Download as a probability wave

Dilemma
• Existence of quanta could no longer be questioned
• e/m radiation exhibits diffraction => wave-like
• photoelectric & Compton effect
=> localized packets of energy => particle-like
• “wave-particle duality”
Wave properties
Photon
detector
clicks
when a
photon is
absorbed
Probability Waves
• Photon detector might be a photoelectric device
• at any point, the clicks will be randomly spaced in
time
• cannot predict when a photon will be detected at any
point on the screen
• if we move the detector, the click rate increases near
an intensity maximum
• the relative probability that a single photon is
detected at a particular point in a specified time
 Intensity at that point
• intensity  Em2
Probability of detection  Em2
Probability Wave
• Probability (per unit time) that a photon is detected
in some small volume is proportional to the square
of the amplitude of the wave’s electric field in that
region
• Postulate that light travels not as a stream of
photons but as a probability wave
• photons only manifest themselves when light
interacts with matter
• photons originate in the source that produces the
light wave (interaction)
• photons vanish on the screen (interaction)
Single-photon version (1909)
• Light source is so weak that it emits only one photon
at a time at random intervals
• interference fringes still build up
• raises the question: if the photons move through the
apparatus one at a time, through which slit does the
photon pass?
• How does a given photon know that there is another
slit?
• Can a single photon pass through both slits and
interfere with itself?
Electrons and Matter Waves
• Light is a wave but can transfer energy and
momentum to matter in “photon” sized lumps
• can a particle have the same properties?
• can it behave as a wave?
• “a matter wave”
• de Broglie (1924) suggested =h/p
•
“de Broglie wavelength”
• a beam of electrons has a wavelength and should
diffract if “slit width” comparable to =h/p
• 1927 Davisson and Germer observed diffraction
of electrons from crystals
Diffraction
• Recall for a single slit, the diffraction minima are at
a sin = m
• if  <<a, then all ~m/a are small and the light
essentially travels as a ray and does not spread out
• need   a for strong diffraction effects
• eg. 1 kg billiard ball moving at 5.0 m/s
• =h/p = h/mv = (6.63x10-34 J.s)/(1kg)(5.0m/s)
= 1.3 x 10-34 m << size of atoms
• => no diffraction
Diffraction
• eg. Electrons with kinetic energy of 54.0 eV
• if K=p2/2me , then p=(2meK)1/2
= (2 x 9.11x10-31 kg x 54eV x 1.6x10-19 J/eV)1/2
=3.97 x 10-24 kg. m/s
• =h/p = (6.63x10-34 J.s)/(3.97x10-24 kg.m/s)
= 1.67 x 10-10 m = .167 nm
• smaller mass => larger 
• typical atom has diameter of 10-10 m
Diffraction
• regular arrays of atoms (crystals) should
diffract electrons!
• Neutrons can also diffract => study
structures of solids and liquids
Double Slit Experiment with electrons (1989)
X-ray beam
(light wave)
Electron beam
(matter wave)
Problem
• Singly charged Na ions are accelerated through
a potential difference of 300 V. What is (a) the
momentum acquired? (b) what is the de
Broglie wavelength?
• Solution: K = qV = (1.6x10-19 C)(300V)
= 4.80 x10-17 J = 300 eV
• m=(22.9898 g/mole)/(6.02x1023 atom/mole)
= 3.819 x10-23 g = 3.819 x 10-26 kg
Solution
• (a) p = (2mK)1/2 = [2(3.819x10-26)(4.8x10-17)]1/2
= 1.91 x10-21 kg.m/s
• (b)  =h/p = (6.63x10-34 J.s)/(1.91x10-21 kg.m/s)
= 3.46 x10 -13 m
What is Waving?
•
•
•
•
If particles behave as waves, what is waving?
Wave on a string => particles in string execute SHM
sound wave in air => air molecules oscillate in SHM
light wave => electric and magnetic fields oscillate
• e.g. E(x,y,z,t) electric field varies from place to
place and with time
• intensity  |E|2
• what varies from place to place for a matter wave?
• Wave function (x,y,z,t)
“psi”
Schrödinger Equation
d  8 m

(
E

V
)


0
2
2
dx
h
2
•
•
•
•
•
2
Schrödinger Equation 1926 H=E
(x,t) is a solution of this equation
the wave equation for matter waves
probability waves
probability density is P(x,t)= (x,t) *(x,t)
= |(x,t)|2
A harmonic wave has
a definite value of k
but extends to infinity
A wave packet has a spread
of k- values but is localized
k.x  1
.t  1
Example
• Radar transmitter emits pulses of electromagnetic radiation
which last 0.15 s at a wavelength of  = 1.2 cm
• (a) to what central frequency should the radar receiver be set?
• (b) what is the length of the wave packet?
• (c ) how much bandwidth should the receiver have?
• (a) f0 = c/0 = 3 x 108 m/s /1.2 x 10-2 m = 26 GHz
• (b) x = ct = (3 x 108 m/s)(.15 x 10-6 s) = 45 m
• (c ) f = /2 = 1/(2t) = 1/(6.28 x .15 x 10-6 s)= 1.1 MHz