Quantum Phenomena 2 Flashcards
problems with Rutherford’s model
if electrons orbited around the nucleus, they should have lost energy doing so and circle down into the nucleus.
As electrons orbited down, their angular speeds would change continuously so energy emitted would constantly change frequency but spectra has distinct lines?
Bohr’s solution
electrons orbit nucleus at fixed distance and do not radiate energy.
definite energy associated with each available stable orbit and electron only emits energy when it moves from one orbit to another.
energy emitted in an electron transition is in the form of
a photon
Ephton=hf=Einitial-Efinal
angular momentum of the electron
an integer multiple of h/2pi ie quantised
L=mvr=nh/2pi (n=1,2,3…)
principle quantum number
the value of n for each orbit
for the radius to remain constant in the Bohr model
electrostatic force must provide exactly the radial motion force
ie Fc=Fe
equations for kinetic energy for an electron in a given orbit
Fc=Fe
rearrange for v and plug into 1/2mv^2
taking the Bohr model further
applying Schrodinger equation to find the wave functions for states with definite energy values for the hydrogen atom.
issue with mass, strictly speaking
the electrons do not orbit the proton, they both orbit their common centre of mass.
use reduced mass.
mr=m1m2/m1+m2
spherical coordinates to solve Schrodinger equation
r, θ, Φ
r- distance of orbiting electron from nucleus
θ - angle the line 0-r makes with z-axis
Φ - angle the same line makes with the y-axis
why is spherical coordinate system useful?
potential energy only depends on r
solutions of Schrodinger equation
obtained by separating variables involved
wave function expressed as a product of three functions
(R depends only on r etc)
how are physically acceptable solutions to Schrodinger obtained?
applying boundary conditions
R(r) tends to 0 as r increases
phi(phi) must be periodic
solving with boundary conditions
produces relation for energy levels, identical to those predicted to the Bohr model
En=-13.60eV/n^2
orbital angular momentum
vector quantity, denoted by L
magnitude of orbital angular momentum
magnitude can take values determined by theta being finite
possible values L=root l(l+1) h bar
orbital angular momentum quantum number
l
an integer, l=0,1,2,..,n-1
permitted values that a component of the vector L can take are determined by
the requirement that phi is periodic.
eg: z component, Lz=mlhbar
orbital magnetic quantum number
ml
also called orbital angular momentum projection quantum number
takes values m=-l,…,0,…,l
comparing Lz with L itself
the component Lz can never be quite as big as L itself (unless both zero)
minimum angle between the overall angular momentum vector and the z-axis
theta l = arccosLz/L (draw out to show)
if we knew the direction of the orbital angular momentum, then we could
define that direction to be the z-axis i.e. Lz=L
only in the x-y plane
if all motion of the particle is in the x-y plane
z component of linear momentum would be zero and carry no uncertainty.
therefore, from the uncertainty principle, uncertainty in Z would be infinite
this is impossible so conclude that we never know precise direction
wave functions for the hydrogen atom are determined by
the values of the three quantum numbers: n,l,ml
n determines energy values En
l sets magnitude of the orbital angular momentum
ml fixes the value of the z-component of angular momentum
degeneracy
the existence of more than one distinct state with the same energy
letters used to label states with various values of l
l=0 s state
l=1 p state
l=2 d state
l=3 f state
l=4 g state
l=5 h state
and so on, alphabetically
spectroscopic notation
eg: if n=2 and l=1 this is 2p state
n=4,l=0 this is 4s state
n=1
k shell
n=2
L shell
n=3
M shell
n=4
N shell
for each n, different values of l correspond to
different subshells eg n=2 contains 2s and 2p subshells
to find total number of distinct states in atom
eg: n=4
list all possible l values and then ml values
count the number of possible ml states for each l and add up
total angular momentum
vector sum of the two components of angular momentum (orbital and spin)
electron carries a charge so its spin creates
current loops and a magnetic moment
spin angular momentum
possible values sz=+/-h bar/2
magnitude of the spin angular momentum
expression equivalent to orbital angular momentum
s=root 1/2 (1/2+1) h bar = root3/4 h bar
ms
quantum number to specify electron spin orientation
takes value 1/2 or -1/2
sz=ms h bar
spin up
z component is + h bar/2
spin down
z component is -h bar/2
in quantum mechanics, the specific Bohr orbits are replaced by
probability distributions
electron is point-like, spin is an intrinsic property of particles that mathematically behaves like angular momentum.
total angular momentum
defined by J
J=L+S
possible values of the magnitude of the total angular momentum J
given in terms of another quantum number j
J=root j(j+1) h bar
j=|l+/-1/2|
l + 1/2 state
case which vectors L and S have parallel z components
l-1/2 state
L and S have anti-parallel z components
spectroscopic notation using j quantum number
superscript is the number of possible spin orientations
capital P indicates state with l=1 (or S, D etc)
subscript is the value of j
eg: 2P1/2
issues with applying Schrodinger equation to the general atom
complexity is so extreme that it has not been solved for even Helium.
number of variables of interaction is too large (electrons with each other and electrons with every proton)
simplest approximation for Schrodinger to the general atom
assume that when an electron moves, it ignores the effects of all other electrons and only feels the influence of the nucleus, which is taken as a point charge.
now have nuclear charge of Ze so every factor of e^2 in wave function is replaced by Ze^2
central field approximation
better option]
think of all the electrons together as making up a charge cloud that is on average spherically symmetric
take each electron to be moving in field due to nucleus and averaged out cloud
difference in Schrodinger equation to equation for hydrogen
1/r potential energy function is replaced by different function U(r)
(only a function of r so phi and theta are exactly as before)
all quantum numbers and z-components same as before
radial wave functions and probabilities are different than for hydrogen because of
change in U(r) so the energy levels are no longer given by previous equation
in general, energy of a state now depends on both n and l rather than just n
why is uncertainty principle needed?
would expect gradual changes as more electrons in each atom
but properties of elements vary widely in order of atomic number
eg: halogens form compounds by acquiring additional electron, alkali metals lose electrons and noble gases do not form compounds at all
since we do not get gradual changes in properties, in the ground state of a complex atom
all the electrons cannot be in the ground state
Pauli exclusion principle
no two electrons can occupy the same quantum-mechanical state in a given system
i.e. no two electrons in an atom can have the same values of all four quantum numbers n,l,ml,ml
chemical properties of an atom are determined principally by
interactions involving the outermost (valence) electrons
chemical behaviour due to electronic configuration
just the same as higher/advanced higher chem
eg: noble gas filled shell, alkali metals ‘noble gas plus one’, halogens ‘noble gas minus one’
3d and 4s
3d and 4s have similar energies
in potassium, additional electron goes to 4s state as energy lower than 3d (transition metals)
(same again starting Z=57 and Z=89)
in classical physics we describe the interaction of charged particles in terms of
coulomb’s law forces
in quantum mechanics, we describe interaction in terms of
emission and absorption of photons
eg: two electrons repelling each other. Think of it as one throwing out a photon and other catches
uncertainty principle in terms of t and E`
ΔEΔt>/= h bar/2
virtual photon
uncertainty allows the creation of a photon with energy ΔE provided it lives no longer than Δt.
(like borrowing energy from a bank, you can have it as long as you pay back within time frame)
nuclear potential energy between two nucleons
in this equation, f represents strength of the interaction and r0 is the range
r is the distance at which the potential is measured
comparing coulomb potential energy with Yukawa potential energy (e^-r/r0 / r)
two functions are similar for small values of r but Yukawa potential energy drops off much more quickly for larger values of r
how to predict the approximate lifetime of a particle
must live long enough, Δt to travel distance comparable to range of nuclear force. This range is of the order 1.5fm. Assuming speed is comparable to c, its lifetime must be of the order 5*10^24s
alternative mass unit
MeV/c^2 (using E=mc^2)
all particles are created through
the interactions between other particles and involve the exchange of virtual particles that exist due to borrowed energy allowed by the uncertainty principle.
In decreasing order of strength, four interactions are
- strong interaction
- electromagnetic interaction
- weak interaction
- gravitational interaction
spin of photon
1
spin of graviton
2
strong interaction
responsible for nuclear force and production of pions (and other particles)
units of constant f^2
energy times distance
basis for comparison with other forces
dimensionless ratio f^2/h barc
called the coupling constant for the interaction
weak interaction
responsible for beta decay (eg neutron into proton, electron and anti-neutrino)
W+,W- and Z0 are short lived, have spin 1, have mass
result of mediating particles for weak force having mass
range much shorter than string force so weaker by a factor of 10^9
hadrons
includes mesons and baryons
femions
half integer spins
obey exclusion principle
bosons
zero or integer spins
do not obey the exclusion principle
the 6 leptons
electron, muon and tau and their associated neutrinos (all has a distinct anti-particle)
spin of leptons
all have spin 1/2 so are fermions
lepton conservation principle
3 lepton numbers Le,Lµ and L𝜏
electron and electron neutrino have Le=1 and antiparticles have Le=-1
same idea for muon and tau
*in all interactions, each lepton number is separately conserved**
quarks
spin half fermions that make up protons and neutrons (and other baryons/mesons)
each baryon consists of three quarks and anti baryon consists of three anti quarks
meson
quark,anti-quark pair
quark charge
have magnitude of 1/3e or 2/3e
principle of conservation of baryon number
analogous to conservation of lepton number
each quark has a value 1/3 for its baryon number and each anti quark has -1/3
conserved in all interactions
spin angular momentum in meson
components parallel to form a spin-1 meson or antiparallel to form a spin-0 meson
spin of baryon
form spin-half or spin-3/2 baryon
quark number conservation
conserved in strong interactions but not in weak interactions
Q/e for quarks
u,c,t = 2/3
d,s,b =-1/3
proton
uud
neutron
udd
pi+
u antidown
quark colour
come in three colours - red, green and blue
baryon contains one of each
gluon contains colour-anticolour pair
emission and absorption of a gluon by a quark
colour conserved
gluon exchange process
changes the colours of the quarks so that there is always one quark of each colour in every baryon
colour of an individual quark
changes continually as gluons are exchanged
mesons
spin 0 or 1
are bosons
there are no stable mesons - they all decay
only stable baryon
proton
criteria for a particle to be its own antiparticle
quark content the same
eg: c cbar = c bar c
conservation of strangeness
conserved in production processes but not usually when strange particles decay individually
general rule is conserved in strong interactions
bottomness
similar rule to strangeness (conserved in strong)
3 families of particles in the standard model
- 6 leptons
- 6 quarks
- mediating particles
what does the standard model not include
gravity
why is Higgs boson needed
for non-zero masses
electroweak theory
at low energies, electromagnetic and weak interactions behave differently due to the mass difference between the photons and the bosons
this disappears at high energy and merges into single interaction
grand unified theories
assumes that at very high energies, strong interaction will merge with electroweak interaction
some theories predict protons are not stable which violates conservation of baryon number
supersymmetric theories and theory of everything
ultimate goal to combine all four under one theory.
Need a space-time continuum with more than 4 dimensions with extra dimensions rolled up into tiny structure we do not notice
redshift
receding sources shifted to longer wavelengths
speed of recession
found by rearranging equation for wavelength
v=(λ0/λs)^2 -1 /(λ0/λs)^2+1 c
parsec
1/3600 degree = 1 arcsecond
estimation of age of universe from hubble’s law
14 billion years
assumes all speeds constant after Big Bang, ignores change in expansion rate caused by gravitational interactions
inflating balloon Universe
as balloon gets bigger, radius gets bigger but the specific coordinates of a point on the surface do not change
distance between points increases, as does the rate of change of distance
taking inflating balloon idea into 3D space
need the curvature of space
scale factor R describes size of the universe, R0 denotes scale factor today
(without subscript is any value past, present or future)
further an object is
the longer it takes light to get to us and the greater the change in R and λ
whether universe continues to expand indefinitely depend on
average density of the matter in the Universe
denser=lot of gravitational attraction to slow and eventually stop expansion and contract again
critical density
needed to just stop the expansion continuing indefinitely
ways of working out density of universe
- counting number of galaxies in patch of sky, average mass of star and average number of stars in average galaxy
- study motion of galaxies within galaxy clusters by monitoring redshift to get an idea of speeds. Speeds related to gravitational force exerted on each galaxy by other cluster members which is related to mass.
dark matter
average density of all matter in the universe is around 26% of critical density. Average density of luminous matter is only 4%
majority of the Universe does not emit electromagnetic radiation - i.e. dark matter
candidates for dark matter
- WIMPs - subatomic particles heavier than others
- MACHOs - black holes that might form halos around galaxies
proof that universe is continuing to expand
if expansion was slowing, must have been faster in past so would expect very distant galaxies to have greater redshifts than predicted by Hubble law
dark energy
explains why rate of expansion is increasing despite gravitational attractions
energy density of dark energy is nearly 3 times greater than that of matter so expansion will never stop and universe will never contract
planck length
assumed that at high energies and short distances, gravitation unites with the others.
Plancks length is the length as which this happens.
planck time
time required for light to travel planck length
exoergic reactions
release energy, heating up the star
