental theory (unlike classical mechanics, whose properties can actually be obtained as an approximation to a quantum mechanical description of nature).

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

We will be interested in both of these primary goals of mechanics, as they apply to Quantum Mechanics, which seems to govern (with incredible accuracy) the interaction of matter and radiation at very small energy and length scales.

As you perhaps also know, of the various forms of mechanics, Quantum Mechanics is considered a fundamental theory (unlike classical mechanics, whose properties can actually be obtained as an approximation to a quantum mechanical description of nature).

Indeed, if a unique Theory Of Everything is ever successfully developed, there is little doubt that it will ultimately be some kind of Quantum Mechanical theory.

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

Thus, we will be interested in: 1) providing a statement of the rules governing the behavior of Quantum

Mechanical systems, and

2) exploring the predictions that arise from applying those rules to particular examples of such systems, like atoms and fundamental particles.

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

Thus, we will be interested in: 1) providing a statement of the rules governing the behavior of Quantum

Mechanical systems, and

2) exploring the predictions that arise from applying those rules to particular examples of such systems, like atoms and fundamental particles.

But we cannot explore the predictions that arise from the rules, until we know what the rules are. Stating these rules clearly and succinctly will therefore concern us a great deal.

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

Thus, we will be interested in: 1) providing a statement of the rules governing the behavior of Quantum

Mechanical systems, and

2) Exploring the predictions that arise from applying those rules to particular examples of such systems, like atoms and fundamental particles.

But we cannot explore the predictions that arise from the rules, until we know what the rules are. Stating these rules clearly and succinctly will therefore concern us a great deal.

Although not the only way to do it, an efficient way of doing this for a given form of mechanics is by stating the rules in terms of postulates, the number and form of which follow essentially from the definition of the term dynamical system.

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

A dynamical system, roughly speaking, is a system with moving parts that may or may not interact with one another, and whose motion we are interested in observing, measuring, and generally describing.

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

A dynamical system, roughly speaking, is a system with moving parts that may or may not interact with one another, and whose motion we are interested in observing, measuring, and generally describing.

At any instant, any such system can be associated with a

dynamical state

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

A dynamical system, roughly speaking, is a system with moving parts that may or may not interact with one another, and whose motion we are interested in observing, measuring, and generally describing.

At any instant, any such system can be associated with a

dynamical state

which describes both its configuration

where all the parts are, at the moment

Statistical Mechanics

Celestial Mechanics

Continuum Mechanics

Classical Mechanics

Quantum Mechanics

Quantum Statistical Mechanics

A dynamical system, roughly speaking, is a system with moving parts that may or may not interact with one another, and whose motion we are interested in observing, measuring, and generally describing.

At any instant, any such system can be associated with a

dynamical state

as wellas its configuration state of motion

how the parts are moving at each particular instant.

Postulates of Mechanics The postulates that define each form of mechanics, either implicitly or explicitly, provide important information regarding the dynamical systems to which they apply. In particular, the postulates of mechanics describe:

Postulates of Mechanics

1. How to specify the State

The postulates that define each form of mechanics, either implicitly or explicitly, provide important information regarding the dynamical systems to which they apply. In particular, the postulates of mechanics describe: How, mathematically, in a given form of mechanics, the dynamical state of the system is specified or represented, in some sense uniquely and completely.

Postulates of Mechanics

1. How to specify the State

2. The Nature of Observables

The postulates that define each form of mechanics, either implicitly or explicitly, provide important information regarding the dynamical systems to which they apply. In particular, the postulates of mechanics describe: How, mathematically, in a given form of mechanics, the dynamical state of the system is specified or represented, in some sense uniquely and completely.

For a given system, what quantities it is possible to observe and measure, at a moment when the system is in an arbitrary dynamical state.

Postulates of Mechanics

1. How to specify the State

2. The Nature of Observables

3. The Measurement Process

The postulates that define each form of mechanics, either implicitly or explicitly, provide important information regarding the dynamical systems to which they apply. In particular, the postulates of mechanics describe: How, mathematically, in a given form of mechanics, the dynamical state of the system is specified or represented, in some sense uniquely and completely.

For a given system, what quantities it is possible to observe and measure, at a moment when the system is in an arbitrary dynamical state.

What happens in an ideal measurement of an arbitrary observable of the system, at a moment when the system is in an arbitrary dynamical state. In particular, what values can be obtained, and what happens to the dynamical state itself as a result of the measurement process.

Postulates of Mechanics

1. How to specify the State

2. The Nature of Observables

3. The Measurement Process

4. Evolution

The postulates that define each form of mechanics, either implicitly or explicitly, provide important information regarding the dynamical systems to which they apply. In particular, the postulates of mechanics describe: How, mathematically, in a given form of mechanics, the dynamical state of the system is specified or represented, in some sense uniquely and completely.

For a given system, what quantities it is possible to observe and measure, at a moment when the system is in an arbitrary dynamical state.

What happens in an ideal measurement of an arbitrary observable of the system, at a moment when the system is in an arbitrary dynamical state. In particular, what values can be obtained, and what happens to the dynamical state itself as a result of the measurement process.

How the system passes from one dynamical state to another, i.e., the equations of motion.

Postulates of MechanicsExamples:

To Illustrate the way in which postulates of this sort can be used to provide the rules for a given form of mechanics, we consider three classical examples.

Lagrangian Mechanics

Hamiltonian Mechanics

Classical Statistical Mechanics

The rules of Quantum Mechanics can also be stated in the form of postulates.

The rules of Quantum Mechanics can also be stated in the form of postulates.

But there are aspects of it that are very odd when looked at from the point of view of classical mechanics.

The rules of Quantum Mechanics can also be stated in the form of postulates.

But there are aspects of it that are very odd when looked at from the point of view of classical mechanics.

These odd aspects, have been deliberately built into the structure of quantum mechanics to make it agree with experimental observations performed on a large (and growing) class of physical systems.

The rules of Quantum Mechanics can also be stated in the form of postulates.

But there are aspects of it that are very odd when looked at from the point of view of classical mechanics.

These odd aspects, have been deliberately built into the structure of quantum mechanics to make it agree with experimental observations performed on a large (and growing) class of physical systems.

So before we state the postulates, we will identify some of the important new features that are essential to any modern version of Quantum Mechanics.

The rules of Quantum Mechanics can also be stated in the form of postulates.

But there are aspects of it that are very odd when looked at from the point of view of classical mechanics.

These odd aspects, have been deliberately built into the structure of quantum mechanics to make it agree with experimental observations performed on a large (and growing) class of physical systems.

So before we state the postulates, we will identify some of the important new features that are essential to any modern version of Quantum Mechanics.

Following this, we will state and explore the postulates for what was perhaps the first "essentially correct" version of quantum mechanics, namely, Schrödinger’s wave mechanics for a single structureless particle.

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

Essential Aspects of Quantum Mechanics

In summary, the five essential new aspects of Quantum Mechanics are:

The superposition principleEigenstates and eigenvalues

The principle of spectral decompositionProbabilities as the squared magnitude of amplitudes, and

Collapse of the dynamical state during measurement

The Postulates of Schrödinger’s wave mechanics incorporate these essential quantum ideas, and supplement them with additional information that addresses:

1. How to specify any given dynamical state2. Which dynamical states are eigenstates of any given observable3. How the spectrum of a given observable is determined, and finally4. The equation of motion: How the dynamical state of the system evolves in

between attempts to make measurements on it.

In the next lecture, we will carefully state the Postulates of Schrödinger’s wave mechanics and begin to explore their implications.

Following this review of Schrödinger’s wave mechanics, we develop and explore a more general set of postulates that describes the quantum mechanics associated with arbitrary quantum systems that can include one, two, or perhaps many interacting or non-interacting particles, and which even apply to more exotic things, like quantum fields.