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CHEM251 > Quantum Mechanics > Flashcards

Quantum Mechanics Flashcards

1
Q

What is a blackbody?

A

A blackbody is a material that converts heat to light

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2
Q

How does a blackbody emit radiation?

A

It absorbs heat and keeps it in a cavity until it reaches thermal equilibrium, then releases it as radiation through a pinhole

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3
Q

How is the energy density of blackbody radiation affected by temperature (energy)?

A

Higher energy = higher energy density

Temperature increase causes the maxima to shift to a shorter wavelength

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4
Q

Wien’s Law

A

ρ=C1/λ^5 ×1/e^(C2/λT)
Good at short wavelengths
Not good at long wavelengths

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5
Q

Wien’s Law assumptions

A

Makes assumptions about absorption and emission of electromagnetic radiation

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6
Q

Rayleigh-Jean theory

A

Treat electromagnetic field as a lot of oscillators
Calculate number with each energy
Predict’s a curve proportional to λ^-4

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7
Q

Rayleigh-Jean’s Law

A

ρ=8πkT/λ^4
Good at long wavelengths
Terrible at short wavelengths (ultraviolet catastrophe)

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8
Q

Ultraviolet catastrophe

A

Emission increases to ∞ at zero λ
High energies emitted strongly even at low T (literally everything glows in the dark - darkness doesn’t exist)
All bodies lose all energy fast

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9
Q

Planck Distribution

A

ρ=C1/λ^5 ×1/(e^(C2/λT) - 1)

A perfect fit to all λ

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10
Q

Quantum assumptions in the Planck Distribution

A

Walls of the blackbody contain harmonic oscillators with a frequency υ
Energy of each oscillator is quantised
E=nhυ n = 0, 1, 2, …
Oscillators can emit or absorb energy only in discrete amounts
ΔE=hυ

C1=8πch
C2=ch/kB

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11
Q

Planck’s Distribution with quantum assumptions

A

ρ=8πch/λ^5 × 1/(e^(ch/λkBT) - 1)

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12
Q

Photoelectric effect

A

The emission of electrons when light is shone onto specific materials

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13
Q

Wave-Particle Duality

A

Light can act as waves and as particles

Particles can also act as waves

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14
Q

Einstein’s theory about quantisation of light

A

Light consists of particles with energies (photons):
E=hυ
Light energy comes in discrete bundles (quanta):
E∝υ

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15
Q

De Broglie’s theory about wave-particle duality

A

Particles can behave as if they’re waves
λ=h/P
Proven through electron diffraction

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16
Q

Issues with

A

Can’t think of a small object (electron) in terms of trajectory (position and movement) - you have to think of the probability of an object being in a particular place (wavefunctions)

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17
Q

Translational energy of bodies

A

E = 1/2mv^2 + V
or
E=p^2/2m + V

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18
Q

Kinetic energy

A

The energy that the particle will possess as a result of momentum

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19
Q

Potential energy

A

The energy a particle possesses as a result of position

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20
Q

What do you need to exactly predict the trajectory

A

Starting position
Momentum
Potential

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21
Q

Force

A

Rate of change of momentum

F = dp/dt

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22
Q

Newtonian Mechanics

A

To predict trajectory you need to know the starting position, momentum, and potential
Any position and momentum (or energy) are possible
Depends on how long the force is applied for

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23
Q

Hooke’s Law

A

F=-kx

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24
Q

Frequency of an elastic band

A

Frequency is independent of energy
Frequency depends on structure
All vibrational energies are possible

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25
Q

Do classical mechanics apply to small objects

A

No

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26
Q

How does light work?

A

Energy transfer

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27
Q

How does reflection work?

A

Light reflects off of an object an interacts with the eye

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28
Q

What is a wavefunction?

A

A mathematical function that describes where an electron will most likely be at a point in time (its probability distribution)

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29
Q

Quantum Postulate 1

A

In one dimension, Ψ is a function of x Ψ(x)

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30
Q

What can be deduced from postulate 1?

A

Wavefunctions contain all dynamical information about a system
A wavefunction value has no physical meaning and yet Ψ contains all info that can be known about a system
Can get information by manipulating Ψ to provide real values relating to physical properties like:
- Energy
- Position
- Momentum

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31
Q

How is postulate 1 determined?

A

Schrodinger’s equation

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32
Q

Born Interpretation

A

Knowing that wavefunction has a particular value Ψ at a point in space r
The probability of finding an electron in a small volume (atom) is proportional to |Ψ|^2dτ

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33
Q

What is probability density?

A

|Ψ|^2 - square modulus of the wavefunction

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34
Q

An acceptable wavefunction must be:

A
  • Continuous
  • Single-valued (you can’t have multiple probabilities)
  • Have a continuous first derivative
  • Able to square integrate (can’t be infinite)
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35
Q

What is an operator?

A

Something you do to a function, e.g. Ĥ. You can use operators to pull info from Ψ to represent a real value.
Different operators represent different properties:
- One for energy
- One for momentum
- One for position
- Etc.

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36
Q

What are observables?

A

Made up of operators:
position operator x^ and momentum operator px^ in the x-direction of a particle moving in 1 dimension
x^ = x * something - usually Ψ
px^ = ħ/i

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37
Q

Heisenberg Uncertainty Principle

A 
ΔpΔx = ħ/2
ħ = h/2π
38
Q

Schrodinger Equation

A

(-ħ/ 2m)(d^2Ψ/dx^2) + V(x)Ψ = EΨ

ĤΨ = EΨ

39
Q

Ψ equation

A

Ψ - coskx + isinkx
k can take any value
E can take any value
Energy is still quantised because there are only certain functions in this equation that work

40
Q

What are the 3 types of motion?

A

Translational
Vibrational
Rotational

41
Q

True or False: any wavefunction and corresponding energy are possible in simple model systems

A

False - only certain wavefunctions and corresponding energies are acceptable (quantisation)

42
Q

How do molecules store energy?`

A

Through motion

43
Q

How are energy levels observed in IR?

A

Transitions between different vibrational energy levels and that energy is being stored by the system

44
Q

What does the vibrating string analogy show?

A

If 2 ends of a rope are fixed, only certain wavelengths are possible (because only a certain amount of nodes is possible)

45
Q

How are wavefunctions defined in terms of a particle in a 1D space?

A

Ψ(x) = 2πx/λ
where λ is restricted by 2L/n
L = length between the ends
n = 1, 2, 3,…

46
Q

What is the energy potential of a particle at the boundary of a 1D box?

A

Infinite

47
Q

What is the energy potential of a particle at the base of a 1D box?

A

Zero

48
Q

What is the probability of a particle being where energy potential is infinite in a 1D box?

A

Zero - this means that the particle has escaped past the boundaries

49
Q

What are the boundary conditions of a particle in a 1D box?

A

Ψ = 0 at x=0 and x=L

50
Q

How many nodes does Ψn have?

A

n-1

51
Q

Why is n=0 not a solution for Ψ in the particle in a 1D box?

A

Because if it was, Ψ would be zero everywhere and |Ψ^2 would also be zero, which isn’t possible because |Ψ^2 = 1

52
Q

True or False: a particle in a 1D box is able to have zero energy

A

False, a particle can’t be at rest because then its momentum would be exactly defined and its position would be totally undefined - but its position is defined because it’s in the box

53
Q

What are nodes in particle distribution diagrams?

A

Areas where there is zero probability of a particle being at a particle point along the line

54
Q

True or False: if n is large, particle distribution is uniform for a 1D box

A

True, if n isn’t large, distribution isn’t uniform

55
Q

True or False: a particle can have 0 energy

A

False, because n can’t be zero, the lowest energy a particle can possess isn’t zero

56
Q

Zero-point energy equation

A

E1= h^2/8mL^2

57
Q

ZPE explanation 1

A

Particles must have non-zero kinetic energy - this is in relation to the Heisenberg uncertainty principle

  • the particle is confined to a finite region
  • so the particle location can’t be completely indefinite
  • thus the momentum can’t be precisely zero
58
Q

ZPE explanation 2

A

Ψ=0 at walls but is smooth, continuous, and non-zero everywhere else - this means there must be a curve which implies possession of kinetic energy

59
Q

Equation for separation between energy levels

A

ΔE=(2n+1)h^2/8mL^2

60
Q

Relationship between energy and change of mass and length

A

ΔE will decrease as mass and/or length increases and increase as they decrease
ΔE is very small at macroscopic dimensions

61
Q

Is translational motion in atoms and particles in a beaker treated as quantised or not quantised?

A

Not quantised

62
Q

Correspondence Principle

A

Behaviour of quantum mechanical systems becomes more and more classical in the limit of large quantum numbers (n is very big)

63
Q

Can dye colour be predicted?

A

For dye to be coloured, it has to be absorbing in the EM spectrum
There is an interaction between light and electrons of molecules
Absorption of light results in a change of the quantised energy
The object is absorbing a certain wavelength or frequency of white light - what can be seen is what’s left over

64
Q

How to predict dye colour

A

Form a box around the conjugated double bonds before they’re disrupted (usually by heteroatoms), leaving an empty bond on either side
Count the number of bonds in the box using the average C-C bond length (1.39A)
Count the number of π electrons (2 for every double bond)
Fill in the energy levels (2 electrons per n) to find the HOMO and LUMO
Calculate ΔE between the HOMO and LUMO

65
Q

How is a particle able to pass through a barrier?

A

The E of the particle must be less than the potential energy (V) otherwise it would just pass over the barrier - (V-E is positive)

66
Q

What happens to the energy of a particle that passes through a barrier?

A

It experiences exponential decay because it experiences no oscillations while it passes through the barrier

67
Q

What happens to a particle once it has passed through a barrier?

A

Its wave continues at a significantly reduced amplitude

68
Q

What is the wavefunction equation for a particle after passing through a barrier?

A

Ψ=Ae^ikx - there is no reverse momentum component

69
Q

What is the wavefunction equation for a particle at a barrier?

A

Ψ=Csinkx + Dcoskx

Or Ψ=Ae^ikx + Be^-ikx

70
Q

Equation for k (particle at a barrier)

A

kħ=sqrt(2mE)

71
Q

Requirements for a particle at a barrier

A

Ψ must be continuous

dΨ/dx must be continuous entering and leaving a barrier

72
Q

What is tunneling?

A

A consequence of the wave character of matter - waves can pass through barriers

73
Q

What influences the ability of a particle to be able to tunnel?

A

Particles with that are small that pass through thin barriers have a high probability of having tunneling
Probability decreases with thickness of the wall and mass of the particle
This important mostly for electrons > protons&raquo_space; heavier particles

74
Q

What is a real world application of tunneling?

A

Scanning tunneling microscopy (STM) - a Tungsten needle scans the surface of a conducting solid - this happens because an electron tunnels between the surface and the tip of the needles while a tunneling current is run through the solid

75
Q

Hooke’s law

A

Restoring force is proportional to the displacement from equilibrium

76
Q

Restoring force equation (Hooke)

A

F=-kx

77
Q

Potential energy equation harmonic motion

A

V=1/2kx^2

78
Q

How does the value of the spring constant affect the parabola in a harmonic oscillator?

A

Small k = wide well

Large k = narrow well

79
Q

SWE for a harmonic oscillator

A

-ħ/2m d^2Ψ/dx^2 + 1/2kx^2 = EΨ

80
Q

Boundary conditions of a harmonic oscillator

A

The particle is trapped by infinite walls which leads to quantisation
Walls don’t rise vertically so Ψ and Ψ^2 don’t come to exactly zero at any point

81
Q

True or False: a particle can be anywhere in a harmonic oscillator, whatever its energy

A

True, even in chemically forbidden (shaded) areas because of tunneling

82
Q

Equation for energy levels of a harmonic oscillator

A

E= (v+1/2)ħw

83
Q

Equation for w

A

w=sqrt(k/m)

84
Q

Energy levels for a harmonic oscillator

A

Energy levels are divided into a series of level

  • equally spaced - remains consistent as energy rises
  • spacing: Ev+1 - Ev = ħw
85
Q

Why doesn’t a harmonic oscillator work for large objects

A

Large mass so ħw is negligible because w is so small

86
Q

What is an example of a harmonic oscillator?

A

An X-H bond where the X is stationary and the H atom is so small that it vibrates as an oscillator

87
Q

What is the diatomic interaction potential when r

A

There’s compression between the atoms which causes them to repel each other - energy increases as distance decreases

88
Q

What is the diatomic interaction potential when r>req?

A

There is stretching between the atoms to a certain point where the bond dissociates (breaks)

89
Q

What is equilibrium dissociation energy?

A

Actual dissociation energy from the energy of the molecule when the atoms are at equilibrium distance from each other to plateau before the molecule no longer exists

90
Q

True or False: the harmonic oscillator parabola is a good model for the behaviour of a diatomic molecule

A

False: it can monitor the bottom of the parabola quite well but as energy increases, accuracy decreases, particularly concerning the plateau

91
Q

Equation for parabolic potential energy

A

V=1/2kx^2

x=r-req

CHEM251 (2 decks)

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