Transcript Quantum Mechanical

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
In each of these different disciplines, the practitioners involved are interested in
two primary goals:
Celestial Mechanics
Continuum Mechanics
Classical Mechanics
Quantum Mechanics
Quantum Statistical Mechanics
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
In each of these different disciplines, the practitioners involved are interested in
two primary goals:
Celestial Mechanics
Continuum Mechanics
Classical Mechanics
Quantum Mechanics
Quantum Statistical Mechanics
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
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).
Quantum Mechanics
Quantum Statistical Mechanics
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
Thus, we will be interested in:
Celestial Mechanics
1) providing a statement of the rules governing the behavior of Quantum
Mechanical systems, and
Continuum Mechanics
Classical Mechanics
Quantum Mechanics
Quantum Statistical Mechanics
2) exploring the predictions that arise from applying those rules to particular
examples of such systems, like atoms and fundamental particles.
Statistical Mechanics
Thus, we will be interested in:
Celestial Mechanics
1) providing a statement of the rules governing the behavior of Quantum
Mechanical systems, and
Continuum Mechanics
Classical Mechanics
Quantum Mechanics
Quantum Statistical Mechanics
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
Thus, we will be interested in:
Celestial Mechanics
1) providing a statement of the rules governing the behavior of Quantum
Mechanical systems, and
Continuum Mechanics
Classical Mechanics
Quantum Mechanics
Quantum Statistical Mechanics
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
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
Classical Mechanics
Quantum Mechanics
Quantum Statistical Mechanics
which describes
both its
configuration
where all the parts are,
at the moment
Statistical Mechanics
Celestial Mechanics
Continuum 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
Classical Mechanics
Quantum Mechanics
Quantum Statistical Mechanics
as well
as 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
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:
1. How to specify the State
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
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:
1. How to specify the State
How, mathematically, in a given form of mechanics, the dynamical state of the
system is specified or represented, in some sense uniquely and completely.
2. The Nature of Observables
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
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:
1. How to specify the State
How, mathematically, in a given form of mechanics, the dynamical state of the
system is specified or represented, in some sense uniquely and completely.
2. The Nature of Observables
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.
3. The Measurement Process
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
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:
1. How to specify the State
How, mathematically, in a given form of mechanics, the dynamical state of the
system is specified or represented, in some sense uniquely and completely.
2. The Nature of Observables
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.
3. The Measurement Process
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.
4. Evolution
How the system passes from one dynamical state to another, i.e., the equations
of motion.
Postulates of Mechanics
Examples:
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 principle
Eigenstates and eigenvalues
The principle of spectral decomposition
Probabilities 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.
2.
3.
4.
How to specify any given dynamical state
Which dynamical states are eigenstates of any given observable
How the spectrum of a given observable is determined, and finally
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.