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151 Cards in this Set

  • Front
  • Back
What proves particles are a wave?
double slit experiment
What proves light is a particle
photoelectric effect
Defining property of a wave

a wave can be at the same point in space and time
Energy of a photon


E=hv (v = greek letter nu)



What is ?Planck's constant
h=6.626x10-34Js
Equation for de Broglie wavelength

What describes the wave nature of a particle


What is a wavefunction Wavefunction ψ

function contains all the in formation about a system defined by the Schrodinger equation
What is the Schrodinger equation

What is the kinetic energy equation

e=1/2mv²
What is Heisenberg's uncertainty principle equation

What is the usual Hamiltonian operator


What is an Operator

Mathematical operation which is performed on a function
How do we get quantised kinetic energy
Take d²/dx² of wavefunction and multiply by -ℏ²/2m
What makes up the total energy of a system.

curvature of wavefunction multiplied by -ℏ²/2m




the potential energy of the system multiplied by the wavefunction


What does the behaviour of curvature of the wavefuction depend on?

Whether V(x)> E or V(x) < E

How is magnitude of the curvature defined?

(V(x) -E) or kinetec energy
Where is the normal regime?

where V(x) < E

Where is the classically forbidden regime?

V(x) > E
When V(x)< E and the wavefunction is positive what is the curvature?

negative (curves towards the x axis)

When V(x)

Positive (curves towards the x axis)

When V(x) = 0, what does this mean?

particle or wave is moving under no external forces
When V(x) >E, and wavefuction is negative what is the curvature?
negative(curves away from x axis)
When V(x) >E, and wavefuction is positive what is the curvature?
positive (curves away from x axis)
What restrictions must a wavefunction must adhere to


Must me continuous


single valued


Have a continuous derivative

How do we get probability of finding a particle in space from wavefuction

Intergrate the square of the modulus of the wavefunction between two points. (modulus because the square of complex numbers can be negative) ∫|ψ|² dx

∫|ψ|² dx between ±∞ does not equal 1 what do we do?

we normalise it (set it equal to 1)
Units of probability?

Unitless

what are the units of ψ(x)? and why?
Edit: for one dimensional etc


What is the kinetic energy and momentum of a free particle where V(x) = 0.

Ek = ℏ²k²/2m, momuntum is defined exactly as p=ℏk with k = 2π/λ

What is the problem with knowing the momuntum of a particle exactly?


You can't know anything about its position
What does the superposition principle do?
Allows us to localise a particle by linear combination of multiple wavefunction?
How does the superposition principle work?
Linear combination of multiple wavefunctions with different momentum values gives uncertainty in momentum meaning we should know more about its position (uncertainty principle), this is because different waves interfere giving localised regions where they are more likely to be than others

What is the basis set?

Set of wavefunctions that can be linearly combined denoted ψi



If there is more than one solution to schrodinger equation for a particular situation what do we do?

true wavefunction will be the linear combination of all possible functions
Definitions of potential for Particle in a box system?


V(x) =0 inside the box


V(x) = ∞ outside the box

How do we write schrodinger equation for a particle in a box?

What potential solutions to the schrodinger equation for a particle in the box are there?
Asin(kx) or Bcos(kx)
What are the second derivatives of the two possible wavefunctions?


d²/dx²(Asin(kx)) = -k²(Asin(kx))


d²/dx²(Bcos(kx)) = -k²(Bcos(kx))

What is the 'true' wavefunction for a particle in a box?

Linear combination of all potential solutions


=

ψ(x) = Asin(kx) + Bcos(kx)

What are boundary conditions?

They impose certain requirements on a wavefunction so that only specific solutions are acceptable. (this is why quantisation occurs)

What are the boundary conditions for a particle in a box?
ψ(x) = 0 when x=0 or x = L (or anywhere else in the classically forbidden region)

What is the wavefunction of a particle in a box once ψ(x) = 0 when x = 0, has been considered?


ψ(x) = A sin(kx)


because ψ(x) when x = 0 therefore B (from Bcos(kx)) must be zero because cos(0) = 1

What is the wavefunction of a particle in a box once ψ(x) = 0 when x = L and 0, has been considered?


ψ(x) = A sin(nπx/L) where n = 1,2 ...




because sinkx is only zero at x =L when kL is a multiple of π. (n = 0 is not a valid solution because then wavefunction would equal 0 at all x and then would not exist.

What is k equal to after applying boundary condion ψ(x) = 0 when x = L?


kL=nπ ∴ k = nπ/L
What is a node when plotting a particle in the box solution?


Where wavefunction cross zero. (at 0 and L wavefunction does not cross zero it merely ends at zero therefore these are not nodes)

How many nodes are there for a particle in a box?

N-1 nodes

Do we need to normalise the particle in the box solution?
yes
How do we get energies from the normalised solution to the particle in a box wavefunction?


Plug the wavefunction back into the Schrodinger equation or remember that k² = 2mE/ℏ² and k=nπ/L


∴E = ℏ² π²n²/2mL² = h²n²/8mL² where n = 1, 2....

zero-point energy for a particle in a box?


Where n = 1, (gonna have to work it out for yourself :P)

What is a stationary state?

A pure quantum state corresponding to a state with a single definite energy (instead of a quantum superposition of energy energies).


They are quantised

What is the lowest energy called and why?
It is called the zero-point energy, it is a non-zero energy therefore a particle at 0K still has energy
What form does all the energy in a particle in a box take?
Kinetic energy as V(x) is defined as 0.

What is vibrational motion modled as ?
A harmonic oscillator
Hooke's law?


F = -kx


where x is displacement

What is the potential energy of a hormonic oscillator?
kx²/2 [(1/2)* kx²]
Why is reduced mass used?

Changes a two-body problem into a one-body problem
How is reduced mass calculated?

How do we write the Schrodinger equation for a harmonic oscillator system

What is the solution to the harmonic oscillator system?


Ev = hve(v+1/2) where v =0, 1, 2...


where v in ve = greek 'nu' and is the molecular constant of vibrational frequency of the oscillator (units s⁻¹)


and v = italic v, and is the vibrational quantum number

What is the vibrational frequencey of an oscillator defined as? (equation)


ve = √(k/µ)



How are the energy levels in an harmonic oscillator spaced?

Energy levels are equally spaced unlike the particle in a box in which the energy levels become further apart an n increases.
What is the zero-point energy for harmonic oscillator?


Ezp = (1/2)*hve


=(hve/2)

What does the solution of the harmonic oscillator schrodinger equation mean?
A diatomic molecule can only vabrate at specific energies and these are called the vibrational levels (for that electronic state) hence why IR spectrum has well-defined lines
What is the origin or quantisation in a hormonic oscillator system?
The wavefunction penetrates into the classically forbidden region (V(x) > E) but only solutions that tend to zero but always curve away from the x axis will be allowed solutions
What do the wavefunctions of the harmonic oscillator look like for its different energies?

What does the plot of a high vibrational quantum number look like for a harmonic oscillator? + and its probability distribution

What does the plot of a probability distribution for a harmonic oscillator system tell us about the most probably inter-nuclear distance?
The most probably inter-nuclear distance is either at the inner turning point or outer turning point

Where is the greatest kinetic energy of a harmonic oscillator system?

when the bond is at its equilibrium value (in the middle where curvature is at its greatest of a probability distribution plot. not at the turning points)
What is the correspondence principle?

States that in the limit of very large quantum number, the system is classical. This is th case for the macroscopic world.
What is significant about ψ(x) at V(x) = E?
The curvature must change sign as for V(x) < E curvature is towards the x axis but for V(x) > E the cuvarture is away from the x axis. (there is a point of inflection)
What is quantum mechanical tunnelling?

When a wavefunction enters the classically forbidden region it quickly tends to zero however if the 'barrier' has a finite width it is plausible that the wavefunction penetrating the barrier emerges on the other side before it has fallen to zero, in this case it has gone through the potentail barrier. The probability of tunnelling is small and only important for very light particles suchas electrons.

How do we consider the rotation of a diatomic molecule in Quantum mechanics?

model it as a particle on a ring

How do we reduce the two-body problem of a rotation of a diatomic particle to a one-body problem?

Transforming the system into a centre-of-mass system in which the reduced mass rotates on a ring of radius ro
What assumption is made for a particle on a ring

Free rotation
If we assume free rotation for a particle on a ring system how does that help us?

potential energy is zero (or constant)
How is the schrodinger equation in the xy-plane expressed as for a particle on ring system?

What is the polar coordinates version of the schrodinger equation in the xy-plane expressed as for a particle on a ring-system?


It is the angular wavefunction


Capital Phi (phi (some form))

What is the angular wavefunction?
A one dimenional function representing the properties of a particle on a ring system
How do we express the schodinger equation for a particle on a string to allow us to solve it?

What are the solutions to the angular Schrodinger equation for a particle on a string?

sine and cosine waves

What is the superposed solution to the schrodinger equation?


Boundary conditions for a particle on string?

After one rotation the wave must be in phase with itself (heads and tails must match up)
How do we apply the boundary conditions for a particle on a string?

Is ∅ = 0 allowed for particle on a ring? why?

Yes, for ∅ = 0, into Φ(∅) = A cos(0) + Bsin(0) = A, therefore non-zero therfore can be normalised
What is the quantisation defined for a particle on a string quantum system?

What quantum number is used for particle on a ring? + values?


m, m= 0, ±1,±2...



Quantised energy levels of a particle on a ring equation?

How many energy levels are there for every m for a particle on a ring?

0 has 1, everything else has 2 (clockwise and anti clockwise)
Zero point energy for a particle on a ring?
When m = 0, E = 0, therefore no zero-point energy in rotation
How many nodes does a wavefunction for a particle on a ring have?

|m| nodes



What is the particle on a sphere schodinger equation given in?


spherical polar coodrinates (r, θ,∅)



Angular wavefunctions for particle on a sphere?

Quantum numbers for a particle on a ring?
l and ml
What values can the particle on a sphere quantum numbers take?

l = 0, 1 , 2 ....


ml = 0, ± 1, ± 2, ... ±l

What is the equation for energy levels of the particle on a sphere?

How many ml quantum numbers are there for a l quantum number?
2l + 1 ml quantum numbers
Rotational constant of a molecule?

Which quantum number represents the quantum aspect of θ for a particle on a sphere?

l
Which quantum number represents aspect of ∅ for a particle on a sphere?
ml
How does nodes work with particle on a sphere?

There are always l nodes total, however there are l - ml nodes on the z axis (θ spanse represented by l) and ml nodes on xy plane (∅ spance represented by ml )
What is the potential energy of a H atom?

Radial wavefunction quantum numbers? (atomic orbitals)

n and l
Angular wavefunction quantum numbers? (atomic orbitals)

l and projection of l onto the z-axis ml
What are the quantum numbers for atomic orbitals? (+ allowed values)


n - Principal quantum number (1, 2 ...)


l - angular momentum quantum number (0, 1 .. n-1)


ml - orientation (magnetic) quantum number (0, ±1,±2...±l)



What equation defines the energy levels for atomic orbitals?

How many degenerate energy levels does each n state have?

n x (2l + 1)
How is the degeneracy different for non-hydrogen atoms?

low l value wavefunctions are influenced by the core electrons
Born oppenheimer Approximation?

electrons move much faster than the nuclei hence electrons instantaneously adjust themselves to changes in the coordinates of the nuclei
What is Ψmol made up?


Ψel electronic wavefunction (which is an atomic orbitals)


Ψnucl nuclear wavefunction (which is a harmonic oscillator)

How is a molecular potential energy curve obtained?

solving schrodinger equation for the electrons at each internuclear separation
What does the molecular potential energy curve look like?

Why is a molecular potential energy curve called a potential energy curve?

The nuclei are a fixed distance apart and thus have zero kinetic energy therefore all the energy must be potential
What is Linear combination of Atomic Orbitals?


we determine Ψ - from the superposition of atomic orbitals ψi as basis functions weighing each with a coefficient ci



Ψ = c1ψ1 + c2ψ2 ...
in LCAO how do we find the probability of finding the electron in an orbital with the properties associated with atomic orbital ψi
|ci
What is the overlap densitiy in LCAO?

|2Ψ1Ψ2|
How is the overlap density obtained?

How does |2Ψ1Ψ2| vary with internuclear distance?

overlap decreases with increasing internuclear distance
What is the variation principle?

method for estimating the energy and shape of a MO
What assertion is the variation principle based on?

The energy calculated using a trial wavefunction is always larger than the true energy
How does one get the best approximation to the real wavefunction?

using coefficients that lead to the lowest energy and using as many atomic orbitals as reasonably possible ( the more the better)
Expectation value?
used to find energy of the system
How do we get the secular equations?


Take partial derivative of expectation value with respect to the coefficients



What are the secular equations for a diatomic molecule?

What does E, αa, αb, β, Sab mean from the secular equations for a diatomic molecule?


E = total energy of MO


αa, αb = energies of the atomic orbitals ψa, ψb


β = resonance or interaction energy (energy gained through bonding)


Sab = overlap energy (spatial overlap of atomic orbitals)

How do we know when β and Sab are non-zero (they are from secular equation)?

Symmetry arguments- if ψa ψb are of the same symmetry then β<0 and Sab>0



How do we simplify secular equations? (diatomic)
assume negligible overlap of the atomic orbitals (Sab = 0)
Simplified secular equation in matrix form?

How do we know if there is a nontrivial solution to a matrix?
Secular determinant = 0
How do we get energies from secular determinant?

set secular determinant to zero to give a polynomial in E then the roots are the energies



What must we normalize on the LCAO wavefunction?

only the coefficients because the wavefunctions are already normalised (because they are atomic orbitals)
Huckel assuptions/approximations?


-Consider only π orbitals which are formed from pz as these are responsible for most of the chemistry and spectroscopy


-α is the same for all atoms


-β is only relevant between neigbouring atoms otherwise β=0


-The overlap integrals are zero: S = 0



What are the frontier orbitals?

HOMO and LUMO
How does chain length affect HOMO LUMO gap

it decreases causing excitation energy to decrease giving longer wavelengths of light
How do we find the probability of finding an electron localised on a specific atom in a specific orbital?

square of the coefficient multiplied by number of electrons in orbital
Total energy for a free rotation?

Angular momentum?

Moment of inertia I?
equivalent to mass - is measure how resistant a object is to change
Alternate angular momentum equation?

1D magnitude of the angular momentum?

Angular momentum of an electron in an atomic orbit (particle on a sphere)?

Why is the orbital angular momentum quantised?

The energy is quatised therefore angular momentum is quantised
What does ml indicate about electron, what does l indicate about electron?
ml quantum number indicates the directionality while l quantum number indicates the 'speed' of the electron in the orbit
How do the orbital motion of electron affect each other?

Generate magnetic field which influence (couples to) other electrons in an atom.


Total angular momentum?


How many degenerate levels or projections are there for Total angular momentum L.

2L+1
Spin wavefunction for ms = +1/2 (spin up)?
α

Spin wavefunction for ms = -1/2 (spin down)?
β
Magnitude of a spin s = 1/2 particle?
|s| = √s(s+1) =√1/2(1/2 +1) = 0.866
Fluorescence vs phosphorescence?
S = s1 + s2 . during excitation Fluorescence ∆S = 0, phosphorescence ∆S=1