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Definition of a black body
An object which absorbs 100% of the incident radiation whose emission spectrum is determined by its temperature
Equation for determine the peak of a black body emission spectra
LambdaT = 0.0029 mK
Equation for total power radiated by a blackbody: Stefan Boltzmann Law
P = sigmaT4 W/m²
equation for possible number of nodes for a wave of wavelenght lambda within a 1D cavity
n = 2a/lambda
What was the ultraviolet catastrophe
Classical theory predicted that as lambda goes to zero, the number of possible wavelength nodes goes to infinity, leading to infinite power emission
What idea solved the ultraviolet catastrophe
the quantization of energy
planck’s equation for the quantization of energy
hf = hc/lambda
what theories does the photoelectric effect explain
the KE of electrons is not dependent on light intensity, the KE is dependent on wavelength and reaches a cutoff above which there is no emission, electrons are emitted instantaneously
equation for quantized orbital angular momentum postulated by Bohr
L = nhbar
does the Bohr model of an atom correctly predict energy levels for all atoms?
no, only H and H like atoms
how did Planck use quantization to explain experimental observations of blackbody radiation
the nodes of a blackbody wavelength in a cavity is quantized such that ndeltaE = nhf, fit this to experimental data and found h
how did Einstein use quantization to explain experimental observations of the photoelectric effect
takes planck further and says that all EM radiation is quantized, solved problem of wavelength cutoff by determining the cutoff is based on frequency: hf = W + KE → fmin = W/h = c/lambdacutoff
How did Bohr use quantization to explain experimental observations of atomic spectra
the angular momentum of electrons in an atom is quantized as L = nh/2pi, this means only certain orbits of certain radii are allowed, matched H atom spectra
explain the stern gerlach experiment
passed a beam of atoms through a changing magnetic field, the atoms were deflected to two specific spots showing that muz could only have two values, intrinsic angular momentum that could be spin up or spin down
explain compton scattering
light can be treated as particles with momentum that can engage in inelastic collisions with free electrons, lose different amounts of energy based on scattering angle
what three postulates were needed in the compton scattering experiment
photons have momentu p = h/lambda, electrons are free (if photon energy is » binding energy), and relativistic momentum and energy are conserved
what kind of wavelengths give observable results in the compton scattering experiment
high-energy: x-rays or gamma
equation for the de Broglie wavelength
lamda = h/p
what does de Broglie’s postulate for particle waves explain
why quantum effects aren’t noticeable (h and lambda are small) and why electrons in Bohr model don’t loose energy while accelerating in their orbits (form standing waves)
How could you prove that light behaves like a wave?
diffraction and interference testing by passing xray and beam of electrons through crystal lattice
equation of a plane wave used to describe particles
Aexpi(kx - wt) where k: wavenumber (p/hbar) and w: E/hbar
what is a wavepacket and how is it produced
the superposition of a number of waves of varying wavelengths results in a beating pattern → center is constructive interference and going away from there it tends to 0
how does the creation of a wavepacket reduce certainty in momentum
the more wavelengths we add to the superposition, the more the uncertainty in delta x is decreased, the more wavenumbers k are involved, the more uncertain k is, and as delta k = delta p/hbar, the more uncertain p is
equation for finding uncertainty of position compared to momentum
deltap * delta x = h/2
what form of the uncertainty principle relates energy and time: i.e the uncertainty of energy depends on how long it exists/is observed
delta E * delta t = hbar
how to combine wavelength equations to find quantized energy levels for particles within a box
n = 2a/lambda and lambda = h/p and E = p2/2m → E = n2h2/8ma2
what facts are true for the quantized energy levels of a particle confined in a 1D potential well
the narrower the confinement (smaller a) the greater the separation of energy levels, as delta p decreases the minimum KE needed to jump increases, the longest wavelength is related to the lowest energy level and the lowest momentum
differences between a classical potential well and a quantum well
classical wells have continuous energy levels and a constant probability density and the lowest energy state is E=0, quantum wells have quantized energy levels, variable probability densities, the lowest energy level is non zero, and a wave function that goes to zero at the edges of the well and at nodes
what equations developed by Einstein and de Broglie respectfully are necessary for creating a particle wave equation
E = hf = hbarw and p = hbark
equation for the probability of finding a wavefunction anywhere
P = 1 = integral from -infinity to +infinity of the wavefunction * wavefunction* dx
equation for finding the expectation value <x> for a particle’s position in a wave function (its average position)
<x> = (integral from -infinity to +infinity of wavefuntion*xwavefuntion dx)/(integral from -infinity to +infinity of wavefuntion*wavefuntion dx)
what is required of wavefunctions and their derivatives in order for the wavefunctions to be “well-behaved”
finite, single valued, continuous, and the wavefuntion goes to zero as x goes to ±infinity
what is the general solution for the TISE in 1D where E>V
Aexp(+ikx) + Bexp(-ikx) or A’coskx + B’sinkx (from Euler)
what is the general solution for the TISE in 1D when E<V
Cexp(alphax) + Dexp(-alphax) where alpha2 = 2m/hbar2 (V0 - E)
what is the difference between solutions to the TISE in 1D for E>V and E<V
for E>V the solutions are waves, for E<V the solutions are an exponential decay to 0
what is the odd parity solution to the TISE for an infinite square well with boundaries at x = -a/2 and x = a/2
A=0, B≠0, wavefunction = Bsin(ka/2) = 0, k = npi/a, plug into wavefunction
what is the even parity solution for a TISE in an infinite square well
A≠0, B=0, wavefunction = Acos(ka/2), k = npi/a, plug into wavefunction
what do odd and even parity solutions reflect
the symmetry of the wavefuntion: even is symmetrical when flipped across y axis, odd is symmetrical when flipped across y and x
equations for the allowed wavenumbers and energies for an infinite square well
k = (2mE/hbar2)1/2 = npi/a (can rearrange to find E)
what do even and odd equations do when calculating the expectation value of a wavefuntion in an infinite well
integrating odd functions between equal and opposite limits gives 0, integrating odd functions between equal and opposite limits means you can multiply just the integral from 0 to the positive side and multiply by 2
equation for momentum when solving for the expectation value of momentum
phat = -ihbar(d/dx)
what are main differences between infinite well solutions and finite well solutions for the TISE
energy eigenvalues are lower in the finite well compared to the infinite, as the wavefunctions extend into classically forbidden areas the wavelengths are longer meaning they have lower momentum and energy, there are a finite number of levels in the finite well and they reduce as V0 reduces, always at least one bound state
equation for potential energy in a simple harmonic oscillator
V = ½ kx2
for a simple harmonic oscillator, what is the genera solution and the separation between energy levels
E = hbarw(n + 1/2) with separatioin hbarw
qualitative facts of potential wells where E>V
solution is a wave, nodes is equal to n-1, as KE (E-V) decreases momentum decreases and wavelength increases
qualitative facts about solutions for potential well where V>E
solution is exponential decay, the smaller the depth of the well (V-E) the greater the barrier penetration, alpha2=2m/hbar2 (V-E), and the wavefunction is proportional to exp(-alphax2)
definition of probability flux
the probability of finding a particle per unit area and per unit time, J = hbark/m |wavefunction|2
equations for reflection and transmission coefficients when E>V0
R = Jreflected/Jincident and T = Jtransmitted/Jincident
how are the reflection and transmission coefficients affected as E grows larger than V0
more waves are transmitted, less are reflected
for V0>E, what are the reflection and transmission coefficients
R = Jreflected/Jincident and T = 0 (still finite probability of wave penetrating the step)
how are wavefunctions changed at each barrier
wavelength stays the same, amplitude decreases
What odd effect is apparent from the transmission and reflection coefficients for a wave passing through a barrier when E>V0
transmission can go to 1 when reflected waves off each barrier destructively interfere with eachother: when k2a = npi
equations for the eigenfunction (wavefunction) for a particle trapped in a 2D infinite potential well and the eigenstates (energy levels)
wavefunction(x,y) = (2/a)sin(nxpix/a)sin(nypiy/a) and E = (h2/8ma2)(nx2+ny2)
definition of degeneracy
same eigenstate (energy), different eigenfunction
equation for the reduced mass
m1m2/m1+m2
what are the quantum numbers associated with the Hydrogen atom
n, l, and ml
define n, the principle quantum number
n defines the energy of the electron and for each level n there are n degenerate states with l different quantum numbers
define the quantum number l
the orbital angular momentum quantum number: L = hbar(l(l+1))1/2
define ml
represents the component of L along a specific axis: Lz = mlhbar
equation for the angle of precession for an eigenstate (l,ml)
costheta = ml / (l(l+1))1/2
how many ml states are there for ever l level
2l + 1
how many l states are there fore every n levels
n
for probability distributions in hydrogen atoms, what can be said about how quantum numbers affect the graphs
for larger l there is lower probability of being close to 0, the number of nodes (P(r) = 0) is equal to n - l - 1
definition of a macrostate
specified by the global properties of the system, the thermodynamic state of the system
definition of microstate
a set of single particle states (the set of parameters that describe an individual particle)
definition of statistical weight (t)
the number of microstates associated with a distribution
equation for the total number of available microstates
Omega = sum(t{ni})
equation for the statistical weight
t{ni} = N!/multi(ni!)
what is the Central Postulate of Statistical Mechanics
all accessible microstates in an isolated system are equally probable
equation for calculating the average distribution across energy levels for single particle states
(sum(number of particles * t))/(total number of microstates)
equation for calculating average energy of a single particle state
average distribution * energy
what is boltzman’s definition of entropy
S = kBlnOmega
from boltzman’s definition of entropy, what rules can be derived for two identical systems
Omegatotal = OmegaA2 and Stotal = 2SA
what is the relationship between entropy and the number of microstates
larger Omega means larger S, this dependence is logarithmic
for large systems, how does the definition of the average distribution changed
for large systems, the statistical weight of most distributions is negligible such that the highest statistical weight is equivalent to the average distribution
equation for single particle state populations derived from the boltzman distribution
ni = (giN/Z)exp(-ei/kBT)
equation for the partition function
Z = sum(giexp(-ei/kBT))
equation for the probability of being in level i
pi = ni/N = (gi/Z)exp(-ei/kBT)
equation for the internal energy of a system
U = sum(niei) = (N/Z)sum(eigi(exp(-ei/kBT)))
how does the probability of an electron being in a certain energy state affect the photon emitted
energy levels that are twice as likely to be populated will emit photons of twice the intensity
what linear combinations satisfy the requirement that the probability density of |wavefunction(x1,x2)|² = |wavefunction(x2, x1)|² as required for indistinguishable particles in quantum states alpha (x1) and beta (x2) respectively
wavefunction = C[wavefunctionalpha(x1)wavefunctionbeta(x2) ± wavefunctionalpha(x2)wavefunctionbeta(x1)] for symmetric and antisymmetric cases respectively
define fermions
particles with wavefunctions that are antisymmetric under particle exchange, obey the Pauli Exclusion principle, have half-integer spin values, and examples of which are electrons, protons, and neutronsd
define bosons
particles with wavefunctions that are symmetric under particle exchange, do not have to obey the pauli exclusion principle, have integer spin values, and examples of which are photons and alpha particles
for systems with indistinguishable particles and very large number of states, what three cases can be used and when
fermi-dirace statistics (for fermions), bose-einstein statistics (for bosons), and dilute gasses (when number of particles is small compared to number of states)
equation for the total statistical weight for a particular distribution under Fermi-Dirac statistics
t{ni} = product(gi!/ni!(gi-ni)!)
equation for the total statistical weight for a particular distribution under Bose-Einstein statistics
t{ni} = product((gi+ni)!/ni!gi!
equation for the total statistical weight for a particular distribution under Dilute Gas statistics
equivalent to boltzman statistics, t{ni} = product(gini/ni!)
equation for the overall distributions of indistinguishable particles for each of the three cases
ni = gi / Cexp(ei/kBT)+delta where delta = 1,0,-1 for FD, B, and BE statistics respectively and C = exp(-alpha)
average occupancy of states at a given energy for the three indistinguishable particle cases
fi = ni/gi
for systems with closely spaced energy levels, how are the distributions and partition functions naturally adjusted
by representing the levels as a continuous function, n(e)de = (Ng(e)/Z)exp(-e/kBT)de and Z = the integral from 0 to infinity of g(e)exp(-e/kBT)de
equation for the number of states with k values between k and k+dk based on the density of states in 1D
g(k)dk = (a/pi)dk
equation for the number of states with k values between k and k+dk based on the density of states in 2D
g(k)dk = (a2/2pi)kdk
equation for the number of states with k values between k and k+dk based on the density of states in 3D
g(k)dk = G(V/2pi2)k2dk where V = a3 and G is the degerenacy G = 2S+1
how can the density of states be manipulated to be in terms of other variables
g(k)dk = g(x)dx : can define the counting as the number of particles with any variable between x and x+dx
equation for the density of states in a 3D space in terms of energy
g(e)de = G(V/4pi2)(2m/hbar2)3/2e1/2de where G is the degeneracy G = 2S+1
equation for the number of particles in a system with energy between e and e+de
n(e)de = f(e)g(e)de
equation for the partition function for a dilute gas
Z = V(2pimkBT/h2)3/2
equation for the dilute gas distribution
n(e)de = (2N/(kBT)3/2)(e/pi)1/2exp(-e/kBT)de
equation for the fraction of particles in a dilute gas with energy between e and e+de
p(e)de = n(e)de/N
equation for the Maxwell Boltzman Speed Distribution for a dilute gas
p(v)dv = 4pi(m/2pikBT)3/2v2exp(-mv2/2kBT)dv