Quantum Phenomena 1 Flashcards
Black body
Idealisation
Perfect absorber and perfect emitter of radiation
Analogy for black body
Hollow box with a small aperture
Light that enters the box is eventually absorbed
Only allowed certain wavelengths- normal modes
Hotter
Higher peak at shorter wavelength
Stefan Boltzmann law
I=sigma T^4
I=total intensity
Spectral emittance
I(lambda)
Intensity per wavelength interval
Rayleigh jeans law
I(lambda)=2pickT/ lambda^4
Agrees with experiment at long wavelength but fails at short
Predicts intensity per wavelength that tends to infinity (UV catastrophe)
Planck hypothesis
Assumes oscillator with frequency f can only have energies E=nhf
Planck’s radiation law
Used cavity model for black body but his hypothesis that energy was quantised
According to Maxwell Boltzmann distribution, for a system in thermal equilibrium at T
A state with energy E has population
nE=Ae^-E/kT
n1/n0=e^-hf/kT
High frequency oscillator
Bigger gaps between energy levels so most likely in ground state
From Planck’s law, can derive
Wiens law by finding lambda for I(lambda) is minimum
Stefan Boltzmann Law by integrating over lambda
Ray,eigh jeans law
Comments on Planck hypothesis
Assumed oscillators emitting radiation could only take quantised values so energy of EM field emitted by black body was quantised
Photoelectric effect
Forcing electrons out of surface by shining light
Must supply enough energy to overcome work function
Photoelectric effect experiment
Electrons given kinetic energy
Can change intensity of light, frequency of light and potential difference
She notes for diagram but basically just anode and cathode with potential difference across and light shining on
Work done in moving charge q across potential difference
E=qV
Work energy theorem
Change in electron Ek= Wtot=-eV=eVac
Stopping voltage
Vac=-V0 at which no electrons reach the anode
Delta Ek=0-kmax=-eV0
K ax=eV0
Photo current does depend on frequency
For given material, light with frequency below threshold frequency produces no photo current, regardless of intensity
Above threshold photo current is proportional to intensity for large positive V
What wave model cannot explain about photoelectric effect
Fact there is no time delay before photo current detected (cumulative energy)
Would expect stopping potential to increase with increasing light intensity ie more energy, more Ek so more voltage to stop
Einstein’s photon explanation
A beam of light is made up of discrete packages of energy (photons) each with energy E=hf
Work function
Amount of energy an electron needs to escape surface
Relating work function and threshold frequency
hf the= work function (just E=hf)
Above threshold frequency, maximum kinetic energy of an electron
Given by excess energy
eV0=hf- work function
Applications of photoelectric effect
Photon momentum eg comets (photons from sun have sufficient energy)
Photon momentum
P=E/c=hf/c=h/ lambda
X ray production experiment
Heated cathode accelerate e to anode
Electrons crashing into anode emits radiation in the form of X-rays
What were X-rays produced as
Bremsstrahlung “breaking radiation”
When electrons are slowed abruptly or deflected
Bremsstralung spectra
Vertical axis is x ray intensity per unit wavelength
Horizontal axis is x ray wavelength
Shows that eVac=hf max which is what the photon model predicted
Compton scattering
X-ray source
Photons incident on target
Scattered towards detector
Compton scattering
Change in wavelength depends on
Angle at which the photons are scattered
Compton scattering- wave model predicts
Scattered light has same frequency as incident light
Compton scattering
Particle model predicts
Scattered light has lower frequency than incident light (think snooker balls)
Photons scattered from tightly bound electrons
Undergo a negligible wavelength shift
Photons scattered from loosely bound electrons
Undergo a wavelength shift given by eq lambda’ - lambda = h/mc (1-cos theta)
m is mass of electron
Single photon diffraction
Monochromatic light incident on slit
Shine onto screen with movable photomultiplier detector - can count individual photons
De Broglie wavelength for massive particles
If waves can behave like particles, can particles behave like waves
Lambda =h/p=h/mv
Relativistic particles
Lambda=h/ gamma mv
Electron diffraction experiment
Heated filament emits electrons
Electrons accelerated by electrodes and directed at a crystal
Electrons strike nickel crystal
Detector can be moved to detect scattered electrons at any angle
Peak in intensity of scattered electrons is due to
Constructive interference between electron waves scattered by different surface atoms
Transmission electron microscope
High voltage supply
Cathode where electron beam starts
Accelerating anode
Condensing lens
Objective lens
Projection lens
Final image in image detector
Superposition of two waves
Algebraic sum
Spatially finite wave packets are formed
Relating wave number and momentum
p=h/ lambda = hk/2 pi = h bar k
Heisenberg’s uncertainty principle
Trade off in how well position and momentum can be simultaneously defined
Overall size of atom
Of the order of 10^-10m
Thomson’s plum pudding model
Electrons embedded in sphere of positive charge
Offered an explanation of line spectra (atoms collide, e oscillates around equilibrium at characteristic frequency and emits light at this f)
Rutherford’s experiment
Alpha particles emitted by radioactive element such as radium
Small holes in pair of lead screens create a narrow beam of alpha particles
Alpha strike foil and are scattered by gold atoms
Scattered alpha produces a flash of light when it hits scintillation screen, showing direction of scattering
Rutherford’s model of the atom
Large angle scattering led Rutherford to develop model where mass and + charge concentrated in nucleus
Classical predictions and problems with Rutherford model
Atoms should emit light continuously
Should be unstable- if radiating energy should lose energy, orbit get smaller
Should emit at all frequencies, further out e less energy, closer = more energy
Atoms are stable and therefore must have
Ground level
Each wavelength in spectrum corresponds to
Transition between two specific energy levels
Bohr model assumption
Allowed energy levels correspond to circular orbits of electron around nucleus
ie Fcoulmb=Fcentripetal
Quantisation of angular momentum
On=mvnrn=no/2pj = n h bar
De broglie wave in an allowed orbit
Standing wave
Fixed number of wavelengths should fit into circle
Bohr model kinetic and potential energies
Using quantisation of angular momentum and circular orbits gives relationships for radius and velocity
Bohr radius given by n=1
Can be used in EK and EP formulae
Energy levels predicted by Bohr model
En=Kn+Un
Reduced mass of atom
Assuming nucleus at rest
Using reduced mass (ie both orbiting centre of mass) gives mr=m1m2/m1+m2
Wave function
¥(x,y)
Actual symbol is psi, ¥ was closest on keyboard
¥(x,t)
Describes distribution of a particle in space
|¥( x,t)|^2
Probability distribution function
|¥(x,t)|^2 dx
Probability of finding a particle between x and x+dx at time t
Mathematical properties of the wave function
Wave function and it’s derivative must be continuous
The sum of all probabilities must be 1 (ie integral between negative and positive infinity of erdivative of wave function dx =1)
Why is continuity important
Idealised: object hits barrier and instantaneous change in velocity
Real: velocity changes quickly and continuously, not abruptly
If velocity was discontinuous
Acceleration would be undefined
Derivative of wave function
Momentum
Second derivative of wave function
Kinetic energy
Potential energy diagrams
U(x) against x
Looks like valley
Particles attracted to places of
Low energy
Wave function outside box
0
Wave function at edges of box
0
Continuous Si cannot suddenly go to 0 outside box
Allowed wavelengths in box
n lambda/ 2=L
Integer number of half wavelengths
Lambda = 2L/n
Time independent Schrodinger equation
Mathematical description of the wave nature of particles and is a statement of conservation of energy
KE+PE=Etot
H bar
H/ 2pi
Capital psi
x and t
lower case psi
for just x
Simplest example to discuss complete solution to time independent Schrodinger equation
Infinite potential well
Potential given by
U= infinity when x is between -infinity and zero
U=0 when x is between 0 and L
U=infinity when x is between L and infinity
Energy barriers at walls is infinite so
Particle cannot escape
Potential barrier
Particle does not have enough energy to make it over the barrier
Non zero portability of finding it on other side thiugh
Analogy: frustrated total internal reflection
EM field falls off at boundary between two media but it is not discontinuous
Potential barrier: limiting cases
Zero probability of finding particle where L tends to infinity
Continuous wave when l tends to zero
Probability as a function of barrier width
Probability walls off exponentially as a function of barrier width
Transmission coefficient
When T«1
T=Ae^-2kL
A and k constants given
Quantum tunnelling
A particle on the left of the barrier has a non zero probability of being found on the right of the barrier, not possible classically
Scanning tunnelling microscope
Sharp needle kept at positive potential relative to surface
If needle close enough to surface electrons can tunnel across and be detected as current
Needle moves across surface and perpendicular to it to maintain constant tunnelling current
Alpha decay
Made possible by quantum tunnelling
Potential seen by alpha in nucleus is due to strong nuclear force
Alpha particle does not have sudpfficient energy classically
