when I first learned about the atomic
structure life was simple electron was
orbiting the nucleus kind of like the
planets going around the Sun everything
made sense but then I entered high
school and learned that this is wrong a
better way to think about it is in terms
of quantum numbers and there were a few
quantum numbers they had their own rules
and that led to electrons doing this
orbital thing 3D structures which had
this beautiful shapes but why do they
have these shapes and what exactly are
they well none of that made any
intuitive sense until now see my goal is
always to not just understand things but
to try and ReDiscover them because I
feel that's a best way to learn
something and that's exactly the
ambitious goal of this video so we're
going to start by pretending we don't
know anything about quantum numbers or
orbitals or their shapes and just by
using intuition step by step let's see
how much we can ReDiscover it together
so if you're ready for this let's begin
since all this comes from shinger
equations let's talk to shinger and
let's start at the very beginning
schinger what's the problem with this
planetary model it's so simple so
beautiful what's the problem with that
and schinger says Mahesh it's terrible
it doesn't explain anything for example
we have a charge that is accelerating as
it goes around and according to
electromagnetic wave theory accelerated
charges should lose energy they should
radiate light and so if it was losing
energy it would spiral into the nucleus
and it would you know it will crash into
the nucleus within a few Nan seconds so
it doesn't explain why the atom is
stable more importantly it doesn't
explain the atomic Spectra for example
if you heat up hydrogen and look at the
light that comes out of it and analyze
it you will find that it has very
specific frequencies why why those
specific frequencies it doesn't explain
that it also doesn't explain anything
about chemistry so it's just a terrible
model and I'm like all right okay fine
how do we update this model well the
short reminds us that the biggest
problem over here is that we are
thinking of electrons as tiny balls and
that's not true we know that they are
quantum particles they have Associated
wave function with them and I'm like
yeah I understand that we've talked
about that in our previous videos and
stuff we can think of them as a wave
it's a mathematical wave nothing is
really going up and down but what's
concrete over here is that the square of
the amplitude represents the probability
of finding the electron at that
particular spot so just from looking at
this diagram we can say that hey there's
a very high probability of finding the
electron somewhere over here low
probability of finding it over here and
almost zero probability of finding it
over here so that kind of makes sense to
me all right so okay shinger how can we
use this and think of a model of the you
know electron inside electron with a
proton so how can we model the hydrogen
atom over here and Stringer says Okay so
this means we need to think about a wave
that is confined in the 3D space right
and there's a beautiful powerful analogy
where we have a wave that is confined
and that is the wave confined on a St
stretched string and I'm like I
understand that shinger we know about
the waves on a stretched string if I
have a string that is connected at the
ends and if I just pluck it it's going
to produce standing waves and the beauty
about that is that it you can't get any
frequency you want the fundamental
frequency is where half the wav length
or half the wave fits the string the
next possible frequency is twice of that
when one full wavelength fits the string
and then you can have one and a half
wavelength fitting the string even
higher frequency and so on and so forth
so I understand that that's beautiful so
we can see right in front of your eyes
the first thing is that if you confine a
wave we get discrete frequencies are
allowed no frequencies in between are
allowed because at those frequencies
waves would just destructively interfere
and cancel each other so we get discrete
frequencies when you confine waves
beautiful right now sh asks an important
question Mahesh what about their
energies can you comment on that and I'm
like yeah we can comment on that see for
the fundamental wave
um it's vibrating with the lowest
frequency so it's oscillating very
slowly so it has the lowest energy it's
oscillating much faster higher energy
higher and higher energy and so on and
so forth so the energy increases as we
go up and shinger is like that's great
that's great for physical waves but what
about mathematical waves like you know
for the probability waves that we're
talking about remember nothing is
oscillating up and down over here how do
we think about energies when it comes to
probability waves I'm like that's an
interesting question how do we think
about energy now well shinger says
here's the thing over here if this was a
probability wave then we get very low
amplitudes over here so less probability
of finding the electron very high
probability of finding the electron so
we can kind of see the electron is
somewhere over here now look at this
particular wave we have a node over here
that means the amplitude is zero in
between so the probability of finding
the electron over here is zero which
means if I were to look at the
probability clouds over here it would
look like this so notice over over here
the electrons are much more confined to
these two regions compared to over here
and from the uncertainty principle you
know that if you try to confine the
electrons and reduce the uncertainty in
its position then its uncertain in
momentum increases and so because of
that you now end up with higher kinetic
energy so you have higher energy because
higher uncertainty momenta gives you
higher kinetic energy that's how if you
have a nde in the wave function that
represents a higher energy so this helps
us understand something beautiful the
first excited state whatever that looks
like we don't know it yet but the first
excited state of the hydrogen should
have one node in it because at higher
energy you get a node and the same thing
continues over here you have two nodes
and so the waves are even more confined
three nodes even more confined and
therefore the momentum uncertainty keeps
increasing even more kinetic energy
keeps increasing so the key idea we get
from just by looking at the standing
waves over here is that the more the
excited state more the energy more the
Norse okay short Ranger so how can we
use all of this to start thinking about
the electrons in three dimension around
the proton let's start with the ground
state in the ground state what do we
find well if this was a wave function
mathematical probability wave then the
probability over here is zero the
probability increases to a maximum and
then goes back to zero again that's what
it looks like if it was a confined wave
but where is it confined between where
is our electron w confined between well
we know that one end could be the
nucleus the other end well Infinity we
have all of space so we could say that
one end is the proton the other end is
all of space so as we radially go
outwards from the proton all the way
towards Infinity the probability should
first increase it should start with zero
close to the nucleus should increase
reach a maximum and then decrease now of
course the the shape of exactly how the
probability increases very reach is the
maximum all of that that's not going to
be like this obviously for that will
require the mathematical rigor but
qualitatively the pattern remains U the
probability should increase reach a
maximum and then gradually decrease off
as we go towards Infinity so let's try
to simulate that and see what it looks
like so here's our nucleus and let's
start with very close over here we're
going to use dots to represent
probability so you can kind of imagine
if there are 100 dots and if there are
three dots in a particular region so you
can say the probability is three 3 divid
by 100 3% of finding probability of
finding the electron over there so we
can use dots to represent probability
the number of dots is a direct
representation of how much probability
is more dots more probability make sense
okay so very close to the nucleus the
number of dots should be very low so in
my simulation you probably can't see it
but there are a few dots over here and
it turns out to be 11 now let's go
radially outwards for each radius let's
look at how many dots we should have so
we're right now over here as I go
radially outwards the number of dots
should increase okay so let's look at
that look at that that's exactly what's
happening as I go farther and further
away for different different radi the
number of dots increases and in my
simulation at about this particular
radius it Peaks out at 600 dots this is
where you have maximum probability of
finding the electrons and then if I go
further farther out um the probability
should decrease and so let's see we are
going here the probability will decrease
decrease decrease you can see number of
dots are decreasing decreasing
decreasing and eventually it should
taper off it should eventually go to
you know as I go towards Infinity it
should tend to become zero so now let's
look at what will the whole thing look
like well it looks like this isn't that
beautiful we've just built this from the
vibrating string but I'm sure we have so
many questions over here the first
question is wait a second we might say
shinger I thought the probability over
here is supposed to be minimum then
reach maximum and then decrease off but
it looks like the this is the bright
spot over here why do we have so many
bright spots at the center itself well
Turing that's a really good question
we'll think about it okay if I were to
put eight dots at the center then the
probability is corresponding to that
much right and if I have let's say 41
dots over here clearly more dots more
probability but look look at the density
because those eight dots are confined in
a very small space the density is much
higher so it looks um very bright so
shinger says what matters to us is not
the density how crowded the dots are
what matters is at any given radius how
many total dots are there that
represents the probability so we don't
have shouldn't be thinking shouldn't be
worried visually when you see the bright
spots what you have to think about is
the number of dots that's what matters
another reason why the center appears
bright is because remember this is a 2d
projection of three dimension if
something is like this over here when
you look at it from the front it appears
to be at the center and so that's the
reason why this looks very bright okay
all right so that kind of explains
what's you know visually going on over
here but now comes the main question
shinger what is this what am I looking
at over here for that he says let's take
an analogy imagine you have a kid who
you know goes out in the afternoon and
he goes out to play let's say and you
want to know what exactly where exactly
your kid is in the afternoon so you
decide to follow him secretly and take a
picture at say exactly 2: p.m. you want
to figure out you want to know exactly
where your kid is at 2 p.m. you start
taking pictures every single day so the
first day you take a picture and your
kid is over here next day exactly at 2:
p.m. she's over here let's say and then
next day she's over here so you keep
taking more and more picks you at see at
first it seems like a little random
later on as you take more and more picks
you get a beautiful map and you realize
oh wait a second your kid is mostly
found somewhere in this region so
tomorrow if I had to if you had to
predict at 2: p.m. where your kid is
going to be you can say that hey you
know what my kid is going to be
somewhere in this region isn't it this
is what orbital is we call this as an
orbital and I'm like wait a second this
has nothing to do with orbits the kid is
not orbiting at all and shinger says
exactly that's not a really great name
and orbital is kind of like a
probability map if you can look at that
map and you can kind of predict what are
the chances where you know you can
predict what is You can predict where
it's most likely to find your kid the
same thing is going on over here this is
a probility map so by looking at this
you can say that hey there's a very high
chance that my electron is going to be
somewhere in this region somewhere in
between inside this region I know that
if you got very far away the probability
of finding is very low but that is all
that is it does not say that the
electron is orbiting it doesn't tell you
what the electrons are doing all you can
talk about is the probability of finding
the electron and that's the orbital and
again I'm so I'm so happy that we were
able to figure this out like we were
able to build this out just by looking
at the stretch string intuitively
wouldn't it be so awesome if you could
ReDiscover all Concepts like this well
guess what you can to some extent using
you'll also be supporting me all right back to the the video so this represents
the ground state of the hydrogen atom
what happens now if you were to excite
it well let's see again going back to
the string analogy we know that the
first excited state should have a node
in between so shinger asks mes if you
now think about the probability as you
go from here to here what should happen
well I'm like okay the probability
should start with zero the probability
should increase to a maximum because you
know negative amplitude doesn't matter
probability is amplitude squared so the
negative amplitude positive amplitude
doesn't matter so starts with zero goes
to a maximum goes to zero again goes to
maximum again and then eventually goes
to zero and again we can confine this
between the nucleus and rally going
outwards towards Infinity so again let's
simulate this and see what it looks like
so here's our nucleus again it's very
tiny and right at the nucleus I should
have very low probability and you might
not be able to see this but in my
simulation there are 10 dots over here
again we're going to go further away
radially as I go radially further and
further away look the probability goes
to a maximum um about 409 dots over here
and then as I further go further away it
should reduce and you can see it reduces
reduces and it goes to zero this is our
node this is our first this is the node
that we get over here all right further
if I go outwards it should again
increase to a maximum all right here it
goes here it goes here it goes and again
this is a qualitative pattern okay we
don't know like we're not able to figure
out how much is this maximum compared to
this maximum all of that requires math
qualitative not quantitative okay but
anyways we get a maximum over here and
again as I now go farther away towards
Infinity it should slowly taper off it
should slowly go to zero this is what it
would look like right okay so if you
build it out what is it going to look
like well this is what we would get
isn't that beautiful look now we have a
region over here which is a node which
means the probability of finding the
electron over here must be zero and
again compare this to the ground
state look the ground state is much
smaller and for this first excited state
you can see the electrons have much
higher energy because they have a nde
because they're more confined they have
much higher energy and so they can be
farther out away from um the nucleus
beautiful isn't it so we can build an
energy map now this is the ground state
and this is the first EX cided State and
you might identify them as s orbitals
spherically symmetric s orbitals but
remember we're pretending we don't know
anything about them so let's keep the
names um side at the side for now and at
this point I'm like shinger I think I
got it for the next excited way uh next
excited state I should have two nordes
and so on and so forth right and shinger
says mahes hold your horses because
things are about to get interesting
mahes over here the nde is a sphere this
is a spherical node but the question is
could we have other shapes for nodes and
I'm like yeah could be but how do we
think about that well here's how in one
dimension a node is just a point there's
no other way to have a node in two
dimensions a point becomes a straight
line in three dimensions that straight
line becomes a plane which means I can
have a planar Nord and I'm like okay how
do I how do I visualize this so he says
let's take our excited state that we
just got let's get rid of that spherical
node use a plane to slice this right at
the center and you get a node a plar
node right through the center and
immediately my question is well why am I
slicing it at the center why can't I
slice it for example over here to get a
node over there and shinger says mahes
you have to respect the Symmetry I mean
showing an equation the kum's potential
over here is symmetric so you need to
cut it in a symmetric way think about
cutting a cake you cut it right in
between to get two halves the same thing
over here you have to cut it in a
symmetric way and so this is another way
to build the orbital for the first
excited state you can have a cut a slice
in between so this means the probability
goes to zero over here now of course in
reality the probability cannot abruptly
go from here to here so the shape will
not be exactly like this if you actually
find what the shape looks like it goes
somewhat like this and I'm like oh my
God that that is the p
orbital this is why P orbitals have a
dumb bell shaped because at the heart of
it what's going on is we're taking a
sphere and we're chopping it to get a
plan our
Nord this is where it all comes from I
mean sure we couldn't get the exact p
orbital but we understood the skeleton
of it why we have two loes oh that is
incredible now at this point we could
ask well but wait a second shinger why
slice it horizontally I mean there are
three axes so I could slice it this way
I could also slice it this way and
shinger says exactly you slice in three
different ways and that's why you don't
get one p orbital you get two
more and that's why there are total four
ways of getting the first excited state
where you can have a spherical Nord or
you can have planar Nord isn't that
wonderful and I'm like at this point
should be like okay are there more ways
to slice and dice It Well turns out no
there are no other ways to slice it
symmet Al the key idea is we should do
it symmetrically and I'm like wait a
second why can't I slice it diagonally
well shinger says diagonal slicing uh
gives you an orbital which can be
thought of as a linear combination of
you know all you know of of these three
think of it as a vector mahes if you
have three bases vectors then any other
Vector in any other direction can be
written as a linear combination of you
know these basis vectors that's what we
write it as i jcap k cap and stuff like
that right so it's same idea so these
are the three so you can think of this
as the three bases orbitals that we have
and so total we get four
orbitals wow okay so now let's look at
some properties of the nodes over here
well over here the node is spherically
symmetric there is no directional
property right but here you can see some
kind of a directional property it's
oriented this way this is oriented this
way this is oriented you know this way
okay that's it's a little hard to see it
it's it's coming out of the screen you
can imagine this way so it gives you a
directional property so look the
spherical node does not give you any
directional property and so we give a
name to this we call this the radial
node so in the first excited state you
can have one radial node a radial node
because you know the node has is at the
same distance from the center anywhere
you go but another way to set up that
node it gives you a directional property
we call that an angular node so you can
either have the node to be radial or you
can have the node to be angular if you
have the angular node that's where the
plan are the plane slicing it that's
where you get the three p orbitals um
again let's not use the word p curse of
knowledge but uh the the earlier one the
ground state has zero notes and now we
can go one step further and we can be a
little bit faster if we go to the next
excited state well we're going to have
two nodes and so as we go radially
outwards we should have two nodes over
here and so that is the next excited
state with two radial nodes and now
again Shing your ask Mahesh could there
be more ways to build it because now we
have two nodes things get even more
interesting so let's figure this out
together how else can we have these
nodes well we can make both these nodes
angular or we can also make one radial
one angular let's try that and let see
what we get so we have let's make one
radial and one angular so let's get rid
of one of the radial nodes and convert
that into an angular node meaning slice
it and if you slice at the center we
should again get a p orbital a dumbbell
shaped p orbital but this time it also
has a radial node in between isn't this
making sense so I know that I should get
three of these definitely possible and
you get these three if you have one
radial node and one angular node in the
second excited state but the last way
which which we can set up these orbitals
is if both of them are angular so let's
do that let's take both these radial
nodes and convert it into angular nodes
how do we have two angular
nodes well slice it with two planes says
shinger but how do we slice it with two
planes well again we have to respect the
symmetry so again think about a cake if
you had to make two slices symmetrically
how would we do it oh we would do it
this way we would do it this way so
we'll slice one this way and we'll slice
the other like this and and that will
give me an orbital with two angular
nodes and what do you notice again of
course probability Cannot adly drop out
so the shape will not be exactly like
this but the shape will turn out to be
this way this is the D orbital a this is
why the D orbital has four loes because
there are two angular two plan on nodes
and that's how we have sliced it again
yeah I want to keep the de orbital the
word the name deorbit to the side but
this makes sense isn't it and again the
big question is how can we Orient this
now my intuition says well just like
with the p orbital double double shaped
orbital um we should be able to orient
this in three different ways right so
one way this way another one like this
probably depending on how we slice it U
but this is where we actually run out of
intuition it turns out that there are
more ways to orient this now again
that's not very straightforward that's
not very intuitive but it kind of makes
sense that if you had one angular node
there are three ways to orient it if you
have two angular nodes there are more
ways to orient it right now this is
where we run out of intuition we hit a
hall and this is where we just accept
that it turns out that there are other
ways to set up these notes complicated
stuff so it's okay turns out that there
are two extra ways of building this so
in total we get five ways of building
this orbital and that's the reason why
the D orbital we call them has five five
there are five different D orbitals but
what's important for us is because
they're all built with two angular nodes
and at least it gives us some intuition
for why they have these four loopes that
is beautiful if you ask me and now let's
try to ReDiscover our quantum numbers
okay so the first question we could ask
over here is how do I identify where my
electron is I need to use numbers to
identify that the first number I can use
to identify where my electron is is the
energy level I could say that my
electron is either in the nals 1 level
or Nal 2 level level or Nal 3 level so I
need a number to represents which energy
level it belongs to and that is the
principal quantum number and that's why
it can takes integer values um the first
one means ground state second one means
the first excited state third one means
the next excited and so on and so forth
but that's not enough for example if I
know that my electron is in this you
know first excited state or n equals 2
how do I know whether it's over here or
it's over here in other words I need to
know the shape of the orbital it belongs
to what decides the shape of the orbital
if you look carefully you
notice the number of angular node
decides the shape of the orbital think
about it if you have zero angular nodes
then the shape of the orbital is going
to be spherically symmetric we call that
the S orbital s doesn't stand for
spherically symmetric I mean there's
some history behind where these s PDF
came from well not we had some sharp
lines and we had some diffused Lin
that's where the word SP come D comes
from but let's not worry about that
what's important is that if you have
zero angular nodes um then you have uh s
orbital if you have one angular node
look we get dumbbell shaped orbital the
same shape we call that the p orbital
and if you have two angular nodes we get
again the same shape the D orbital and
so on and so forth the shape of the
orbital depends upon the number of
angular nodes and that could be our next
quantum number and that's exactly what a
zimal quantum number is a zimal quantum
number is basically telling you how many
angular nodes are present in your
orbital and let's look at its values now
um for the first one you can only have
zero because there are zero nodes in the
second one you can have either zero or
one angular node in the third one you
can have 0 1 2 so look it runs from 0 to
n
-1 why is it run to n minus one well
because the first ex first state Nal 1
represents the ground state ground state
has zero nodes Nal 2 has one node so
maximum one angular node Nal 3 has two
nodes so maximum two angular nodes and
that's why the maximum is n minus one
nodes isn't that making beautiful sense
now okay but just knowing this it's not
enough because if I know the shape of
the orbital I still need to know which
orientation it belongs to and that's why
I need another quantum number to
represent the orientation for that and
that is the next quantum number the
magnetic quantum number it tells me
which orientation the electron belongs
to and this is where our intuition
stopped remember for the p orbital it we
could intuitively see why there are
three orientation because of the three
axes but for D orbitals we couldn't
explain why there are five orientations
um there is some connection between the
angular the number of angular nodes and
the number of orientations you can have
and to and there is a hand wavy
intuition that we can build it's not
rediscovery but here's here's something
that helps me okay helps me sleep at
night I could say all right so here's a
different perspective okay so let me
just try to look at this in a slightly
different perspective consider one
angular node okay how can you set it up
in a spinning sphere I know like looks
random but it make sense okay one way is
to set it up set up the angular node as
an equator like your slicing it in the
equator okay I'm just going to draw a
sphere over here and now let's say you
spin it along the
z-axis okay you can spin it in two ways
you can spin it clockwise you can spin
it counterclockwise what's happening to
the node the node is spinning like this
and it doesn't look like anything is
happening right it looks stationary
whether you spin like this or you spin
like that it looks the same stationary
we will call this uh
orientation um the zero state so the ml
will be equal to zero but there's
another way to orient this node instead
of Equatorial we can Orient it polar
pole to pole and again we can spin it in
two ways now if we spin it we actually
speed see one node spinning we see it
going this way and now clockwise and
anticlockwise we can differentiate
clockwise will be this this way
anticlockwise will be this way and so
because we see one node moving around
now we can call this as m equal plus one
if it's say clockwise and we can call it
minus one if it's
anticlockwise any other way you Orient
that node and you make it spin can be
written as a linear combination of these
two and therefore we have total three
different states so if you have one node
we have three states they don't directly
map to the shape of the orbitals you
have to combine them and stuff but let's
not worry about it at Le it gives you
the number why it runs from minus1 to+
one now let's consider what happens if
you have two angular nodes well how can
we set them up well two plane slicing
for the sphere the way we can do that is
one we can have two equatorial nodes um
and again it is symmetric so it's at the
same distance from the center again if
you make it spin in either ways the
nodes don't appear to be moving at all
so this is the zero State now you can
have one equatorial and one
polar and so we get a plus one and minus
one because the pole to pole one will
spin like this and spin like that or you
can have both pull to pole not exactly
pull to pole longitudinal wise okay and
now if you spin them both spin this way
or that way and that's why I get plus
two and minus two this is the reason why
the orbital the magnetic quantum numbers
run from minus L to plus l i yeah I
agree this is very hand Wy it's
connected to something called the
spherical harmonics and I really didn't
want to get into the details of it we're
running towards the end of the video
hopefully this gives you some intuition
for why if you have two angular nodes
there must be five orbitals and you can
now figure this out if you have three
angular nodes we should have seven
orbitals it'll run from minus 3 to+
three so this is where the third quantum
number comes from it runs from minus L
to plus L it tells you the shape of the
orbital these three are enough for the
hydrogen so just from this we can now
explain everything all the problems that
the earlier model had first of all look
the electrons are no longer orbiting
they are forming these stationary wave
waves so there is no accelerated charges
and therefore they don't have to spiral
they don't lose energy and therefore
they are stable secondly what happens
why do we get the atomic spectrum why do
we get the hydrogen spectrum that way
well imagine we have hydrogen in the
ground state if it absorbs energy and
let's say jumps to the first excited
state over here when it jumps back look
it goes from here to here and releases
the energy which is exactly the
difference between the two states and
that particular energy corresponds to a
particular frequency according to to e
is equal to HF and I think this is the
part that you probably know we've
studied that in B's Theory and again it
could now next time it gets excited to
let's say this state it can either
directly jump back to the ground state
and give another Photon of energy or it
can jump to this state and give another
Photon of energy so because there are
discrete energy levels that's why we're
getting these transitions it explains
the hydrogen Spectrum
beautifully but guess what it can
explain more to do that for multi-
electron systems like helium and
everything else in the periodic table we
need one last quantum number that does
not come from the short Rangers
equations that comes from Po's Exclusion
Principle which says that two electrons
cannot have the same quantum numbers and
because you know by now that electrons
have two kinds of spin spin up and spin
down that means each orbital has a
maximum capacity of two electrons and
that gives us the last quantum number
isn't that a amazing we've discovered
quantum numbers the shape of the
orbitals what the orbitals means all
from this one single idea of standing
waves and thinking about nodes I find
that absolutely beautiful but guess what
shinger says this is just the tip of the
iceberg because we just talked about the
hydrogen but now with all of this we can
start talking about other elements and
in fact we can start answering the most
fundamental mystery that I always had in
my chemistry course the octet rule why
does everything have octet rule why do
noal gases have octat and all of those
things why everybody loves to have octed
and why certain elements can even break
the octet rule to form bonds all of that
can be understood using this ideas and
that's something that we'll explore in
the next video