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Principles of Quantum Mechanics
What is Quantum Mechanics?
QM is the theory of the behavior of very small objects (e.g.
molecules, atoms, nuclei, elementary particles, quantum fields,
etc.)
Why Quantum Mechanics?
One of the essential differences between classical and quantum
mechanics is that physical variables that can take on continuous
values in classical mechanics (e.g. energy, angular momentum)
can only take on discrete (or quantized) values in quantum
mechanics (e.g. the energy levels of electrons in atoms, or the
spins of elementary particles, etc).
QM is weird!
QM deals with concepts that are counter intuitive, and impossible
to visualize in classical mechanical terms. Some concepts defy
common sense, e.g. a) superposition (of states, quantum systems can be in more than
one discrete state at a time)
b) non-locality (spooky action at a distance)
c) non determinism (QM is essentially stochastic)
d) non reality (some “interpreters of QM” claim that QM implies
that there is no independent reality)
e) uncertainty (one cannot have full simultaneous knowledge
of certain physical quantities, e.g. the position and momentum
of an elementary particle)
f) wave-particle duality (waves behave like particles, & vice versa!)
The next few slides deal with experimental phenomena which
historically motivated the introduction of basic quantum concepts.
The most famous one is arguably the “double slit experiment”
which illustrates several of the weirdo phenomena mentioned
earlier. Here it is.
Bullets
hits
Machine Gun
Double Slit Screen
Absorbing Screen
Waves
intensity
Wave Generator
Double Slit Screen
So what kind of pattern would you expect if you shot electrons
through the double slit? Would it be bullet-like or wave-like?
Its wave-like! But how? Electrons are particles. They leave
a flash on the absorbing screen at localized points. How do they
interfere? Which slit do they go through? Mysterious!
What happens if the electron emission rate is so low, that only one
electron goes thru the slit at once? Will there be a wave-like
pattern? Yes. Somehow the single electrons are interfering
with themselves! What! How? Does a single electron go through
both slits (to cause the interference)? How can an electron that
causes a localized flash on the absorbing screen go through both slits?
Can an electron be in two places at once? Counter common sense.
As Feynman said “No one understands quantum mechanics!”
Lets try to determine which slit the single electron goes thru, by
shining light on it and detecting the deflected photon. The light
has to have a very high frequency, to have a very small wavelength.
The smaller the wavelength, the bigger the kick of the photon on
the electron. This kick causes the wave-like pattern to disappear
and become a bullet-like pattern! We know which slit the electron
went thru but we lost the interference (the wave-like) pattern!
So we cant have both the knowledge of which slit the electron
went thru AND have the interference pattern, so how can we
ever know which slit it went thru when there IS a wave-like
pattern? Maybe we can NEVER know?
So, QM postulates that there two wavelike phenomena (one
from each slit) that interfere at the absorbing screen and that
these two “waves” influence where the electron will hit the screen.
These “waves” were thought at first to be real physical waves
(de Broglie, Schrodinger, (later)Bohm, etc), but most QMechanics
interpret these waves as “probability waves” or “probability
amplitudes”, whose squares gives the probability distributions (a la
statistics) of finding the electron in a given region of space, etc.
The mathematical formulae of QM is generally accepted (they work!),
NOT the interpretation of what the mathematical formulae mean.
100 years of controversy - still!
Further Quantum Phenomena
e.g.
a) Einstein’s Explanation of the Photo-Electric Effect
b) de Broglie’s idea that particles have wave properties
c) Heisenberg’s Uncertainty Principle
d) The Stern-Gerlach Experiment
e) Schrodinger’s wave equation
Newton (17th century) thought that light consisted of particles.
Then Young’s double slit experiments with light (19th century,
i.e. interference fringes) implied that light was a wave phenomenon!
Then Einstein came along (20th century) and said that light consisted
of “photons” i.e. localized discrete bundles of light energy (i.e.
particles again!)
The Photo-Electric Effect.
Experiments around the turn of the 19th/20th centuries showed some
weird phenomena going on with the so-called “photo-electric effect”.
This effect could NOT be explained with classical physics. It is a
primary example of quantum physics whose explanation earned
Einstein the Nobel Prize.
The effect is that shining light on certain metal surfaces, caused
electrons to be emitted.
From classical physics one would argue that increasing the intensity
of the light, would mean the energy of the emitted electrons would
increase, but NO. The electron energy would only increase if the
frequency of the light increased! Below a certain frequency, NO
electrons were emitted, independently of the intensity of the light???!
To explain this puzzling phenomenon, Einstein introduced the
Concept of the “photon” (= light particle). He postulated that
A photon had an energy proportional to its frequency f. Hence
E = hf
(where h is a constant of proportionality called Planck’s constant).
For an electron to be emitted, it has to be given a minimum energy
by the photon to overcome the attraction of the metal. Call this
minimum energy M. Hence the energy En of the emitted electron is
En = hf - M
If f is too small, En < 0, so no electrons are emitted!
Hence light consists of photons, i.e. particles!
de Broglie’s idea that particles have wave properties
In the 1920s the Frenchman Louis de Broglie was pondering on
the particle nature of waves (e.g. photons) and hit on the idea
that maybe the reverse was also true, that maybe there was a kind
of symmetry in nature, i.e. waves behave like particles, so maybe
particles behave like waves!!!???
When de Broglie was defending his PhD thesis with this weirdo
idea, his examiners found it too radical, so they sent the thesis
to Einstein, who loved it.
de Broglie used ideas from Einstein’s relativity, to derive a formula
for the wavelength of the wavelike phenomenon depending on the
particle’s momentum.
From relativity for a photon,
p = E/c
For a photon, E = hf, but c = lf (where l is the wavelength)
So p = E/c = hf/c = hf/lf = h/l
p=h/l
de Broglie postulated that the same formula would apply to a
particle, i.e. l = h / p where l is the wavelength of the particle’s
associated wave. “Matter waves”.
This formula was tested experimentally in the late 1920s by the
son of J.J. Thompson (the discoverer of the electron). He obtained
interference patterns from electrons reflecting off a metal surface.
Hence J.J. discovered the particle nature of electrons, and his son
discovered the wave nature of electrons - the ironies of history!!
This wave-particle duality is an important part of quantum physics!
Heisenberg’s Uncertainty Principle
This famous principle of quantum physics says basically that
there are physical quantities whose values cannot be known
accurately simultaneously, e.g. the momentum and position of
an electron.
Try to measure the momentum and position of an electron. This
would not be difficult conceptually using classical physics thinking,
Since light would be considered as having continuously varying
Energies and hence minimal disturbance on the electron being
Observed.
But in quantum physics, light consists of photons which have
discrete (quantized) energies, with a kick! Bouncing a photon off an
electron will give the electron a kick and disturb its momentum.
To obtain an accurate measurement of the position of the electron
the photon used must have a small wavelength (a large wavelength
would not have enough resolution to locate the electron, hence
there would be an uncertainty in the electron’s position Dx).
But a small wavelength for a photon, means a large energy and
hence momentum (p = E/c), which gives the electron a kick and
Hence an uncertainty in the electron’s momentum measurement.
The measurement (using a photon - what else is there to use?!)
disturbs the electron. Measurement is NOT gentle as in classical
physics.
There is thus a trade off between knowing accurately (and
simultaneously) the momentum of the electron and its position.
A more quantitative analysis (see the handout) shows that the
product of the two uncertainties, Dx and Dp is a constant DxDp ~ h
where h is again Planck’s constant
(Planck was the first person to discover (1900) the value of this
constant, so it is named after him).
It appears that the uncertainty principle is purely a measurement
phenomenon. Quantum physicists think that undisturbed
particles (wavicles) just don’t have exact positions and momenta!
A consequence of the HUP is that particles do NOT have trajectories!
So what happens to the notion of an external reality with QM????!!