Semester 2: Questions Flashcards
(161 cards)
1
Q
What is the wavefunction for a particle in a 1D infinite potential well?
A
A = normalisation constant
ψ = wavefunction
k = nπ/L = wavenumber
2
Q
What is the wavefunction for a particle in a 2D infinite potential well?
A
A = normalisation constant
ψ = wavefunction
k = nπ/Lₓ or lπ/Lᵧ = wavenumber
3
Q
What is the wavefunction for a partible in a 3D infinite potential well?
A
A = normalisation constant
ψ = wavefunction
k = nπ/L_x or lπ/L_y = sπ/L_z = wavenumber
4
Q
What are the quantised energy levels of a particle in a 1D box?
A
E = energy level
k = wavenumber
m = mass
L = box width
5
Q
What are the quantised energy levels of a particle in a 2D box?
A
E = energy level
k = wavenumber
m = mass
L = box width
6
Q
What are the quantised energy levels of a particle in a 3D box?
A
E = energy level
k = wavenumber
m = mass
L = box width
7
Q
Define the partition function for a particle in a box
A
The sum of all energy levels from one to infinity for a particle in a box.
8
Q
What is k-space?
A
A representation of the spatial frequency of a system.
9
Q
What is the equation for the partition function for a particle in a 1D box?
A
Z = partition function
10
Q
What is the equation for the partition function for a particle in a 2D box?
A
Z = partition function
11
Q
What is the equation for the partition function for a particle in a 3D box?
A
Z = partition function
12
Q
What is the equation for the density of states in 1D k-space?
A
D(k) = density of states
N = number of states in given distance
L = length
13
Q
What is the equation for the density of states in 2D k-space?
A
D(k) = density of states
N = number of states in given area
L*L = area
14
Q
What is the equation for the density of states in 3D k-space?
A
D(k) = density of states
N = number of states in given volume
LLL = volume
15
Q
What length does an energy level occupy in 1D k-space?
A
Length = π/L
16
Q
What area does an energy level occupy in 2D k-space?
A
17
Q
What volume does an energy level occupy in 3D k-space?
A
18
Q
How can the partition function be re-written in terms of the density of states in 1D k-space?
A
Z = partition function
19
Q
How can the partition function be re-written in terms of the density of states in 2D k-space?
A
Z = partition function
20
Q
How can the partition function be re-written in terms of the density of states in 3D k-space?
A
Z = partition function
21
Q
Define energy density of states
A
The number of states per unit energy, D(E).
22
Q
What is the equation for the energy density of states?
A
N = number of quantum states whose energy is between E and E + ∆E
23
Q
What is the equation for the energy of a particle in a 1D, 2D, or 3D box?
A
E(k) = energy
k = wave vector magnitude (wavenumber)
m = mass
24
Q
What is the shape of the plot of energy against wave vector for a particle in a box?
A
25
How does the number of states in a given energy range relate to the number of states between k and k + ∆k?
D(k) = density of states
∆k = number of states
26
How does the wave vector range, ∆k, relate to its corresponding energy range, ∆E?
∆k = range of states
∆E = range of energies
27
What is the equation for the energy density of states in terms of the density of states in k-space?
D(E) = energy density of states
D(k) = density of states in k-space
28
What is the equation for the energy of a particle in a graphene sheet?
E(k) = energy
s = electron speed = 10⁶ m/s
k = wave vector
29
What is the equation for the energy density of states in a 1D box?
D(E) = energy density of states
L = length
m = mass
30
What is the equation for the energy density of states in a 2D box?
D(E) = energy density of states
L*L = area
m = mass
31
What is the equation for the energy density of states in a 3D box?
D(E) = energy density of states
L*L*L = volume
m = mass
32
What is the equation for the partition function of a particle in a box in terms of the energy density of states?
Z = partition function
E = energy
D(E) = energy density
33
What is the equation for the allowed energy levels of a single particle in a 3D box?
E = energy levels
m = mass
n, l, s = quantum states
34
How can the distribution of the energies (or speeds) of particles in a gas be found?
Using the Boltzmann probability distribution.
35
What is the equation for the Boltzmann probability distribution?
p(E) = probability of being in a given energy state
E = energy
T = temperature
Z = partition function
36
What is the equation for the number of particles in a given state?
nₐ = number of particles in a state
Nₐ = total number of particles
p(E) = probability of being in a given energy state
E = energy
T = temperature
Z = partition function
37
What is the equation for the average number of particles with energy between E and E + dE?
dN = average number of particles
nₐ = number of particles in a state
D(E) = energy density of state
Nₐ = total number of particles
Z = partition function
38
What is the equation for the partition function in terms of the thermal de Broglie wavelength?
Z = partition function
V = volume
λ = thermal de Broglie wavelength
39
What is the equation for the average number of particles with energy between E and E + dE (fully expanded)?
dN = average number of particles
Nₐ = total number of particles
λ = thermal de Broglie wavelength
E = energy
40
What is the graphical form of the energy distribution of a gas using the equation for the average number of particles with a given energy?
41
How can the Maxwell speed distribution be derived?
1) Take the equation for the average number of particles with a given energy.
2) Using the equation for kinetic energy and differentiate so that the energy interval can be replaced with a speed interval.
42
What is the equation for the Maxwell speed distribution?
n(u) = number of particles per unit speed with a speed between u and u + du
Nₐ = total number of particles
λ = thermal de Broglie wavelength
u = speed
43
What is the equation for the Maxwell speed distribution (eliminating the thermal de Broglie wavelength)?
n(u) = number of particles per unit speed with a speed between u and u + du
Nₐ = total number of particles
u = speed
44
What is the graphical form of the Maxwell speed distribution of a gas?
45
Why is the Maxwell speed distribution a classical distribution?
- The equation is independent of ħ.
- The exponential in this equation is also classical (from the Maxwell-Boltzmann distribution).
- The density of states term was assumed to be continuous as they were so close together.
46
What is the Maxwell speed distribution used for?
It can be used to find the mean speed of a gas because the mean value of a quantity that depends on speed can be found by integrating the product of that quantity and the Maxwell distribution then dividing it by the number of particles.
47
How can the mean speed (or mean square speed) of a Maxwellian gas be found?
1) Set a quantity that depends on speed, A(u), equal to the speed (or square of the speed).
2) Substitute it into the equation for the Maxwell speed distribution.
3) Integrate by substitution (let s² = value in the exponential).
4) Simplify the equation and divide it by Nₐ (the total number of particles).
48
Describe the graph of the Maxwell speed distribution in terms of the Maximum speed, mean speed, and root mean square speed
49
Give 2 examples of technological applications of narrow beams of particles in Physics
- Ultrasound atoms
- Molecular beam epitaxy (MBE)
50
How does molecular beam epitaxy work?
Atoms are heated in ovens (known as effusion cells) with each oven containing a different type of atom. A beam of these atoms emerges through a hole in each oven and lands on a substrate.
51
What is the equation for the fraction of atoms that emerge from an MBE oven in a given time?
F = fraction of atoms that emerge
u = speed
θ = angle of cylinder to the z-axis
52
How can the fraction of atoms that emerge from an MBE oven in a given time be calculated?
1) Assume the atoms follow a Maxwell distribution.
2) Only a fraction of atoms will escape, these atoms are found in a cylinder with length l = ut where u is the velocity and t is the time period.
3) The fraction of atoms that escape is the cylinder volume divided by the overall volume.
53
What coordinate system should be used when calculating the velocity of particles in an oven, given they escape out of a circular hole?
Spherical polar coordinates
54
What are the spherical polar coordinate ranges used when calculating the fraction of atoms that emerge from an MBE oven?
u: from 0 to infinity (depends on x, y, and z)
θ: from 0 to π/2 (for atoms approaching the hole)
φ: from 0 to 2π
55
What velocities and angles need to be required for atoms escaping an MBE oven?
Velocities between u and u + du, and angles between θ + dθ and φ + dφ. These are the atoms that can leave.
56
What is the equation for the flux leaving an MBE oven?
f = flux
Nₜₒₜ = total number of atoms
Nₐ = number of atoms per unit time
t = time
A = area
u = mean speed
V = volume
n(u) = number of particles per unit speed
57
How is the equation for the flux leaving an MBE oven derived?
58
What is the equation for the mean kinetic energy of a Maxwellian gas?
E = energy
m = mass
u = mean speed
59
What is the equation for the mean speed of a Maxwellian gas?
u = mean speed
T = temperature
m = mass
60
Define black body
An object that absorbs all radiation that is incident on it; none is reflected.
61
If a black body is in thermal equilibrium with its surroundings, its temperature is ________. Hence, it must both ___________ and ___________ all radiation that is incident on it, otherwise it would get warmer.
Constant
Absorb
Re-emit
62
What happens as a body gets hotter?
- The emitted photons have more energy (so a shorter wavelength)
- More total energy is emitted
63
At room temperature, where on the EM spectrum are most emissions?
Infrared
64
At high temperatures (~1000K), where on the EM spectrum are most emissions?
Visible light
65
At low temperatures, (a few degrees K), where on the EM spectrum are most emissions?
Cosmic microwave background radiation
66
How can a black body be modelled?
It can be modelled as an oven of volume, V, whose exterior is coated with a perfect thermal insulator. This means that radiation is only emitted into the interior of the oven.
67
What is the equation for the fraction of emitted radiation from a black-body oven in a given time?
F = fraction emitted
c = speed of light
t = time
V = volume
68
How can the fraction of emitted radiation be calculated for a black-body oven with a small hole in it?
1) Assume the photons follow a Maxwell distribution.
2) Only a fraction of atoms will escape, these atoms are found in a cylinder with length l = ct where c is the speed of light and t is the time period.
3) The fraction of atoms that escape is the cylinder volume divided by the overall volume.
69
What are the spherical polar coordinate ranges used when calculating the fraction of photons that emerge from a black-body oven?
u: only c
θ: from 0 to π/2 (for atoms approaching the hole)
φ: from 0 to 2π
70
What velocities and angles need to be required for atoms escaping an MBE oven?
Only angles between θ + dθ and φ + dφ. These are the atoms that can leave.
71
What is the equation for the flux leaving a black-body oven?
F = flux
c = speed of light
t = time
A = area
V = volume
72
How is the equation for the flux leaving a black-body oven derived?
F(θ) = fraction of the photons that escape in time, t
f(θ, φ) = fraction of photons with velocity vectors in the appropriate interval
73
What is the equation for the energy of radiation in the black-body oven?
ε = total energy of radiation
u = total energy density
V = volume
74
What is the equation for the rate of emission of photons out of a black-body oven?
Q = energy emitted in a given time
t = time
u = total energy density
A = area of hole
75
What is the equation for Stefan's law?
Q = energy emitted in a given time
t = time
u = spectral density
A = area of hole
λ = wavelength
T = temperature
76
Describe the shape of the graph for Stefan's law
77
What is the equation for Weins' law?
λ = maximum wavelength
T = temperature
78
What is the Rayleigh-Jeans theory for spectral density?
The assumption that the energy per radiation mode is equal to the average energy of each oscillating particle in the wall of a black-body oven (i.e. k_B*T). They developed a formula to explain this.
79
Why was the Rayleigh-Jeans theory incorrect?
- It predicted infinite power output which is impossible.
- The spectral density tended to infinity as the wavelength tended to 0.
80
What is the general formula for spectral energy density?
u(λ) = spectral energy density
E = energy
λ = wavelength
81
What was Planck's idea for the spectral energy density?
He assumed that the energy within each EM mode of frequency, ω = 2πc / λ, originated from particles performing simple harmonic motion (corresponding to the vibration of atoms within the solid) of the same frequency. He also assumed that the energy for this was quantised in discrete energy levels separated by ħω.
82
What was Planck's idea for the spectral energy density?
He assumed that the energy within each EM mode of frequency, ω = 2πc / λ, originated from particles performing simple harmonic motion (corresponding to the vibration of atoms within the solid) of the same frequency. He also assumed that the energy for this was quantised in discrete energy levels separated by ħω.
83
What is the equation for the energy levels of a simple harmonic oscillator?
E = energy
ω = mode of frequency
84
What is the equation for the energy levels of a simple harmonic oscillator (ignoring the zero point energy)?
E = energy
ω = mode of frequency
85
What is the equation for the partition function of a simple harmonic oscillator?
Z = partition function
E = energy
ω = mode of frequency
86
What is the equation for the average energy of each oscillating particle in a simple harmonic oscillator with frequency, ω?
E = energy
Z = partition function
ω = frequency
87
What is the equation for Planck's distribution (spectral energy density equation)?
u(λ) = spectral energy density
E = energy
λ = wavelength
88
What were the successes of Planck's spectral density model?
- The shape of the emission spectrum was correctly predicted as it matched Wein's scaling law.
- The total power emitted was correctly predicted as it matched Stefan's law.
89
When does the peak of Planck's distribution (for spectral density) occur?
This is when du/dλ = 0.
90
What is the value of the peak of Planck's distribution? How is is found?
x ~ 4.965
It is found using a computer or iteration.
91
How can Planck's distribution show Wein's scaling law?
By dividing the distribution by T⁵
92
How can Planck's distribution show Stefan's law?
If it is substituted into the integral for Stefan's law it produces the same answer, including the factor of T⁴.
93
Define specific heat
The amount of heat required to raise the temperature of 1mole of a body by 1K.
94
Classically, what are the equations for the internal energy and specific heat of a solid?
95
What was Einstein's model of specific heat?
He assumed that each atom performed SHM with quantised energy in each direction and that each atom is independent of one another. This meant that he used the partition function for a 1D oscillator to write equations for the total internal energy and specific heat.
96
What were Einstein's equations for total internal energy and specific heat?
97
Why was Einstein's model of specific heat incorrect?
It neglected inter-atomic coupling so it predicted that specific heat tended to 0 exponentially as T tended to zero (rather than as T cubed tended to zero).
98
Atoms couple to one another and oscillate coherently to create _________ sound waves.
Propagating
99
What are phonons?
Quantised sound waves that oscillate in each direction.
100
What is a longitudinal phonon?
Atoms that oscillate along the quantisation direction.
101
What is a transverse phonon?
Atoms that oscillate perpendicular to the quantisation direction.
102
Define Debye's model of specific heat in solids
A model that includes coupling of oscillators, which generates quantised sound waves within the solid.
103
How can the internal energy of phonons be found?
By calculating each atom's contribution to the internal energy (using the energy equation with the partition function) and multiplying that by the number of modes in the frequency interval. This equation should then be integrated.
104
What is the equation for the number of modes in a frequency interval?
105
What is the equation for the Debye frequency?
106
What is the equation for the Debye wavevector?
107
What is the equation for the Debye energy?
108
What is the equation for the Debye temperature?
109
What is the specific heat due to phonons in the high and low temperature limits?
High: ~ 3Nk_B
Low:
110
When does Debye's model for specific heat stop working? Why?
Down to ~10K. It stops working after that because the theory neglects the contribution of conduction electrons to internal energy and specific heat.
111
When does chemical equilibrium occur?
When entropy is maximised (dS = 0)
112
As a system approaches equilibrium its evolution is governed by _______________________________.
The need for entropy to increase
113
What is the equation for the change in entropy when particles are transferred between two systems (A and B) but no particles are added to the overall system (A + B)?
114
What is the equation for chemical potential?
µ = chemical potential
S = entropy
115
Define chemical potential
A measure of the rate at which a system's entropy changes as particles are added.
116
What is the equation for the change in entropy when particles can enter or leave two systems?
Hence, dS = 0 and µ = 0.
117
What are stoichiometric constants?
Constants that specify how many particles of each type are used or produced.
118
What is the equation for change in entropy in terms of chemical potential?
dS = change in entropy
µ = chemical potential
119
What is the equation for the first law of thermodynamics in terms of chemical potential?
E = energy
S = entropy
µ = chemical potential
N = number of molecules
120
What are the 3 equations for the different ways to calculate chemical potential?
µ = chemical potential
S = entropy
E = energy
F = Helmholtz free energy
121
What is the equation for chemical potential in terms of Gibbs free energy?
µ = chemical potential
G = Gibbs free energy
122
What is a distinguishable particle?
A particle that can be identified by some unique feature (e.g. people, coloured balls, etc.).
123
What is an indistinguishable particle?
A particle that has no features that allow us to tell them apart (e.g. red snooker balls, electrons, protons, neutrons, etc.).
124
What is the general partition function for N distinguishable particles?
Z = partition function
N = number of particles
125
What is the general partition function for N indistinguishable particles?
Z = partition function
N = number of particles
126
What is the equation for the chemical potential of a gas in terms of the partition function?
µ = chemical potential
Z = partition function
g = spin degeneracy
n = N/V = particle density
∆ = constant
127
Does the configuration of a system change if distinguishable particles are swapped?
Yes so it produces a measurable change in the physical properties of the system.
128
Does the configuration of a system change if indistinguishable particles are swapped?
No so it produces no measurable changes in the physical properties of the system.
129
Does the probability density of indistinguishable particles change if their position is swapped?
No
130
What are bosons?
Particles with symmetrical wave functions. They have integer spin quantum numbers (s = 0, 1, 2, ...). Photons are an example of bosons.
131
What are fermions?
Particles with antisymmetric wave functions. They have half-integer spin quantum numbers (s = 1/2, 3/2, ...). Electrons, protons, and neutrons are examples of fermions.
132
How many fermions can occupy a given quantum state?
Only one due to the Pauli exclusion principle.
133
How many bosons can occupy a given quantum state?
There are no restrictions on the number of bosons in a given quantum state.
134
What is the grand canonical ensemble?
The set of all possible quantum states of an isolated system that consists of a system, A, and a reservoir (heat bath). Heat can flow between the two systems.
135
What is the equation for the number of micro states in the reservoir of the grand canonical ensemble?
W = number of micro states
U = internal energy of reservoir
136
What is the equation for the probability of system A in the grand canonical ensemble being in a given state?
p = probability
W = number of microstates
Ξ = grand partition function
137
What is the equation for the grand partition function?
Ξ = grand partition function
138
Define the grand potential
The grand potential, Φ, is an analogue of the free energy (F = E - TS) for systems with a variable number of particles.
139
What can the grand potential be used for?
It can be used to derive the thermodynamic properties of systems with a varying number of particles.
140
What is the equation for the grand potential?
E = energy
N = mean number of particles
Ξ = grand partition function
Φ = grand potential
141
What is the equation for a small change in the grand potential?
Φ = grand potential
142
What is the equation to derive pressure using the differential form of the grand potential?
p = pressure
Φ = grand potential
Ξ = grand partition function
143
What is the equation to derive the mean number of particles using the differential form of the grand potential?
N = mean number of particles
Ξ = grand partition function
Φ = grand potential
µ = chemical potential
144
What is the equation to derive entropy using the differential form of the grand potential?
S = entropy
Ξ = grand partition function
Φ = grand potential
145
Define the Fermi-Dirac distribution for fermions
The probability that a single particle state with energy, ε, is occupied at given T and µ. It equals the average number of fermions in a given state.
146
What is the equation for the Fermi-Dirac distribution?
p(E) = probability of a given energy value
ε = state energy
147
Define the Bose-Einstein distribution for bosons
The average number of bosons in a given state of energy, ε.
148
What is the equation for the Bose-Einstein distribution?
N = average number of particles
ε = state energy
149
How does the shape of the Fermi-Dirac distribution change with increasing T?
The step function softens with increasing T. At T=0 it is a sharp step.
150
What is the equation for the number of particles in a 3D Fermi gas?
N = number of particles
V = volume
E(K) = energy of states
151
What does analysis of the pressure of a 3D Fermi gas at high temperatures show?
It shows that pressure is equal to the ideal gas law.
152
What is the equation for the Fermi wave vector?
k = Fermi wave vector
n = N/V = particle density
153
What is the equation for the Fermi energy?
E = Fermi energy
n = N/V = particle density
154
What is the equation for the average internal energy of a 3D Fermi gas at absolute zero?
E = average internal energy
155
What is the equation for the isovolumic heat capacity of a Fermi gas at low temperatures?
C = heat capacity
D(E) = density of Fermi energy
156
What does analysis of the pressure of a 3D Fermi gas at low temperatures show?
The pressure is equal to quantum pressure, so is independent of T.
157
What is Bose-Einstein condensation?
The transfer of bosons into the lowest energy level.
158
What is the consequence of Bose-Einstein condensation?
Near absolute zero, atoms begin to act collectively as a giant quantum wave.
159
What is the critical temperature?
The temperature at which bosons begin to fall into the lowest energy level (when Bose-Einstein condensation occurs).
160
What is the equation for the critical temperature in Bose-Einstein condensation?
161
What type of particle are photons and phonons?
Bosons, meaning that they can condense.
