Quantum physics

1
Quantum physics
Anyone who is not shocked by the quantum theory
has not understood it. Niels Bohr, Nobel Price
in 1922 (1885-1962)
I can safely say that nobody understand
quantum physics Richard Feynman Nobel Lecture,
1966, (1918-1988)

• PHY232
• Remco Zegers
• zegers_at_nscl.msu.edu
• Room W109 cyclotron building
• http//www.nscl.msu.edu/zegers/phy232.html

2
quiz (extra credit)

• a distant star moves at a velocity of 0.5 times
  the speed of light away from us. It emits light
  that can be detected in earth-based telescopes.
  What is the speed of the light of the radiation
  we receive from these quasars? Ignore the fact
  that the telescope is in air and not in vacuum.
• a) it is less than the speed of light in vacuum
  (3x108 m/s) by a factor 1/??(1-v2/c2)?(1-(0.5c)2
  /c2)0.87
• b) it is larger than the speed of light in vacuum
  (3x108 m/s) by a factor ?1/?(1-v2/c2)1/?(1-(0.5c
  )2/c2)1.15
• c) it is equal to the speed of light in vacuum
  (3x108 m/s)

3
so far

• we have treated light as being waves and used
  that formalism to treat optics and interference
• we have seen that under extreme conditions (very
  high velocities) the Newtonian description of
  mechanics breaks down and the relativistic
  treatment designed by Einstein must be used.
• Now, we will see that the description of light in
  terms of waves breaks down when looking at very
  small scales. In addition, we will see that
  objects that we usually refer to as particles
  (like electrons) exhibit wave-phenomena.

4
photoelectric effect

• when light hits a metal, electrons are released.
  By providing a voltage difference between the
  metal and a collector, these electrons are
  collected and produce a current.
• if light is described in terms of waves one would
  expect that (classical description)
• independent of the frequency of the light,
  electrons should be emitted if one waits long
  enough for sufficient energy to be absorbed by
  the metal
• the maximum kinetic energy depends on the
  intensity (more energy absorbed)
• the kinetic energy of the electrons is
  independent on the frequency (wavelength of the
  light) and only depends on the intensity
• electrons take a little time to be released since
  sufficient energy needs to be absorbed

5
however

• one observes that
• if the frequency of the light is too low, no
  electrons are emitted
• the maximum kinetic energy of the electrons is
  independent of the intensity.
• the maximum kinetic energy increases linearly
  with frequency
• the electrons are emitted almost instantaneously,
  even at very low light intensities
• These observations contradict the classical
  description. It suggest that energy is delivered
  to the electrons in the metal in terms of
  well-localized packets of energy. The photons in
  the light beam are thus seen as particles that
  deliver packets of energy (so-called energy
  quanta) to the electron it strikes.

6
photo-electric effect
The energy carried by a photon Ehf h plancks
constant (h6.63x10-34 Js) f frequency with
cf? The energy is localized in the
photon-particle The maximum kinetic energy of a
released electron KEmaxhf-? with ? the
workfunction (binding of electron to the
metal) So only if hfgt? will electrons be
released from the metal fc?/h fc is the
cut-off frequency ?cc/fc(hc)/? the cut-off
wavelength
see table 27.1 for work functions for various
metals
7
example

• light with a wavelength of 400 nm is projected on
  a sodium metal
• surface (?2.46 eV).
• what is the energy carried by a single photon?
• what is the maximum kinetic energy of the
  released electrons?
• what is the cut-off wave length for sodium?
• what happens if light with a wavelength of 600 nm
  is used?

a) Ehfhc/?6.63x10-34Js x 3x108/(400x10-9)4.97x
10-19 J in eV (1 eV1.6x10-19 J)
3.11 eV b)
KEmaxhf-?3.11-2.46 eV 0.65 eV c)
?cc/fc(hc)/?6.63x10-34x3x108/(2.46x1.6x10-19J)
505x10-9 m the cut-off wavelength is 505
nm. d) if light with a wavelength of 600 nm is
used no electrons are emitted Note if fltfc no
electrons emitted if ?gt ?c no electrons
emitted
8
particle-wave dualism

• So, is light a wave or particle phenomenon?
• answer it depends!

9
question

• light from a far-away star is used to perform a
  double slit experiment. Approximately once per 10
  minutes will a single photon from the star arrive
  at the double slit setup on earth. Which of the
  following is true?
• a) since light is a wave-phenomenon, an
  interference pattern will be seen on a screen
  placed behind the double slits.
• b) since only one photon arrives every 10
  minutes, interference is not possible since one
  can hardly think of the light coming in as waves

10
question

• instead of a light source, an electron gun firing
  electrons at high speeds is used in a double-slit
  experiment. which of the following is true?
• a) since electrons are massive particles, an
  interference pattern is not produced
• b) electrons are similar to photons they exhibit
  both wave and particle phenomena. In this case,
  electrons behave like waves and an interference
  pattern is produced.

11
interference pattern
P1
B
A
P1
P2
P2
If one of the slits in a double slit experiment
is closed one sees only a diffraction pattern
from a single slit (P1). If the other slit is
opened and the first one closed, one sees only
the diffraction pattern from the other slit
(P2). If both are opened, one does not simply
see the sum of P1 and P2 (like in A), but the
double-slit interference pattern (like in B). The
reason is the following Remember that the
intensity (I) is proportional to the E-field
squared IE2E02cos2?. In A, it is assumed that
the intensities add IsumI1I2 . However, one
should add the E-fields (which can be positive or
negative) and than squared, like in B
Isum(E1E2)2 where E1 and E2 are treated as
vectors.
12
and if you think that youve seen it all
B
A
lets assume I determine through which hole the
photon (or electron) goes by placing a detector
before the slits. Would I still observe an
interference pattern like in B? Answer no! By
measuring the location of the photon, we have
turned the light-wave into a particle and the
interference pattern gets lost.
13
Schrödingers cat
A cat is placed in a closed box. Inside the box a
radioactive source is placed in which on average
once per hour a radioactive decay takes place.
If the decay takes place, a bottle of poison
breaks, killing the cat. In quantum-physical
sense, the cat is 50 dead and 50 alive after
half an hour. Since we cant see it, it is in a
superposition of those two states and there is a
certain probability of being in one of either
states. Only when we open the box, do we
determine what state the cat is in. The
observation is crucial to determine the state of
the cat.
14
(No Transcript)
15
heisenbergs uncertainty principle
If we want to determine the location and velocity
(momentum) of an electron at a certain point in
time, we can only do that with limited
precision. Lets assume we can locate the
electron using a powerful light microscope. Light
scatters off the electron and is detected in the
microscope. However, some of the momentum is
transferred and observing the electron means we
can only determine its velocity (momentum) with
limited accuracy.
note hh/(2?)
?x?p?h/(4?) with ?x precision of position
measurement with ?p precision of momentum (mv)
measurement this can also be expressed in terms
of energy and time measurements ?E?t?h/(4?) with
?E precision of energy measurement with ?p
precision of time measurement
16
example
The location of an electron is measured with an
uncertainty of 1 nm. One also tries to measure
the velocity of the electron. What is the
(minimum) uncertainty in the velocity
measurement? The mass of the electron is
9.11x10-31 kg.
use the uncertainty principle ?x?p?h/(4?)
with ?x1x10-9 m, h6.63x10-34 Js so ?p?
6.63x10-34/(4? x 1x10-9)5.28x10-26 kgm/s ?vmin
5.28x10-26/9.11x10-315.79x104 m/s (use pmv)
note that the uncertainty principle works for the
three dimensions separately ?x?px?h/(4?)
?y?py?h/(4?)
?z?pz?h/(4?)
17
photons as particles and quanta

• Some other examples of where the particle nature
  of
• light plays a role
• Photo-electric effect
• Black-body radiation
• bremmstrahlung
• Compton effect

18
black-body radiation
A black body is an object that absorbs all
electromagnetic radiation that falls onto it.
They emit radiation, depending on their
temperature. If Tlt700 K, almost no visible light
is produced (hence a black body). The energy
emitted from a black body P?T4 with ?5.67x10-8
W/m2K4
The peak in the intensity spectrum varies with
wavelength using the Wien displacement
law ?maxT0.2898x10-2 mK Until 1900, the
intensity distribution, predicted using classical
equations, predicted a steep rise at small
wavelengths. However, the opposite was determined
experimentally
(classical)
19
Planck to the rescue

• Max Planck devised a theory for a simple black
  body that could describe the measured spectra.
• He assumed that the walls consisted of little
  radiators that only emitted light at certain
  discrete energies Enhf
• f the frequency of the light (Hz)
• h plancks constant (6.63x10-34 Js)
• n integer.
• His achievement was really the first success of
  quantum theory
• In essence, his theory showed that because the
  energy is quantized, it is hard to emit light of
  small wavelengths (high frequency) since a lot of
  energy is required

20
example
Hot lava can be considered as a black body
emitting radiation at a variety of temperatures.
If temperature of molten lava is about 1200
0C, what is the peak wave length of the light
emitted?
21
X-rays

• when energetic electrons are shot on a material,
  photons with small wavelengths (0.1 nm) are
  produced.
• The spectrum consist of two components
• broad bremsstrahlung spectrum
• peaks at characteristic wavelengths depending on
  the material (see next chapter)
• the bremsstrahlung (braking radiation) is due to
  the deflection of the electron in the field of
  the charged nucleus.
• a light quantum is produced when the electron is
  deflected. It takes away energy from the electron

22
bremmsstrahlung

• assume electrons are accelerated in a potential
  of V Volts.
• their kinetic energy is EeV with e1.6x10-19 C
  and V the potential
• If the electron is completely stopped in the
  material, all its kinetic energy is converted
  into the photon with maximum frequency fmax and
  hence minimum wavelength ?min
• if it merely deflected, the frequency f is
  smaller than fmax and its wavelength ? larger
  than ?min.
• so eVhfmaxhc/?min

23
example

• an X-ray spectrum is analyzed and the minimum
  wavelength is found to be 0.35 angstrom (1
  angstrom 10-10 m). What was the potential over
  which the electrons were accelerated before the
  interacted with the material?

eVhfmaxhc/?min Vhc/(?mine)
6.6x10-34x3x108/(0.35x10-10x1.6x10-19)
3.55x104 V
24
Braggs law
X-rays scattered off atoms in e.g. will interfere
and the interference pattern can be used to
identify/study materials.
25
question

• X-rays are sometimes used to identify crystal
  structures of materials. this is done by looking
  at the diffraction pattern of X-rays scattered
  off the material (see ch 27.4). Why are X-rays
  used for this and not for example visible light?
• a) the wavelength of X-rays is close to the
  spacing between atoms in a crystal
• b) since the frequency (and thus energy) of
  X-rays is much larger than that of visible light,
  they are easier to detect
• c) X-rays are much easier to produce than visible
  light

26
compton effect

• When photons (X-rays) of a certain wavelength are
  directed towards a material, they can scatter off
  the electrons in the material
• If we assume the photon and the electron to be
  classical particles, we can describe this as a
  normal collision in which energy and momentum
  conservation must hold.
• after taking into account relativistic effects
  (see previous chapter) one finds that

27
compton scattering

• ???-?0h/(mec) x (1-cos?)
• with ? wavelength of photon after collision
• ?0 wavelength of photon before
  collision
• h/(mec) Compton wavelength (2.43x10-3
  nm)
• me mass of electron
• ? angle of outgoing X-ray relative
• to incoming direction

?
?0
28
example
???-?0h/(mec) x (1-cos?)

• A beam of X-rays with ?010-12 m is used to
  bombard a material.
• a) What is the maximum shift in wavelength that
  can be observed due to Compton scattering?
• b) What is the minimum shift in wavelength that
  can be observed due to Compton scattering?
• c) What are the minimum and maximum kinetic
  energies of the struck electrons, ignoring
  binding to the material they are in.

• maximum shift occurs if cos?-1 (?1800). This is
  usually referred to as
• Compton backscattering. in that case
• ??2h/(mec)2x2.43x10-124.86x10-12 m
• b) minimum shift occurs if cos?1 (?00) in which
  case essentially no
• collision takes place ??0
• c) gain in kinetic energy by electron is loss in
  energy of x-ray
• case b) no kinetic energy gained by electron
• case a) energy of X-ray before collision
  hfhc/?01.98x10-13 J
• energy of X-ray after collision
  hfhc/(?0??) 3.38x10-14 J
• kinetic energy gained by electron
  1.64x10-13 J1.0 GeV (giga)

29
applications