Transcript Quantum Physics - University of Sheffield

Quantum Physics
• Waves and particles
• The Quantum Sun
• Schrödinger’s Cat and the Quantum
Code
Waves and Particles
• Waves
– are continuous
– have poorly defined
position
– diffract and interfere
• Particles
– are discrete
– have well-defined
position
– don’t (classically)
diffract or interfere
Light is a wave
• Thomas Young (1773–1829)
– light undergoes diffraction and interference
(Young’s slits)
– (also: theory of colour vision, compressibility of
materials (Young’s modulus), near-decipherment
of Egyptian hieroglyphs—clever chap…)
• James Clerk Maxwell (1831–79)
– light as an electromagnetic wave
– (and colour photography, thermodynamics, Saturn’s rings—incredibly
clever chap…)
Light is particles
• Blackbody spectrum
– light behaves as if it came in packets of
energy hf (Max Planck)
• Photoelectric effect
– light does come in packets of energy hf
(Einstein)
– used to measure h by Millikan in 1916
Photoelectric effect
• Light causes emission of electrons from
metals
– energy of electrons depends on frequency of
light, KE = hf – w
– rate of emission (current) depends on intensity
of light
– this is inexplicable if light is a continuous wave,
but simple to understand if it is composed of
particles (photons) of energy hf
Millikan’s measurement of h
h = (6.57± 0.03)
x 10-27 erg s
(cf h = 6.6260755 x 10-27)
Electrons are particles
• JJ Thomson (1856–1940)
– “cathode rays” have
well-defined e/m (1897)
• RA Millikan
– measured e using oil drop
experiment (1909)
Electrons are waves
• GP Thomson (1892–1975)
– electrons undergo diffraction
– they behave as waves with
wavelength h/p
• JJ Thomson won the Nobel Prize for
Physics in 1906 for demonstrating that
the electron is a particle.
• GP Thomson (son of JJ) won it in
1937 for demonstrating that
the electron is a wave.
• And they were both right!
Electrons as waves & light as
particles
• Atomic line spectra
– accelerated electrons radiate light
– but electron orbits are stable
– only light with hf = DE can induce transition
• Bohr atom
– electron orbits as
standing waves
hydrogen lines
in A0 star
spectrum
The Uncertainty Principle
• Consider measuring position of a particle
–
–
–
–
hit it with photon of wavelength l
position determined to precision Dx ~ ±l/2
but have transferred momentum Dp ~ h/l
therefore, DxDp ~ h/2
(and similar relation between DE and Dt)
• Impossible, even in principle, to know position
and momentum of particle exactly and
simultaneously
Wavefunctions
• Are particles “really” waves?
– particle as “wave packet”
• but mathematical functions describing particles as waves
sometimes give complex numbers
• and confined wave packet will disperse over time
• Born interpretation of “matter waves”
– Intensity (square of amplitude) of wave at (x,t)
represents probability of finding particle there
• wavefunction may be complex: probability given by Y*Y
• tendency of wave packets to spread out over time
represents evolution of our knowledge of the system
Postulates of Quantum
Mechanics
• The state of a quantum mechanical system is
completely described by the wavefunction Y
– wavefunction must be normalisable: ∫Y*Ydt = 1
(particle must be found somewhere!)
• Observable quantities are represented by
mathematical operators acting on Y
• The mean value of an observable is equal to
the expectation value of its corresponding
operator
The Schrödinger equation
• non-relativistic quantum mechanics
2


2

– classical wave equation  2 Y     Y
 l 
lh p
– de Broglie wavelength
E  12 m v2  V
– non-relativistic energy
– put them together!

2
2m
2
 Y  VY  EY
Barrier penetration
• Solution to Schrödinger’s equation is a plane
wave if E > V
• If E < V solution is a negative exponential
– particle will penetrate into a potential barrier
– classically this
would not
happen
2
1.5
1
0.5
0
0
-0.5
-1
-1.5
0.5
1
1.5
2
2.5
3
3.5
4
The Quantum Sun
• Sun is powered by hydrogen fusion
– protons must overcome electrostatic repulsion
– thermal energy at core of Sun does not look high
enough
– but wavefunction penetrates into barrier (nonzero
probability of finding
proton inside)
– tunnelling
– also explains a
decay
The Pauli Exclusion Principle
• Identical particles are genuinely
indistinguishable
– if particles a and b are interchanged, either
Y(a,b) = Y(b,a) or Y(a,b) = –Y(b,a)
– former described bosons (force particles, mesons)
latter describes fermions (quarks, leptons,
baryons)
– negative sign implies that two particles cannot
have exactly the same quantum numbers, as
Y(a,a) must be zero
– Pauli Exclusion Principle
The Quantum Sun, part 2
• When the Sun runs out of hydrogen and
helium to fuse, it will collapse under its own
gravity
• Electrons are squeezed together until all
available states are full
– degenerate electron gas
– degeneracy pressure halts collapse
– white dwarf star
Entangled states
• Suppose process can have two possible
outcomes
– which has happened?
– don’t know until we look
– wavefunction of state includes both possibilities
(until we look)
• e.g. 0  gg
• spin 0  1+1, so g spins must be antiparallel
• measuring spin of photon 1 automatically determines spin
of photon 2 (even though they are separated by 2cDt)
Quantum cryptography
• existence of entangled states has
been experimentally demonstrated
• setup of Weihs et al., 1998
– could send encryption key from A to B
with no possibility of eavesdropping
– interception destroys entangled state
Summary
• Origin of quantum mechanics: energy of light
waves comes in discrete lumps (photons)
– other quantised observables:
electric charge, angular momentum
• Interpretation of quantum mechanics as a
probabilistic view of physical processes
– explains observed phenomena such as tunnelling
• Possible applications include cryptography
and computing
– so, not as esoteric as it may appear!