3b Flashcards
schrödinger equationssss
HY=EY
(T’ + V’)Y = EY
T’ =
(- h.dash/2m) X (2nd derivative) (y)
is the result an eigen function?? a possible solution for Y ??
what’s potential energy,, V’ for a free particle
it’s 0
this is the model potential
what’s a more realistic model
particle in a box
describe the drawing for particle in a box
okay so u have a box in the middle with dashes outside the box.
outside the box,, where it’s dashed: Y and V are infinite.
inside the box: V=0 so H’=T’ (kinetic energy only)
the box prevents the e- from moving for infinity,, the e- can move within the box,, but cannot leave the box.
when V = infinite,, what is Y
0
sin,, cos and exp particle in a box explanation
box for each,, with dashes outside.
x axis: 0 -> L
sin(0)=0 so the graph starts at 0 and ends at 0,, it’s giving sad face,, rainbow that fits perfectly in the box
cos(0)=1 so graph starts on 1 on the y axis and goes down to -1 basically.
exp(0)=1 so graph starts on 1 and goes up up up. like a smiley face slanted to the left kinda.
sin is: eigen, finite, single, continuous and normalised as it starts and ends at 0.
cos and exp aren’t continuous bc they don’t start at 0,, they start at 1.
what’s a boundary condition : particle in a box
it’s when when graph should reach 0 at the corners of the box graph.
the Y should = 0 at the ends of the box.
sin does this but cos and exp don’t. (cos goes down and exp goes up)
bc one sin wave works and is both an eigen function, value and agrees with the born requirements,, will every sin wave do the same
nope!!
although they all start at 0,, they don’t all end at 0. not all of them respect the boundary condition.
bc one sin wave works and is both an eigen function, value and agrees with the born requirements,, will every sin wave do the same
nope!!
although they all start at 0,, they don’t all end at 0. not all of them respect the boundary condition.
quantisation of the sin graph,, bc every n(pi) is a node
aka 2n is a node,, so is 3n
sin(kL) = 0 (node)
kL=n pi ( n as in an integer)
K = n pi / L
n = 1,2,3,4 etc.
when sin(x) is plotted against x (//radians)
in particle in a box,, how many directions can the e- move in
just one
left to right
is given: ‘x’ on the x axis
spider in bath and particle in box explanation
aka the particle can move in the bath aka the box
but will never have enough energy to go over the bath walls,, where the potential energy,, V,, is infinite
inside the box,, the V = what,, and what does that mean
V’ = 0
particle can be anywhere in the box,, just not outside it.
start to end of the box
0 –> L
length of the box
V’ here is 0
where V’ = infinite,, what does Y equal
Y = 0
probs of finding an e- here // outside the box = 0
in the box // between 0–>L,, what does H’ equal
H’ = T’
bc H represents all the energies of that particle and usually = V’ + T’
so if v’ = 0,, only T is left
what gies an eigenfunction when u take the 2nd derivative of it
sin
cos
exp
watch out for the constants tho
why are cos and exp not continuous
bc theres a discontinuity: going from V=infinity and Y=0 to Y=1 in the box
bc cos(0) = 1 and exp(0) = 1 so on the ‘0’ on the x axis,, u will start the line on 1.
whereas for sin,, sin(0) = 0,, so u go from Y=0 outside the box to Y=0 inside the box
what must be seen for a boundary condition to be satisfied
Y=0!!!!
end of graph//box must be where the line starts and ends.
for the sin graph to end and start at Y=0,, what must sin(kL) equal to
and what does kL equal
and describe everything about this
it must equal to 0
kL=n pi
where n is an integer
so L must equal n pi/k
and thid only gives Y=0 when n = integer: 1,2,3,4
bc on a sin wave,, it touches the x axis at pi, 2pi, 3pi, 4pi etc
in radians
different sin waves // waves in the particle in a box are different what
theyre different possible wavefunctions!
different arrangements (how the wave graph looks in particle in a box) have different energies.
aka if the waves look different,, theyre different possible wave functions,, and have different energies,, and different ‘n’ integer values.
in the kL=n pi equation
so Yn (x) is equal to what (aka the wavefunction at a certain integer// different wavefunction) as a function of x
C sin(n pi x) // L
where C is a constant
so for different values of ‘n’ u get different wavefunctions
for different values of ‘n’ ,, in the particle in a box model,, u get what
different va;ues of ‘n’ give u different wave functions
bc Yn(x) = C sin(n pi x) // L
describe the sin wave in particle in a box where: n = 1
little hill
starts at 0 and ends at 0
so X=0 (start of box) and L= X
( end of box) bc these are the bottom corners of the box + theyre both 0
describe the sin wave in the particle in a box where: n = 2
actual wave shape..
starts at X=0,, goes up,, goes down past the x axis,, then goes back up and reaches L=X=0
describe the sin wave in particle in a box where n = 3
starts at x=0,, goes up,, does down, passes x axis,, goes back up past the x axis,, then goes back down and reaches X=L=0
n= 3 and n=2 in terms of how the wave looks
and its properties
n = 3
3= number of half wavelengths
up, down, up down
n = 2
2 = number of half wavelengths
up, down, up
each value of n gives a different wave with the same amplitude but they have a different wavelength due to them having to squish more peaks//troughs in the same distance (the box x axis: x –> L)
kinetic energy is like the second derivative,, what do we mean by that in terms of particle in a box
okay so for different values of n we get different possible wavefunctions
and each of these,, due to having a different ‘n’ value,, will have the same amplitude but different wavelengths and different amount of peaks.
due to the different amount of peaks // number of half waves,, we can see that some waves have a higher energy:
the more peaks + more movement = shorter wavelength// kinetic energy the wave has,, the higher the nergy of the wave
KE is changing more quickly!!
bc inside the box V’=0 so H’ = T’,, the only energy it has is Ke(T’),, so the more it moves, ,the more Ke it has,, meaning the more energy it has .
sin wave is also used to describe what in organic chem
its used to describe molecular orbitals!!
esp for butadiene
there will be a node when the phases change.
all orbitals in phase= n=1 no node,, just the hill.
4 pi electrons (2 for each double bond) so 4 MO
instead of Y,, what do we acc use and what does it mean
Y^2
its the probability of finding a particle at any position
C ,, the normalised constant = what
so what is Y^2(x) equal to
and what shape does this give for n =1
2/L
= (2/L) (sin^2(n pi x /// L)
same shape but more like a graph and not a hill,, lower edges and higher peak.
inegrat of Y^2 dx with limits of 0 and L gives 1
probablistic graph ,,, graph peak =
high probability of finding a particle there
particle in a box,, what does the ‘n quantum number specify
the wavefunction and the energy.
zeropoint energy for n is what value
n=1
n cannot be 0
for n = 1 whats the E1 equation for an e-
E1 = h^2 // 8me L^2
what is the main transition we would see a molecule do
the homo to lumo transition
how do we know which orbital is the homo and which one is the lumo
count the number of pi electrons!!
then do an energy graph of energy being an arrow up
draw lines going up like a ladder and label them 1, 2, 3, 4, etc
each line obvs has 2e- so u fill them up with 8e- in total,, 2 in each line.
homo is the highest occupied one ,, lumo is the lowest unocupied one
this is the energy change // transition we would see.
what is the denominator,, ‘L’
the length of the particle in ‘m’
so the larger the particle,, the greater the denominator bc L^2,, so a lower energy,, + lower energy transition between energy levels
smaller ‘L’ = smaller particle length = larger energy gap,, energy levels space out.
what is tunnelling
a property that quantum particles have
quantum particles can tunnel through objects! think of the memes where ppl try to push through doors!!
what is V(potential energy) when tunneling occurs
V = 0 except at the barrier
kinda opposite of particle in a box as there,, inside the box was v=0,, here,, outside the barrier = V=0
when V=0 what can occur + where is V=0 seen on the tunneling diagram
the particle can move anywhere,, can go to infinity
kinetic energy = H
wavefunction = sin//cosine wave
describe the tunneling experience
so ur a He atom,, and ur a quantum particle and therefore have a wavelength + wave properties: ur norally a sin//cosine wave
so ur waving around and can go anywhere bc V=0 until u reach a barrier,, at the barrier,, the V=is no longer 0. aka at the barrier,, ur energy is not only kinetic energy.
u hit the barrier and need to be continuous: so if u hit it u get dropped onto the x axis,, and this isnt continuous,, u arent a good wavefunction bc ur not continuous,, u kinda just dropped instead of being a wave.
what happens instead is that u experience exponential decay inside the barrier. so u go through the barrier as ur wavefunction extends through it ,, u kinda go down but slowly,, u dont get dropped bc u have a wavelike property. iu get through the barrier and keep on being a sin//cosine wave. ur amplitude decreases a bit tho due to exponential decay!! but u keep on being a wave. THIS IS TUNNELING!!
tunneling: exponential decay through a barrier where V is not 0,, and u pass through and remain being a sin//cosine graph but with a slightly lower amplitude due to the exponential decay.
in the barrier: Y = what
Y = e ^ -kx
what is k in Y = e ^ -kx ,, aka the waefunction in the barrier
its the tunnelling coefficient
k = squareroot( 2m(V-E) //h.dash)
what does the extent of tunneling depend on: if smt is gonna tunnel through a barrier or not
barrier height
barrier width
mass of particle
high barrier: can we tunnel through
low chance
bc Y = e^-kx
and k = root (2m(V-E)//h.dash)
low barrier: can we tunnel
more likely to tunnel yessss
narrow barrier: can we tunnel
yesss
bc less exponentisal decay,, it wont decay to 0
wide barrier : can we tunnel
even if the barrier is short,, if its wide exponential decay can make the particle reach 0,, so harder to tunnel through.
think that exponential decay occurs for longer as u need to tunnel through a greater distance!!
light particle: cancwe tunnel
yessss
less decay occurs
heavy particles: can we tunnel
nope
more decay occurs due to the larger mass!!
cannot tunnel through.
e- , protons, atoms , molecules : rate in terms of tunnelling abilities
e- : can tunnel a long way and through wide barriers
protons
atoms
molecules
due to them getting larger and larger.
before tunnelling. and after tunneling the particle is aaaaa
a free particleeeee ayayayayya
bc V= 0
so its energy = due to kinetic enenrgy ,, how much it moves
H’ = T’
