A
not-so-famous athletics coach once said that “Because of the rotation of the
Earth, an object can be thrown further if it is thrown to the west.”
Using your Preliminary Course knowledge of vectors,
determine whether
this statement is an accurate description of the real world? Explain.
DATA:
The Earth spins on its own axis from west to east at about 1833 km/h at the
equator. An average, red-blooded
Australian can throw a cricket ball at about 100 km/h, if he/she really wants
to. Giant cephalopods are quite
dangerous and are to be avoided at all costs.
Go
to Answer
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GRAPHING EXERCISE
You sketch lots of great graphs in most mathematics
courses. Have a go at these.
Plot graphs to
show the main features of the following functions and relations:
a.
y = x^{3} –
3x^{2} – 4x
b.
x^{2/3} + y^{2/3}
= 2^{2/3}
c.
y = 1.5cos(x) + 6e^{-0.1x}
+ e^{0.07x}sin(3x)
Go
to Answer
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HOLLYWOOD PHYSICS
Warning: This
activity analyses the Physics of movie scenes where someone gets shot & is
subsequently thrown backwards through the air as a result of the impact.
It is NOT meant to glorify such action. It is simply designed to get
students to do some thinking about momentum.
We have all seen it. The bad-guy in the movie pulls a gun and blows
some dude away. The dude is thrown backwards, lifted off his feet and
crashes through the glass shop-front window. Great movie action!
What about the Physics?
Imagine that the scene in the movie is as follows: The
bad-guy is stationary and fires his gun at point blank range (ie extremely
close) to his stationary victim's chest. The bullet has a mass of 0.025
kg, leaves the barrel of the gun at a speed of 500 m/s and comes to rest inside
the victim. The victim has a mass of 100 kg. Assume that all the
momentum of the bullet is transferred to the victim. Determine the speed
of the victim's body as a result of the impact of the bullet. In the light
of your calculations, comment on the accuracy of movie scenes such as that
described above.
Go
to Answer
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BEAM ME UP SCOTTY
Most aspiring Physicists on planet Earth would recognize the command "Beam
me up Scotty". Star Trek's famous "transporter" could
deconstruct people into their sub-atomic essences, send that information over a
given distance and then reconstruct the people from that information at their
destination. What a great way to travel!
Those of you who are studying the "From Quanta To Quarks" Option
will come across the work of Werner Heisenberg, notably the Uncertainty
Principle. How might this principle impact on the
possibility of a Star-Trek-type transporter?
Have you heard of quantum entanglement? How
might this phenomenon enable Physicists to achieve quantum teleportation?
Go to a discussion of these questions.
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Schrodinger’s Cat – the
Paradox
On June 7 of 1935, Erwin Schrodinger wrote to Albert
Einstein to congratulate him on what is now known as the EPR paper, a famous
problem in the interpretation of Quantum Mechanics. Soon thereafter, he
published what was to become one of the most celebrated paradoxes in quantum
theory:
Schrodinger's Cat
A cat is placed in a box, together with a radioactive atom. If the atom
decays, and the geiger counter detects an alpha particle, the hammer hits a
flask of prussic acid (HCN), killing the cat. The paradox lies in the clever
coupling of quantum and classical domains. Before the observer opens the box,
the cat's fate is tied to the wave function of the atom, which is itself in a
superposition of decayed and undecayed states. Thus, said Schrodinger, the
cat must itself be in a superposition of dead and alive states before the
observer opens the box, "observes" the cat, and "collapses" its wave
function. In other words, before
the observer opens the box and observes the cat, the cat is BOTH dead AND alive
at the same time. Cute, eh?
What are your thoughts on the matter?
Go to a discussion of this paradox.
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COMPLEX
NUMBER MAGIC
Complex numbers (imaginary numbers) are numbers of the form:
a + ib, where a and b
are real numbers and i is
the square root of minus one.
Not only are complex numbers extremely useful in
the subject of mathematics, they play an essential role in many areas of
Physics - Electromagnetic Theory, Electronics, AC Circuit Theory, Atomic
& Nuclear Physics, Quantum Physics, Wave Theory and Signal Analysis
just to name a few. In many situations in Physics, the real
solutions to a problem would be inaccessible without the use of complex
numbers. For instance, Schrodinger's Equation, the basic equation
of wave mechanics expressing the behaviour of a particle moving in a
field of force, contains the imaginary number i
(see below).
The following little
exercise is probably best attempted by Extension 2 Maths students after
learning about complex numbers.
The polar form of any complex number w can be
written as:
w = r e^{i}^{q}
The circle with r = 1 is called the unit circle in the complex plane.
All complex numbers lying on this circle are of the form:
w = e^{i}^{q} ,
for q real.
Using the Cotes-Euler formula
e^{i}^{q} =
cosq + i sinq for real q.
This equation encapsulates the essentials of trigonometry in the much
simpler properties of complex exponential functions. Use this
equation to show that:
cos(A + B) = cosAcosB - sinAsinB
sin(A + B) = sinAcosB + cosAsinB
cos3q = 4cos^{3}q
- 3cosq
sin3q = 3sinq -
4sin^{3}q
Go to Answer
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DIOPHANTUS' AGE PROBLEM
The Greek mathematician Diophantus is generally
regarded as the "Father of Algebra". Diophantus first used algebraic
notation and symbols around 250 AD. He wrote a treatise on algebra in his
"Arithmetica", comprising 13 books. Only six of these books have survived.
A mathematician to the very end, Diophantus'
age can be determined from the epitaph on his tombstone. An
English paraphrase of this epitaph reads:
"Diophantus passed one-sixth of his life
in childhood, one-twelfth in youth, one-seventh more as a bachelor; five
years after his marriage a son was born who died four years before his
father at half his father's final age."
How old was Diophantus when he died?
Go to Answer
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SHARING BIRTHDAYS
PROBABILITY PROBLEM
Imagine you are at a party with 22 other people. What is the
probability that any two of the 23 people at the party will share the
same birthday? Assume 365 days in the year.
Go
to Answer
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SOLUBILITY
QUESTION - One for the Chemists
So, you think you are
pretty good at Chemistry eh? Well
try this little sizzler.
The
solubility product of MCO_{3} in water is 25 times as great as
the solubility product of MC_{2}O_{4} in water at the
same temperature. A mixture
of K_{2}CO_{3} and K_{2}C_{2}O_{4}
totalling 2.042 x 10^{-3} mole was dissolved in water.
To this solution was added 2.80 x 10^{-3} mole of MCl_{2}.
Water was then added to give a total volume of one litre.
When precipitation was complete a mixture of MCO_{3} and
MC_{2}O_{4} totalling 2.000 x 10^{-3} mole had
precipitated. Calculate the
solubility product for MC_{2}O_{4}.
Note:
In case you are ranting and raving that this is just too hard and is
obviously university standard – think again.
This was question 2 from the HSC First Level Chemistry Exam back
in 1974. Dear me, how
standards have changed.
Go to Answer - Note
that the answer is provided in pdf format. You will therefore need a
suitable pdf file reader such as Adobe Acrobat reader.
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Time
Paradox
The
following is taken from Michio Kaku's excellent book titled "Parallel
Worlds".
There are several different types of paradox that
can occur when one considers the possibility of time travel. Most
people are familiar with the "Grandfather Paradox" in which
you alter the past in a way that makes the present impossible - eg you
shoot your grandfather when he was just a boy, thus ensuring that your
father was never born and therefore that you were never born.
Then there is the "Information Paradox" where information
comes from the future, which means that it may have no origin. The
"Terminator" movies hinge on an information paradox - a
microchip recovered from a robot from the future is studied by
scientists, who then create a race of robots that become conscious and
take over the world. The design for these robots was never created
by an inventor; it simply came from a piece of debris left over from one
of the robots from the future.
"Bilker's Paradox" is where a person knows what the future
will be and does something that makes the future impossible - eg you
know you are going to marry a particular person, so you deliberately
decide to marry a different person instead.
The "Sexual Paradox" is perhaps the most interesting of all
the different types of time paradox. In this paradox, you go back
into the past and father yourself, which of course is biologically
impossible. In a science fiction story written by Jonathan
Harrison in 1979, a young woman by the name of Jocasta Jones finds a
freezer. Inside the freezer she discovers a handsome young man
frozen alive. She thaws him out and learns that his name is Dum.
Dum tells Jocasta that he has a book that tells how to build a freezer
that can preserve humans and how to build a time machine. Jocasta
and Dum fall in love, marry and have a son, whom they call Dee.
Years later, when Dee has grown to be a young man, he follows in his
father's footsteps and builds a time machine. This time both Dee
and Dum take a trip into the past, taking the book with them.
Sadly, they find themselves stranded in the distant past and running out
of food. Realizing that the end is near, Dee does the only thing
possible to stay alive. He kills his father and eats him.
Dee then follows the instructions in the book and builds a
freezer. To save himself, he enters the freezer and is frozen in a
state of suspended animation.
Many years later Jocasta Jones finds the freezer and thaws its
occupant. To disguise himself, Dee claims to be Dum. They
fall in love, marry and have a baby, whom they call Dee. And so
the cycle continues.
Clearly, there is a serious biological paradox here. Since half
the DNA of an individual comes from the mother and half from the father,
this means that Dee should have half of his DNA from Jocasta Jones and
half of his DNA from his father, Dum. However, Dee is Dum.
Therefore, Dee and Dum must have the same DNA because they are the same
person. But this is impossible, since by the laws of genetics,
half their genes come from Jocasta Jones.
Weird eh?
Top of Page
CHOCOLATE
MATHEMATICS
2017
An
interesting piece of mathematical fun that has appeared on the Internet
in recent years is something called "Chocolate Maths".
While it is fun just to follow the instructions below and marvel at the
fact that it works, it is even more fun to explain mathematically how it
works. It is a good exercise in thinking!
Here we go:
1.
First of all, pick the number of times a week that you would like to have
chocolate. (try for more than once but less than 10)
2.
Multiply this number by 2 (Just to be bold)
3.
Add 5. (for Sunday)
4.
Multiply it by 50
(I'll
wait while you get the calculator)
5.
If you have already had your birthday this year add 1767.... If you
haven't, add 1766
6.
Now subtract the four-digit year that you were born.
You
should have a three-digit number
The
first digit of this was your original number
(i.e., how many times you want to have chocolate each week).
The
next two numbers are .......
YOUR
AGE! ~
(Oh YES, it IS!!!!! )
(Note:
If the last two numbers are not your age, then you have made a mistake
with your calculations. This algorithm DOES WORK! So, try
again.)
On
the Internet this set of instructions often ends with words to the
effect of:
THIS IS THE ONLY YEAR
IT WILL EVER WORK, SO SPREAD IT AROUND WHILE IT LASTS.
TASK:
Explain mathematically how this mathematical
novelty works and comment on whether it will only work this year.
Go To Answer
Top of Page
Answer
to parachute hole question: By
allowing the air to leak slowly out of the top of the parachute, the hole
stabilizes the parachute
and allows for much safer landings than old-style parachutes without the
hole. The old-style parachutes produced a backwards and forwards
pendulum-like motion on the parachutist caused by air leaking out from the edges
of the chute. As the air leaked from one edge, the chute tilted, throwing
the parachutist to one side. As the chute swung back, more air would leak
out from the opposite side, setting up the pendulum motion.
The
hole in the top of the chute also slows down the opening of the chute, causing a
more comfortable opening, especially for male parachutists.
Top of Page
Answer
to Flight of Fancy question: Throwing to the east:
Ball has velocity of 1933 km/h east and the earth a velocity of 1833 km/h east,
giving a velocity of the ball relative to the earth of 100 km/h east.
Throwing to the west: Ball has velocity of 1733 km/h east and the earth a
velocity of 1833 km/h east, giving a velocity of the ball relative to the earth
of 100 km/h west. Clearly, the coach's belief is not an accurate
one. Whether thrown to the east or west, the ball will travel the same
distance in the same time. (Note that as your Physics knowledge develops,
you may like to re-consider this problem.)
Top of Page
Answer
to Graphing Exercise: The following are the graphs you were asked to
sketch.
Note that the horizontal red line y = 3.1596 in the third graph is the mean
value of the function.
Top of Page
Answer
to Hollywood Physics question:
Clearly, the momentum of the bullet is given by
p = m.v
p
= 0.025 x 500
p
= 12.5 Ns (or kgm/s)
If all of this momentum is transferred to the victim, we have
12.5 = 100 x v, where v = speed of victim after impact of bullet
v = 0.125 m/s
So, we have that the speed of the victim after the
impact of the bullet is 0.125 m/s or about 0.45 km/h.
Clearly, 0.45 km/h is not a very high speed. The speed of a person
walking for exercise can be as much as 10 km/h. It is therefore very
unlikely that a person shot as described would be lifted off his feet and thrown
backwards. Thus, the Physics of such scenes in movies is often inaccurate.
Thinking about things from another perspective - when a gun fires a bullet,
the gunpowder in the cartridge explodes and vaporizes. The gas produced
expands rapidly and applies a force to the bullet, pushing it in the direction
in which the gun is pointing. By Newton's Third Law, the bullet applies an
equal but oppositely directed force on the gun and therefore on the
gunman. This is called the recoil of the gun. It stands to reason
that the force the bullet applies to the target can at the very most be equal to
the force applied backwards on the gunman. Otherwise, the bullet has
gained extra momentum from some unknown source. So, if the gunman is not
lifted off his feet and thrown backwards, neither is the victim.
Careful analysis reveals that in reality, the force that the bullet applies
to the target is always less than the force it applies backwards on the person
firing the gun. You might like to think about why this is so.
Note that the stimulus for this Brainteaser came from Dr Karl Kruszelnicki's
book "Dis Information". Also
have a look at the Physics Web article
"Hollywood Physics".
Top of Page
Discussion
related to "Beam Me Up Scotty":
In one form, Heisenberg's Uncertainty Principle states that it
is impossible to know both the position and the momentum of a quantum
particle (eg an atom, electron etc) simultaneously. In attempting
to measure say an electron's position, the technique we use must
interact with the electron to obtain the measurement. In so doing,
we alter the velocity of the electron and therefore its momentum.
So, after the measurement we may know more about the electron's position but
less about its momentum.
Clearly, the Uncertainty Principle seems to forbid the possibility of
a "transporter". Whenever you make a measurement on an
atom, you disturb the quantum state of the atom, so how can you possibly
make an exact copy of that atom?
Quantum entanglement is a phenomenon in which the quantum
states of two or more objects have to be described with reference to
each other, regardless of how far apart the objects may be
spatially. This means that there are correlations between the
observable physical properties of the objects.
Let's consider an example. Experimentally, it is possible to prepare
two particles, say two electrons, in a single quantum state such that their
spins are correlated. In other words, when one electron has its spin axis
pointing up, the other electron has its spin axis pointing down, so that the
total spin is zero. Until we make a measurement however, we do not know
which electron is spin up and which is spin down.
Now imagine that the two electrons move away from each other at close to the
speed of light for several years. By then the two electrons are many light
years apart. Now, if at this time we make a measurement of the spin of one
of the electrons, we see the startling consequence of entanglement. If we
find that the measured electron has spin up, then instantly we know that the
other electron has spin down. Think about this for a moment. By
measuring the spin of one electron, we instantly determine the spin of the other
electron, even though it is many light years away. Information seems to
have travelled faster than the speed of light, in apparent
contradiction of Special Relativity Theory. Albert Einstein, Boris
Podolski and Nathan Rosen proposed this very scenario back in 1935 as a
demonstration that quantum mechanics must be wrong, since it clearly led to
absurd results. This scenario became known as the EPR Experiment and was
the focus of much philosophic and scientific investigation over many years.
The EPR Experiment was actually performed for the first time in the 1980's by
Alan Aspect et al and has since been performed several times over distances as
great as 11 kilometres. The results show that the predictions of
quantum mechanics are accurate. Yes, if you do measure the spin of one
entangled particle, you instantly know the spin of its entangled twin regardless
of how far away that twin is. Entangled particles act as if they are one
single object. What happens to one particle automatically affects the
other, regardless of the distance between them.
Please note, although Einstein was not correct about the implications
of the EPR Experiment for quantum mechanics, he was still correct in
saying that communication cannot occur faster than the speed of
light. The EPR Experiment does allow you to know something
instantly about the other side of the galaxy but it does not allow you
to send a stream of useful information (that is, to communicate) in this
way. You cannot, for instance, send a short-wave radio message in
this manner. There is a distinct difference between knowing a
piece of information and communicating it to someone else. Given
the right piece of evidence you can know a particular fact instantly but you cannot communicate
that information from one point to another faster than the speed of light.
Now, getting back to the "transporter" idea - In 1993,
Physicists suggested that it might be possible to perform some sort of
Star-Trek-type transportation using quantum entanglement. The
technique is called quantum teleportation and aims to send a
quantum state from one place to another. This was first
experimentally demonstrated
in 1997 when the quantum state of a single photon was teleported across
a table top. In 2004 the quantum characteristics of an entire atom
were successfully teleported from one place to another.
How does this work? Let us consider two eminent Physicists,
Angela and Rebecca. Angela & Rebecca produce a pair of
entangled electrons - electrons 1 & 2. Rebecca takes electron
2 to the Moon, while keeping it completely isolated from any other
influences. Angela remains on Earth with electron 1, which she
also keeps isolated from all other influences. One day, Angela
wishes to teleport the unknown quantum state of a third electron -
electron 3 - to Rebecca on the moon. To do this, Angela firstly
allows electron 3 to interact with her entangled electron 1 and measures
the quantum state of the combined system of two particles. This
measurement causes Rebecca's twin entangled electron 2 on the Moon to
adopt a quantum state resembling electron 3. Now in order for
Rebecca to "know" the quantum state of electron 3, she needs
to be informed of the result of the measurement of the quantum state of
the combined electron 1 & 3 system. It is this result that
tells Rebecca how to adjust the state of her electron 2 to obtain an
exact copy of the state of electron 3. Once this is done, the
quantum state of electron 3 has been teleported from Earth to the Moon.
If you would like a more detailed explanation of the actual process
of quantum teleportation, Chapter 23 of Roger Penrose's "The
Road To Reality", specifically section 23.9 on Quantum
Teleportation, is an excellent reference. There are also some good
references on the Internet. Search for quantum entanglement,
quantum teleportation or Bell state measurements for a start.
A few points of clarification are needed! Firstly, note that
quantum teleportation is not a means of transferring physical systems
from one point to another, but rather the information encoded by those
systems. Secondly, note that
the original quantum state of electron 3 in our example, is destroyed in the
process. It is combined with the quantum state of electron 1
during the interaction of those particles. What you end up with at
the other end is an exact copy of the quantum state of electron 3. These two
points taken together imply that if we ever get to the stage of
teleporting a whole person, we will actually be teleporting the entire sum
of quantum states that make up that person NOT the physical person
themselves. In other words, each time Captain Kirk is to be
teleported, he will be scanned to obtain his complete quantum state
information, destroyed in this process and then reassembled from
appropriate raw materials at the other end. Still want to try
this?
Thirdly, the teleportation process must occur at less than
the speed of light. Even though quantum entanglement is involved
here and in our example, the measurement of the quantum state of the
combined electron 1 & 3 system instantly affected the quantum state
of electron 2 on the Moon, Rebecca could not produce the final copy of
electron 3's quantum state until Angela informed her of the result of
her measurement. This would take place for instance via computer,
say by email, or by some other classical (ie non-quantum) communication
channel. It is this fact that ensures that the whole teleportation
process must occur slower than light speed and is therefore not in
contravention of Special Relativity. In terms of the
teleportation of human beings, this means that as quantum state
information was determined for the human being teleported, it would have
to be communicated via a classical communication channel to the device at
the other end, to enable the human to be re-assembled. So, if we
wished to teleport someone to the mars, the quickest it could be
accomplished is about five minutes - not instantaneously!
Now, to what is probably the biggest barrier to the teleportation of
human beings. Humans are incredibly complex systems. One of
the Physicists who actually worked on the first successful teleportation
of the quantum state of a photon, Samuel Braunstein of the University of
Wales, has calculated that even if all the technical barriers to the
teleportation of humans could be overcome tomorrow, it would take at
least the age of the universe just to teleport all of the quantum
information that describes just one human being.
So, I suppose the "Beam-me-up-Scotty Transporter" will
remain in the realm of science fiction for some time to come.
The above discussion was prepared using information contained in
Roger Penrose's "The Road To
Reality" chapter 23 and Michio Kaku's "Parallel
Worlds" chapter 6.
Top of Page
Discussion of
Schrodinger's Cat:
Be aware that Schrodinger's Cat is one of the most
famous & thought about paradoxes in Physics and Philosophy. It
raises many deep questions about the nature of the real world and what
we can know about it. Indeed whole PhD Theses have been written in
an attempt to answer the questions raised by the paradox.
The traditional question to ask at the end of the
Schrodinger cat story is: “Is the cat dead
OR alive or dead AND alive before
we
open the box to check?”. At
the simplest level Schrodinger’s cat is meant to point out to us that
the quantum mechanical answer to this question is that the cat is
BOTH dead AND alive at the same time until we perform a measurement
on the system by opening the box and observing the cat.
Strange, eh? Quantum
mechanical systems (eg atoms, nuclei, quarks etc) actually exist in all
possible states at the same time until we make a measurement on
the system and in so doing “collapse” its wave function, that
is force it to adopt a particular state at that particular time. The wave function of a system is the mathematical equation
that contains all the information about the system that is knowable.
When we speak of a system being in all possible
states simultaneously we say that the system exists in a “superposition”
of states. For example,
let’s consider the spin property of electrons.
An electron can spin up or spin down.
Since electrons are quantum mechanical systems, this means that
all the electrons in each atom of your body, for instance, exist in both
the spin up and spin down states simultaneously.
The only way we can say that a particular electron is in a
particular state is if we measure it at a particular time and
force it to adopt a particular state of spin.
Until then the electron exists in BOTH states at the same
time.
On a deeper level, Schrodinger’s cat asks the
question “At what stage does a system become too big or too complex to
exist as a whole in a superposition of states?”.
For example, our experience of the macroscopic world tells us
that even before we look in the box, the cat is EITHER dead OR alive. Opening the box (measuring the system) cannot force the cat
into one state or the other. Our
experience tells us that the cat is already in one state (dead) or
the other (alive). We
know even before we open the box, the cat can never have been BOTH
dead AND alive in our macroscopic world.
The famous physicist Niels Bohr (1885-1962)
did a lot of work trying to establish where along the chain of
increasing size from atoms to macroscopic objects did it become
unnecessary to use quantum mechanics to give an accurate description of
the state of a system. To
cut a long story short, Bohr came to the conclusion that for systems of
the order of the size of the atom (10^{-10}m) and smaller, quantum
mechanics is essential to accurately describe the system.
For systems bigger than this, ordinary “classical” physics
is adequate to calculate the state of the system.
This idea is called the Correspondence Principle.
So, for anything as big as a cat, although the
atoms of which it is composed will be in all manner of superposition of
states, the actual macroscopic entity that we call the cat always and
only exists in one definite state (dead) or in the other (alive) –
never BOTH at the same time. For
our cat, or any other macroscopic entity, ordinary classical physics is
capable of accurately describing any action that it may (or may not)
perform.
For
those who would like even more detail, read the next section.
Top of Page
The Domain of
Quantum Mechanics:
The following is a
summary of pages 8-11 of Basic Quantum Mechanics by R. L. White
(McGraw-Hill, 1966) – an excellent text commonly used in third year
university Physics courses on quantum mechanics.
How do we know precisely when the use quantum
mechanics is essential? An
accurate criterion for establishing the appropriate domain for quantum
mechanics is based directly on Planck’s constant, h = 6.626 x 10^{-34
}joule-second. Planck’s
constant has the dimensions of energy x time, a quantity known in
classical physics as action.
The dimensions of action can be shown to be the same as momentum
x distance, which is another physical quantity called angular
momentum.
In terms of Planck’s constant, our criterion for
the use of quantum mechanics can be stated as follows:
If the action or angular momentum involved in a
given physical event is of the order of Planck’s constant, then we
must use quantum mechanics to describe the event accurately.
If the action or angular momentum involved in a given physical
event is orders of magnitude larger than Planck’s constant, classical
mechanics will describe the event with satisfactory accuracy.
Quantum mechanics can be used to describe systems
where classical mechanics is valid, but the effort involved would
usually be substantially greater than that required to describe the
system using classical physics. The
reverse, however, is not true. In
general, classical mechanics cannot be extended to provide accurate
descriptions of atomic or smaller systems.
The difference between quantum and classical
mechanics is much more profound, however, than just that of scale. Classical mechanics is deterministic.
Given a completely specified initial state, a set of forces and
constraints, the values of the various system parameters can be
determined at any subsequent time with arbitrary precision.
Quantum mechanics, on the other hand, is
fundamentally a probabilistic theory.
Where classical mechanics will predict a specific result for the
measurement of some dynamic quantity (momentum, velocity, etc), quantum
mechanics will predict that any of several values might be the result of
an accurate measurement of the quantity involved and will assign a
relative probability to the various results.
In some instances the “allowed” results may be grouped
closely together, and the coarseness of the measuring apparatus or the
large size of the dynamic system may mean that it is impossible to tell
the difference between the prediction of the classical theory and the
quantum theory. In this
limit, called the correspondence limit, the two theories merge,
as they must.
The basic philosophic difference between classical
mechanics and quantum mechanics can be understood as follows: All
physical theory is directed at explaining the results of experiments and
at correlating these results in the simplest and most aesthetically
pleasing manner possible. Fundamentally
the theory must explain the results of measurements.
In the classical limit, the measurement on a system can in
principle be made so as to produce a vanishingly small reaction upon the
system. The interaction of a physical system with the measuring
apparatus is a physical event, which must involve action in the
amount of at least Planck’s constant, h, if any change of state of the
measuring apparatus is to be effected – that is, if any reading is to
be obtained. For large
systems the resultant alteration of the total state of the system
(itself involving for example 10^{30 }h) is relatively so small
as to be entirely negligible. So, the measurement is presumed to have no effect whatsoever
upon the state of the system being measured.
For atomic systems, however, the action
involved in the measurement is of the same size as the action
involved in the system’s own evolution.
By making a measurement on such a system, we inevitably
interact with the system in such a way as to significantly alter the
state of the system. The
alteration of the system by the measurement can be treated only on a
statistical or probabilistic basis.
The theory that deals with the results of such a statistical
measurement must also be statistical in character.
It is also worth mentioning here that a statistical
theory is not an intrinsically inaccurate one.
While quantum mechanics cannot generally predict unambiguously
the result of a single measurement on a single atom, it can generally
predict the distribution of results of a large number of measurements.
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Answer
to Complex Number Magic:
To derive the first two trigonometric identities start with:
e^{i(A + B)} = e^{iA} . e^{iB}
and proceed from there.
To derive the three theta identities start with:
e^{i3q} = (e^{iq})^{3}
and proceed from there.
If you have difficulty with this exercise, I suggest you ask your
Maths Teacher for assistance.
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Answer to Diophantus' Age Problem
There are at least two quick ways to solve this problem. The first uses
algebra (in honour of Diophantus). The second uses common factors and in
this case is probably even easier and quicker than the algebraic way.
Have you actually done this problem yet?
If not, go back & do it before you read the answer. It's more fun
that way.
Method 1: Hint - Let
x be Diophantus' age at death.
Solution - pdf file
Method 2:
The only number divisible by 6, 12, and 7, and also a number in the
human lifespan is 84. So, he spent 14 years in childhood, seven in
youth, and 12 as a bachelor. He married at 33, and five years
afterward, at age 38, a son was born who later died at age 42, when
Diophantus was 80. Four years after that, Diophantus himself died at
84, twice the age to which his son lived.
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Answer
to Sharing Birthdays question: To solve the shared birthday
problem, we need to make use of one of the basic rules of probability: the sum
of the probability that an event will happen and the probability that the
event won't happen is always 1. If we can work out the probability
that no two people will have the same birthday, we can use this rule to find the
probability that two people will share a birthday.
P(two people share birthday) = 1 - P(no two people share birthday).
So, what is the probability that no two people will share a
birthday?
The first person can have any of the 365 days in the year as his
birthday and the probability that he has a birthday is 365/365 or 1. The
second person's birthday has to be different. There are 364 different
days to choose from, so the chance that two people have different
birthdays is 364/365. That leaves 363 birthdays out of 365 open for the
third person, and so on.
To find the probability that both the second person and the third
person will have different birthdays, we have to multiply:
(365/365) * (364/365) * (363/365) = 0.9918.
If we want to know the probability that four people will all
have different birthdays, we multiply again:
(364/365) * (363/365) * (362/365) = 0.9836. I have left out the
(365/365) since it is just 1.
Following the pattern we can develop a formula for the probability
that n people have different birthdays:
((365-1)/365) * ((365-2)/365) * ((365-3)/365) * . . . * ((365-n+1)/365).
If you know permutation notation, you can write this formula as
^{365}P_{n}
/ 365^{n}
which can be re-written as:
which when calculated for n = 23 equals 0.493.
So, the probability that no two people share a birthday at a party of
23 people is 0.493. Therefore the probability that two people at a
party of 23 people do share a birthday is (1 - 0.493) = 0.507 or
50.7%.
So, next time you are at a party with more than 23 people, it is more likely
than not that two people at the party share a birthday. Obviously, the
more people present, the higher the probability that two people share a
birthday. Once you get to 32 people present, there is a 75% chance that
two people at the party will share a birthday. With 41 people present
there is a 90% chance that two of them will share a birthday. What good
odds.
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Answer to Chocolate Maths Task
for 2017:
call the number you thought of x
multiply by 2 gives 2x
add 5 gives 2x +5
multiply by 50 gives 100x + 250
then add 1767 gives 100x + 250 + 1767
NOTE that 250 + 1767 = 2017 which is the current year.
Subtracting the year of your birth from the current year just leaves you with your age. That's why you add
1766 if you have not
had a birthday yet this year. Otherwise you would work out to be a
year older than you actually are.
The 100x (that is 100 times the number you first thought of) just allows
the number you first thought of to be in the hundreds column. It
becomes the first digit of the 3 digit number you have finished up with.
The process will work each year. Just keep adding 1 to the 1767 (and
1766) for each new year.
Cute eh? Mathematics is truly very beautiful.
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