Quantum Mechanics Flashcards
What is a blackbody?
A blackbody is a material that converts heat to light
How does a blackbody emit radiation?
It absorbs heat and keeps it in a cavity until it reaches thermal equilibrium, then releases it as radiation through a pinhole
How is the energy density of blackbody radiation affected by temperature (energy)?
Higher energy = higher energy density
Temperature increase causes the maxima to shift to a shorter wavelength
Wien’s Law
ρ=C1/λ^5 ×1/e^(C2/λT)
Good at short wavelengths
Not good at long wavelengths
Wien’s Law assumptions
Makes assumptions about absorption and emission of electromagnetic radiation
Rayleigh-Jean theory
Treat electromagnetic field as a lot of oscillators
Calculate number with each energy
Predict’s a curve proportional to λ^-4
Rayleigh-Jean’s Law
ρ=8πkT/λ^4
Good at long wavelengths
Terrible at short wavelengths (ultraviolet catastrophe)
Ultraviolet catastrophe
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
Planck Distribution
ρ=C1/λ^5 ×1/(e^(C2/λT) - 1)
A perfect fit to all λ
Quantum assumptions in the Planck Distribution
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
Planck’s Distribution with quantum assumptions
ρ=8πch/λ^5 × 1/(e^(ch/λkBT) - 1)
Photoelectric effect
The emission of electrons when light is shone onto specific materials
Wave-Particle Duality
Light can act as waves and as particles
Particles can also act as waves
Einstein’s theory about quantisation of light
Light consists of particles with energies (photons):
E=hυ
Light energy comes in discrete bundles (quanta):
E∝υ
De Broglie’s theory about wave-particle duality
Particles can behave as if they’re waves
λ=h/P
Proven through electron diffraction
Issues with
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)
Translational energy of bodies
E = 1/2mv^2 + V
or
E=p^2/2m + V
Kinetic energy
The energy that the particle will possess as a result of momentum
Potential energy
The energy a particle possesses as a result of position
What do you need to exactly predict the trajectory
Starting position
Momentum
Potential
Force
Rate of change of momentum
F = dp/dt
Newtonian Mechanics
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
Hooke’s Law
F=-kx
Frequency of an elastic band
Frequency is independent of energy
Frequency depends on structure
All vibrational energies are possible
Do classical mechanics apply to small objects
No
How does light work?
Energy transfer
How does reflection work?
Light reflects off of an object an interacts with the eye
What is a wavefunction?
A mathematical function that describes where an electron will most likely be at a point in time (its probability distribution)
Quantum Postulate 1
In one dimension, Ψ is a function of x Ψ(x)
What can be deduced from postulate 1?
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
How is postulate 1 determined?
Schrodinger’s equation
Born Interpretation
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
What is probability density?
|Ψ|^2 - square modulus of the wavefunction
An acceptable wavefunction must be:
- Continuous
- Single-valued (you can’t have multiple probabilities)
- Have a continuous first derivative
- Able to square integrate (can’t be infinite)
What is an operator?
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.
What are observables?
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
Heisenberg Uncertainty Principle
ΔpΔx = ħ/2 ħ = h/2π
Schrodinger Equation
(-ħ/ 2m)(d^2Ψ/dx^2) + V(x)Ψ = EΨ
ĤΨ = EΨ
Ψ equation
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
What are the 3 types of motion?
Translational
Vibrational
Rotational
True or False: any wavefunction and corresponding energy are possible in simple model systems
False - only certain wavefunctions and corresponding energies are acceptable (quantisation)
How do molecules store energy?`
Through motion
How are energy levels observed in IR?
Transitions between different vibrational energy levels and that energy is being stored by the system
What does the vibrating string analogy show?
If 2 ends of a rope are fixed, only certain wavelengths are possible (because only a certain amount of nodes is possible)
How are wavefunctions defined in terms of a particle in a 1D space?
Ψ(x) = 2πx/λ
where λ is restricted by 2L/n
L = length between the ends
n = 1, 2, 3,…
What is the energy potential of a particle at the boundary of a 1D box?
Infinite
What is the energy potential of a particle at the base of a 1D box?
Zero
What is the probability of a particle being where energy potential is infinite in a 1D box?
Zero - this means that the particle has escaped past the boundaries
What are the boundary conditions of a particle in a 1D box?
Ψ = 0 at x=0 and x=L
How many nodes does Ψn have?
n-1
Why is n=0 not a solution for Ψ in the particle in a 1D box?
Because if it was, Ψ would be zero everywhere and |Ψ^2 would also be zero, which isn’t possible because |Ψ^2 = 1
True or False: a particle in a 1D box is able to have zero energy
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
What are nodes in particle distribution diagrams?
Areas where there is zero probability of a particle being at a particle point along the line
True or False: if n is large, particle distribution is uniform for a 1D box
True, if n isn’t large, distribution isn’t uniform
True or False: a particle can have 0 energy
False, because n can’t be zero, the lowest energy a particle can possess isn’t zero
Zero-point energy equation
E1= h^2/8mL^2
ZPE explanation 1
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
ZPE explanation 2
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
Equation for separation between energy levels
ΔE=(2n+1)h^2/8mL^2
Relationship between energy and change of mass and length
ΔE will decrease as mass and/or length increases and increase as they decrease
ΔE is very small at macroscopic dimensions
Is translational motion in atoms and particles in a beaker treated as quantised or not quantised?
Not quantised
Correspondence Principle
Behaviour of quantum mechanical systems becomes more and more classical in the limit of large quantum numbers (n is very big)
Can dye colour be predicted?
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
How to predict dye colour
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
How is a particle able to pass through a barrier?
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)
What happens to the energy of a particle that passes through a barrier?
It experiences exponential decay because it experiences no oscillations while it passes through the barrier
What happens to a particle once it has passed through a barrier?
Its wave continues at a significantly reduced amplitude
What is the wavefunction equation for a particle after passing through a barrier?
Ψ=Ae^ikx - there is no reverse momentum component
What is the wavefunction equation for a particle at a barrier?
Ψ=Csinkx + Dcoskx
Or Ψ=Ae^ikx + Be^-ikx
Equation for k (particle at a barrier)
kħ=sqrt(2mE)
Requirements for a particle at a barrier
Ψ must be continuous
dΨ/dx must be continuous entering and leaving a barrier
What is tunneling?
A consequence of the wave character of matter - waves can pass through barriers
What influences the ability of a particle to be able to tunnel?
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»_space; heavier particles
What is a real world application of tunneling?
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
Hooke’s law
Restoring force is proportional to the displacement from equilibrium
Restoring force equation (Hooke)
F=-kx
Potential energy equation harmonic motion
V=1/2kx^2
How does the value of the spring constant affect the parabola in a harmonic oscillator?
Small k = wide well
Large k = narrow well
SWE for a harmonic oscillator
-ħ/2m d^2Ψ/dx^2 + 1/2kx^2 = EΨ
Boundary conditions of a harmonic oscillator
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
True or False: a particle can be anywhere in a harmonic oscillator, whatever its energy
True, even in chemically forbidden (shaded) areas because of tunneling
Equation for energy levels of a harmonic oscillator
E= (v+1/2)ħw
Equation for w
w=sqrt(k/m)
Energy levels for a harmonic oscillator
Energy levels are divided into a series of level
- equally spaced - remains consistent as energy rises
- spacing: Ev+1 - Ev = ħw
Why doesn’t a harmonic oscillator work for large objects
Large mass so ħw is negligible because w is so small
What is an example of a harmonic oscillator?
An X-H bond where the X is stationary and the H atom is so small that it vibrates as an oscillator
What is the diatomic interaction potential when r
There’s compression between the atoms which causes them to repel each other - energy increases as distance decreases
What is the diatomic interaction potential when r>req?
There is stretching between the atoms to a certain point where the bond dissociates (breaks)
What is equilibrium dissociation energy?
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
True or False: the harmonic oscillator parabola is a good model for the behaviour of a diatomic molecule
False: it can monitor the bottom of the parabola quite well but as energy increases, accuracy decreases, particularly concerning the plateau
Equation for parabolic potential energy
V=1/2kx^2
x=r-req
