Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Thermal and Statistical Physics > Spring > Flashcards

Spring Flashcards

You may prefer our related Brainscape-certified flashcards:
Physics 101 flashcards AP® Physics flashcards
1
Q

In 1D, how is wavenumber related to wavelength?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

For a particle in a box, how is energy level spacing related to L (length of sides of box)?

A

Energy level spacing decreases as L increases (levels get closer together)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Roughly, how is density of states found in 1D?

A

By finding the number of k states between k and k+Δk.
- by dividing width from k to k+Δk by the gap between each state
The taking this number of states and dividing it by the gap Δk

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

How is partition function found using density of k states?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

For 2D and 3D what does this represent?

A

Number of intersections of equally spaced lines which correspond to an allowed energy level

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is E(k)? What varies between dimensions?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

How is the number of states between k and k+Δk related to the number of states between E and E+ΔE?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What is the formula for D(E)?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

For typical E, which dimensions have D(E) independent of E?

A

2D only
D(E) for 3D depends on sqrt E
D(E) for 1D is proportional to 1/sqrt E

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Is D(E) always the same for each dimension?

A

No, D(E) depends on E’s dependence on k so varies e.g. for an electron in a graphene sheet (E=ℏsk)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

What is the equation for Z without E explicitly subbed in?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

How can Z be evaluated using D(E)?

A

Subbing in D(E) and evaluating over dE

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

In words, what is energy density of states?

A

The number of states with energy between E and E+ΔE, divided by ΔE

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

Do both D(E) and D(k) change for materials?

A

No, D(k) always the same for each dimension (1D, 2D, 3D) but D(E) can change for non-parabolic relations between E and k

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

What does the Boltzmann probability distribution tell us? What is the equation?

A

The probability that any one of the particles occupies a particular quantum state of energy E

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

How can the Boltzmann probability distribution be used to find average number of particles in each state? What are we assuming here?

A

Where Na is amount of particles in whole box.

We assume na(E) «1 so we don’t worry about several particles on one level

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

From the average number of particles in a state, na, how can average number of particles with energy between E and E+dE be found?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Z is?

A

The partition function

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

na(E) is ?

A

Number of particles in a particular energy state

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

What is λD? Why do we introduce it?

A

Thermal deBroglie Wavelength corresponding to a particle whose total KE is kBT.
Introduced to simplify Z

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

How does energy distribution relate to volume of gas?

A

It’s independent.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

What is α(E)? (in words and as a product of other values)

A

Number of particles per unit energy between E and E+dE.
= naD(E)
dN=α(E)dE

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

How does α(E) (particles per unit energy) depend on E?

A

Product of an exponential term and a sqrt dependence so gives rise to a peak in dependence.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

On a graph of α(E) against E, roughly what value does the peak take?

A

3/2KBT

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

When deriving the Maxwell speed distribution, what energy equation is used? What kind of energy is this?

A

E=1/2mu2
Just kinetic

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

When deriving the maxwell speed distribution, we start with dN and rearrange the dE term to what?

A

dE = mudu where u is speed

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

What is n(u)?

A

Number of particles, per unit speed, with speed between u and u+du

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

Why is the maxwell speed distribution indpendent of ℏ?

A

Because the density of states used to derive it assumed that quantized energy levels are so close they act as a continuum and therefore effects of discrete energy levels are lost

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
29
Q

What is n(u)du? What is du?

A

Number of particles in the speed interval u -> u + du where du is that difference between either side of the interval

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
30
Q

What exactly is λD?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
31
Q

For a quantity, A(u), that depends on speed, u, how is the average of that quantity found? Assuming it obeys Maxwell-Boltzmann statistics.

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
32
Q

What is the integral of n(u)du from 0 to infinity equal to?

A

Na total number of particles

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
33
Q

What integral would you evauluate to find mean magnitude of the speed of a particle in a Maxwellian gas?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
34
Q

What integral would you evaluate to find mean square speed of a particle in a Maxwellian gas?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
35
Q

For root mean square speed, what order do the mean, square, and root take place in?

A

square, mean, root.
a mean is taken of u2 and then this is rooted

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
36
Q

Does root mean square speed equal mean speed?

A

No

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
37
Q

What question are we attempting to answer with ‘‘molecular beams’’?

A

speed distribution of a gas emerging through a hole in a container wall

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
38
Q

What happens in the process of MBE?

A

Molecular Beam Epitaxy: atoms are heated in ovens and fired at something

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
39
Q

For the problem of a particle escaping through a hole we consider slightly different ranges for spherical co-ordinates, what are these and why?

A

θ from 0 to pi/2 (compared to normal up to pi, because we don’t want the particle going backwards)
φ from 0 to 2pi (same as normal)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
40
Q

How do we find what fraction of particles in a box volume, V, would escape in a given time, t ?

A

All particles within a particular cylinder of length ut (have to travel the length in time) and cross section Acosθ will make it out. (A is area of hole at a slant of angle θ)
therefore

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
41
Q

To find particles leaving a hole:
Once we have found the fraction of particles that would lie in a particular cylinder, what is done next and how?

A

We find the number of cylinders that end in the hole Acosθ on the wall. This is done using small changes in spherical co-ords.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
42
Q

Once we have found this value, how do we find total number of atoms to escape in time t?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
43
Q

How would we find flux of atoms escaping through a hole (number of atoms per time per unit area of the hole)

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
44
Q

How is a black body defined?

A

A black body absorbs all electromagnetic radiation that is incident on it, none is reflected.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
45
Q

As a black body gets hotter what happens to the photons and energy it emits?

A

1) emitted photons have more energy, hence lower wavelength
2) it emits more total energy (across all wavelengths)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
46
Q

How can overall fraction of photon’s escaping a black body be found (explanation and integral)?

A
  • model cylinders that end in exit area A, find fraction of velocities that end in this A point. Integrate over all u
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
47
Q

For a black body, from the fraction of photons that can escape a given hole in time t, how do we find energy emitted in time t?

A

Multiplying the fraction Ftot by total energy of radiation in the cavity.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
48
Q

What is rate of emission / power for a black body? Define terms

A

dQ/dt = cuA/4
c = speed of light
u = energy per unit vol
A = area of hole

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
49
Q

How can we adjust energy per unit volume u to a continuum for a range of wavelengths?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
50
Q

What is u(λ)?

A

Spectral density

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
51
Q

How is energy per unit volume related to spectral density?

A

u=energy per unit vol
u(λ)=spectral density

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
52
Q

How does u(λ) relate to wavelength and T (temp)

A

u(λ), for a given temp T, peaks at a particular wavelength, this wavelength decreases as T increases

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
53
Q

How does rate of energy emission dQ/dt relate to spectral energy density? What law is this?

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
54
Q

What is Wein’s scaling law?

A

That the curve of wavelength against spectral density can be made universal by plotting u(λ)/T5 against λT

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
55
Q

What did Rayleigh and Jeans assume?

A

That energy per radiation mode is equal to the average energy of each particle oscillating in the wall

56
Q

When deriving density of modes, D(λ)dλ is equal to what expression in k?

A

2D(k)dk

-then we can get dk in terms of dλ and use D(k) eq from formula sheet

57
Q

What happens to radiation incident on a black body?

A

It is absorbed and re-emitted.

58
Q

By Wein’s law, plotting what two variables gives a universal curve?

A

u(λ)/T5 against λT

59
Q

What are the two main problems with the Rayleigh jeans model?

A

1) infinite power output
2) spectral density tends to infinity as λ tends to 0

60
Q

How is u(λ)dλ related to E(λ) generally speaking?

A
61
Q

How did Planck explain energy for each EM mode to explain blackbody radiation?

A

Assumed energy for each mode came from simple harmonic motion at a set frequency w and radiation is generated by a transition between these levels.

62
Q

When deriving energy of particles for a harmonic oscillator, what do we assume about En?

A

ignoring 0 point energy -

63
Q

The harmonic oscillator is a rare case we can calculate what?

A
64
Q

How would we find average energy of an oscillating particle with frequency w? (steps)

A

1) Start with equation for calculating partition function, z, from En (shown here)
2) Use En = nh(bar)w
3) use equation for energy from partition function (formula sheet)

65
Q

What does Planck’s model for radiation from an object correctly predict?

A

Shape of the emission spectrum
Total power emitted

66
Q

How do we find peak wavelength of spectra density under Planck’s distribution?

A

Differentiate equation for u(λ) wrt λ and set to 0.
Split into two halves, plot and look for intersection

67
Q

What logic is the classical model of specific heat capacity based on? When does it hold true?

A
  • each oscillation has energy kBT
  • a solid has 3 directions of oscillation
  • total internal energy = Etot=3NkBT
  • therefore CV =

which gives 3R for 1 mole and holds above room temp

68
Q

When does the classical model of specific heat capacity agree with experiments?

A

T larger than or equal to room temp

69
Q

What turned out to be wrong with Einstein’s model for specific heat?

A

It assumed all particles oscillate at the same frequency (Einstein frequency)

70
Q

What does Einstein’s model for specific heat capacity incorrectly predict?

A

That CV would tend to 0 exponentially not proportional to T3

71
Q

What is the Dulong-Petit law?

A

That around room temp and above, CV tends to 3R (molar gas constant)

72
Q

What is the foundation of Debye’s model of specific heat?

A

That atoms couple to each other (coupled to all others) and oscillate coherently - propagating sound waves

The sound waves are quantized and are transverse or longitudinal

73
Q

What is the difference between longitudinal and transverse phonons?

A

Longitudinal phonons only oscillate in 1 direction (quantization direction)
Transverse propagating perpendicular to quantization direction so one of two ways (2 polarizations)

74
Q

How is angular frequency related to speed, v, and wave vector k?

A

w = vk

75
Q

What is the density of modes for all 3 types of phonon?

A
76
Q

If N oscillators are coupled together, how many possible modes of vibration do they have? How many total modes is this?

A

N possible modes of vibration
And 3 directions, so 3N total phonon modes

77
Q

Define the 3 Debye quantities? (other than wavelength)

A
78
Q

Why does Debye’s model fail at T <10K?

A

It neglects contribution of conduction electrons to internal energy and specific heat

79
Q

From Clausius’s principle, the entropy of an isolated system…….?

A

tends to a maximum

80
Q

Is entropy extensive or intensive?

A

Extensive, proportional to number of particles

81
Q

What is the chemical potential for a system with N particles?

A
82
Q

Physically / qualitatively, what is chemical potential ?

A

It measures the rate at which a system’s entropy changes as particles are added.

83
Q

How does particle behaviour depend on chemical potential?

A

Particles flow away from the system with the highest chemical potential

84
Q

If particles are reacting to make new particles, how are their chem potentials related?

A

The sum of the chem potentials of the reacting particles equals the chemical potential of the particles produced

85
Q

Entropy _________ on the approach to equilibrium.

A

Increases

86
Q

What is the general condition for equilibrium in chemical reactions?

A
87
Q

In first term we saw, knowing S also depends on N, how can we adjust this?

A
88
Q

What are two alternative definitions for chemical potential involving E and F?

A
89
Q

What is W in the Bolztmann distribution?

A

Number of ways in which the particles of a system can be distributed amongst the energy levels of the system

90
Q

How can we write chemical potential, utilising the Boltzmann distribution?

A
91
Q

What sign does chemical potential take?

A

Always negative

92
Q

How is F related to Z? What are F and Z?

A

Helmholtz free energy and Partition function

93
Q

How do we convert from Z formula for distinguishable particles to indistinguishable particles?

A

divide by N!, where N is number of particles, to account for the amount of states that are the same when particles are indistinguishable

94
Q

Degeneracy, g, equates to?

A

g distinct quantum states

95
Q

How is chemical potential related to z?

A
96
Q

How does the partition function for N distinguishable and N indistinguishable particles vary?
What has to be the case for this to hold?

A

Only hold when prob of two particles on the same level is low e.g high temp

97
Q

What must be true about probability density for indistinguishable particles?

A

Probability density must stay the same when particles are swapped

98
Q

What is ‘phase factor’, K, what purpose does it serve?

A

The factor that relates two wavefunctions when particles are swapped - disappears when prob density is taken

99
Q

Wavefunctions with phase factors +1 or -1 are called?

A

symmetrical (K=+1)
anti-symmetrical (K=-1)

100
Q

Particles with symmetrical wavefunctions (K=+1) are called..?

A

Bosons

101
Q

What kind of spin quantum numbers do bosons have?

A

Integer spin quantum numbers, s=0,1,2,3,…..

102
Q

Particles with antisymmetric wavefunctions (k=-1) are called?

A

Fermions

103
Q

What kind of spin quantum numbers do fermions have?

A

half-integer s=1/2,3/2….

104
Q

What are some examples of Bosons?

A

Photons, phonons, mesons..

105
Q

What are some examples of fermions?

A

electrons, protons, neutrons

106
Q

If a particle has total spin number s, how many different quantum states does each allowed energy level correspond to?

A

2s+1

107
Q

When two indistinguishable particles are swapped, the wavefunction changes, for this module, how does it change?

A

either the same, or negative

108
Q

For n bosons, how many terms do we need in a wavefunction to maintain symmetry?

A

n!

109
Q

What derivation leads to the pauli-exclusion principle?

A
110
Q

Can two fermions have the same energy?

A

Yes but not the same spin too, if two fermions are in the same energy level they must have different spins.

111
Q

What is the grand canonical ensemble?

A

The set of all possible quantum states of a total system = system A + large reservoir/heat bath

112
Q

For a system, A, connected to a large reservoir - what is the probability of system A being in state B?

A

Equates on the formula sheet to ‘particle number not fixed’

113
Q

Break down the terms in the grand partition function.

A
114
Q

How can we rearrange entropy to be in terms of the grand partition function for systems with variable particle number?

A
115
Q

What is grand potential, qualitively?

A

an analogue of Free energy, F=E-TS for systems with variable particle amounts.

116
Q

How are the 3 thermodynamic variables derived from the equation from the grand potential?

A
117
Q

What units does the grand potential have?

A

Energy

118
Q

What is the fermi-dirac distribution, qualitatively?

A

The probability that a single particle state with energy ei is occupied for a given T and μ (where only 1 particle can occupy, e.g. fermions)

119
Q

What is the Bose-Einstein distribution, qualitatively?

A

The average number of bosons in a single particle state with energy ei

120
Q

Define terms in the grand potential equation:

A
121
Q

How is the probability of a fermion being up spin in an energy level related to its probability of being down in the same level?

A
122
Q

To find total number of fermions across states, we sum probability for each energy across all energies. We then multiply by 2, why?

A

To account for each energy level having two states, spin up and spin down

123
Q

Why are boson/fermion distinctions not significant at high temps?

A

At high enough temps, properties of a gas are independent of whether it’s made from bosons or fermions

124
Q

In a high T limit, why does the probability for the ground state affect the Fermi-Dirac distribution?

A

At high T, probability of lowest state being occupied is much smaller than 1, so the exp on the bottom of the probability equation is much larger than 1. this makes the +1 in the fermi dirac distribution negligible

125
Q

At high T, if we analyse a Fermi gas for pressure, what do we recover?

A

The ideal gas law

126
Q

Define fermi energy.

A

EF = the energy of the highest occupied state when T=0 (all levels below this will be full)

127
Q

What can be notes about energy levels with E<EF?

A

They are full as they are below the fermi energy

128
Q

Define Fermi wavevector.

A

kF = magnitude of the wavevector of the highest occupied state at T=0 (State where E=EF

129
Q

Define fermi temperature.

A

TF=EF/KB

130
Q

Define fermi velocity.

A

vF= h(bar)KF/ m

where KF is the fermi wavevector

131
Q

Why at T=absolute zero, is the pressure (quantum pressure) independent of T?

A

Because no more fermions can be ‘squished’ into the states below EF

132
Q

What is the qualitative meaning of TF?

A

Fermi gases at temperature much lower than the fermi temperature are said to be degenerate and can be used to act like they’re at T=0 with a very small amount of particles just above EF (about KBT over)

133
Q

What are conduction electrons?

A

The few electrons excited to levels within KBT of EF which contribute to specific heat at low T at a different proportion to other particles

134
Q

What do conduction electrons explain?

A

The deviation of Debye’s theory from real data on heat capacity as electrons dominate heat capacity at very low Ts

135
Q

What is the biggest difference between Bosons and Fermions at low T? What does this cause?

A

No limit on number of Bosons in an energy level so we expect a lot to fall into the lowest energy level - Bose Einstein condensation

136
Q

What is the critical temperature TC?

A

The temp, where higher than this no bosons will occupy the lowest energy state

137
Q
A

Thermal and Statistical Physics (3 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions