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00:00:00,000 --> 00:00:03,660
PROFESSOR: So I want
to go a little further

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to try to put resonances in
a more intriguing footing.

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That you can play with and if
you-- at some point interested.

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So let's think of discovering
[INAUDIBLE] that we have.

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We had A s--

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remember the scattered wave was
A s e to the ikx [INAUDIBLE]

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that divided 2.

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And what was A s?

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Well, A s squared--

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the sine square delta.

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So if you remember this was
sine delta e to the i delta.

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So let's stick to that and try
to write it in a funny way.

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Certainly, A s is becoming
large near resonance,

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so let's think when
A s becomes large.

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Well, in another way let's be
a little creative about things,

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It's good sometimes
not to be logical.

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So let's write this
as sine delta--

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I'll do it here--

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sine delta over e to
the minus i of delta .

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And that's sine delta over
close delta minus i sine delta.

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That's all good.

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A s-- let me divide by
sine delta both sides--

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both numerator and denominator.

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So-- no divide it
by cosine delta,

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so I'll have tan delta
over 1 minus i tan delta.

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I divide it by cosine.

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You want A s large?

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You really want it
large, choose tan delta--

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equals to minus i.

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Sounds crazy, but
it's not really crazy.

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The reason it sounds crazy and
it's somewhat strange and not

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very logical is tan
delta is a phase

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and the tangent of any phase
is never an imaginary number.

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So then I would have
think of delta itself

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as a complex number.

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And what would that mean.

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So things are weird.

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But it's certainly the fact
that A s will become infinite--

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not just large-- but infinite.

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A s will become infinite.

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And you say, wow, this
doesn't make any sense.

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But maybe it makes sense
in the following way.

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This is the line of
real phase shifts.

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[INAUDIBLE] are real.

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And here is the world
of complex phase shifts.

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These are the real phase shifts
and there are the complex phase

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shifts.

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Maybe if the phase
shift becomes infinite--

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off the real axis--

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it's just large
on the real axis.

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So actually, if you
wanted it to be very large

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you would have to get
off the real axis.

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If this sounds vague,
it is still vague.

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But in a minute we'll
make it precise.

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So I suggest that we take
this idea seriously--

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that maybe this means something.

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And we can try to argue
that by looking back

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at what resonances do.

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So what I will do is look with
[INAUDIBLE] a resonance here--

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tangent delta.

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So let's look at what A s does.

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We have it there.

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A s is tan delta--
well, tan delta--

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we had it in the
middle of blackboard

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is beta over alpha
minus k, 1 minus i beta

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over alpha minus k, again.

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So that's how A s
behaves in general.

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That's fine, there's no -- at
this moment there's nothing

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crazy about this.

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Because this is
something you all agreed,

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nobody complained
about this formula.

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So A s is given
by that formula--

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that's also legal math, so far.

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So we'll have this.

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And then let's simplify
it a little bit which

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is beta over alpha
minus k minus i beta.

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So this still beta over
alpha minus i beta minus k.

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So we usually would plot
A s as a function of k.

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That's what we're trying to
do, it's a function of k.

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And now here is the formula
for A s as the function of k.

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And here is k.

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But let's be daring now
and not say this is k,

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this is the complex k-plane.

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00:07:03,340 --> 00:07:09,390
And yes, you work with
real k, but that's

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because that has a direct
physical interpretation.

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But maybe the complex
plane has a more subtle

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physical interpretation
and that's

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what they claim
is happening here.

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This quantity becomes
infinite near the resonance.

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Here was the resonance,
what you call the resonance.

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But this becomes really
infinite not at alpha--

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for when k is equal
to alpha, but when k

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is equal to alpha minus i beta.

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Beta was supposed to be
small for a resonance.

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So here is minus i beta and
here is this very unusual point.

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Where the scattering
amplitude blows up.

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It has what is in
complex variables--

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if you've taken 1806
it's called a pole.

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In a complex variable
when you have

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a denominator that vanishes
linearly we call it a pole.

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Things blow up.

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So this carrying amplitude
has a pole off the real axis.

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And interpretation is correct.

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At this point, this
function becomes infinite.

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And what is happening
on the real line

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that A s is becoming
large is just

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the remnant of that
infinity over here

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that is affecting the
value of this point.

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So in the complex plane
you understand the function

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a little better.

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You see why it's
becoming big and you

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can see also with a
little [INAUDIBLE]

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why the phases
shifting very fast

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because you have this point.

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And that's called the resonance.

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And this is the
mathematically precise way

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of searching for resonances.

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If you want to search for
resonances what you should do

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is you have your formula for
delta as a function of k.

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I mean, it's a
complicated formula,

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but now try to solve the
equation tan delta of this

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is equal to minus
i because that's

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what guarantees
that you have a pole

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that indeed it blows
up at some value.

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That's where A s blows up which
we see directly here-- it's

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this value.

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Alpha minus i beta, so alpha
minus i beta is a pole of A s.

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And therefore, you must be
happening when tangent of delta

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is equal to minus i.

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So you have a very
complicated formula

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maybe for tangent of delta.

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But set it equal to minus
i and asked mathematically

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to solve it.

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And a number will come--

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k a equal 2.73 minus 0.003.

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And you will know-- oh, that's
a resonance, it's off the axis.

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And the real part is
the value of alpha.

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And since this is beta
the closer to the axis --

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00:10:46,470 --> 00:10:48,115
if you find more--

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00:10:48,115 --> 00:10:50,642
the more resonant it is.

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00:10:50,642 --> 00:10:53,135
And by the time it's
far from the axis,

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some people call it the
resonance-- some people say,

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no that not the resonance.

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It's a matter of taste.

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But there are important
things which are these poles.

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00:11:05,490 --> 00:11:09,480
So I will not give
you exercises on that,

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00:11:09,480 --> 00:11:12,550
but you may want
to try it if you

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00:11:12,550 --> 00:11:16,950
want to have some entertainment
with these things.

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00:11:16,950 --> 00:11:20,250
I want to say one
more thing about this.

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And it's the reason why this
viewpoint is interesting,

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00:11:28,290 --> 00:11:30,260
as well.

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00:11:30,260 --> 00:11:35,360
We already found that if we
want to think of resonances more

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00:11:35,360 --> 00:11:37,340
precisely.

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00:11:37,340 --> 00:11:39,925
We can think of them
as just an equation.

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00:11:39,925 --> 00:11:42,210
You solve for the equation,
so that it gives you

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00:11:42,210 --> 00:11:43,140
the resonance.

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00:11:43,140 --> 00:11:46,500
And this is the
equation you must solve

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00:11:46,500 --> 00:11:53,875
and you must admit complex k.

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But now you can say,
look actually you

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00:12:01,250 --> 00:12:06,726
have e is equal to h
squared, k squared, over 2m.

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And we have real k's--

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00:12:16,870 --> 00:12:20,920
this is the physical
scattering solutions,

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00:12:20,920 --> 00:12:25,000
complex k's, also resonances.

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00:12:25,000 --> 00:12:28,730
How about imaginary k's?

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00:12:28,730 --> 00:12:33,226
If k is equal to i kappa--

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kappa belonging to
the real numbers--

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then the energy becomes minus
h squared, kappa squared,

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00:12:43,610 --> 00:12:46,845
over 2m and its less
than zero and it

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00:12:46,845 --> 00:12:48,780
could represent bound states.

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00:12:51,960 --> 00:12:57,770
So you'll be then discovering
solutions of real k

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00:12:57,770 --> 00:13:00,400
representing your waves.

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00:13:00,400 --> 00:13:04,240
Now mathematically, you
are led to resonances

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00:13:04,240 --> 00:13:08,140
understood as poles
in the scattering

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00:13:08,140 --> 00:13:09,300
amplitutde we did here.

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00:13:11,970 --> 00:13:22,104
We see that k's in
the imaginary axis

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00:13:22,104 --> 00:13:24,840
would represent bound states.

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00:13:28,004 --> 00:13:31,530
So the complex
k-plane is very rich.

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00:13:31,530 --> 00:13:34,675
It has room for your
scattering solutions,

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00:13:34,675 --> 00:13:37,190
it has room for your
resonance, it even

180
00:13:37,190 --> 00:13:39,990
has room for your bound states.

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00:13:39,990 --> 00:13:42,310
They're all there.

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00:13:42,310 --> 00:13:45,580
That's why it's a
valuable extension.

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00:13:45,580 --> 00:13:48,940
I have now proven for you
that bound states correspond

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00:13:48,940 --> 00:13:50,420
to poles.

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00:13:50,420 --> 00:13:54,550
It's a simple calculation, and
that I would assign it to you

186
00:13:54,550 --> 00:13:56,380
with a little bit of guidance.

187
00:13:56,380 --> 00:14:02,020
And you will see that also
for the case of bound states,

188
00:14:02,020 --> 00:14:05,766
you get a pole in the
scattering amplitude,

189
00:14:05,766 --> 00:14:10,550
and that will complete the
interpretation of that.

190
00:14:10,550 --> 00:14:14,340
Now people go a little
further, actually,

191
00:14:14,340 --> 00:14:18,596
and they invent
poles in this part

192
00:14:18,596 --> 00:14:21,292
and they're called
anti-bound states.

193
00:14:24,370 --> 00:14:27,310
And you'll say, what's that?

194
00:14:27,310 --> 00:14:30,520
If you have a bound state
you match a solution

195
00:14:30,520 --> 00:14:36,010
to a pure decaying exponential
for the [INAUDIBLE] region.

196
00:14:36,010 --> 00:14:40,285
In an anti-bound bound state
you match your solution

197
00:14:40,285 --> 00:14:44,106
to a pure increasing
exponential.

198
00:14:44,106 --> 00:14:46,010
A pure one.

199
00:14:46,010 --> 00:14:47,765
Does that have an
interpretation?

200
00:14:47,765 --> 00:14:50,460
It actually does
have interpretation.

201
00:14:50,460 --> 00:14:52,790
Some nuclear states
are associated

202
00:14:52,790 --> 00:14:55,230
with anti-bound states.

203
00:14:55,230 --> 00:14:57,665
So the mathematical
description--

204
00:14:57,665 --> 00:15:00,800
the rich complex
plane is ready for you

205
00:15:00,800 --> 00:15:03,680
if you just do
scattering amplitude k,

206
00:15:03,680 --> 00:15:06,800
resonances-- complex k.

207
00:15:06,800 --> 00:15:10,385
Normal bound states,
imaginary k-- positive.

208
00:15:10,385 --> 00:15:13,080
Anti-bounds is negative k.

209
00:15:13,080 --> 00:15:15,030
It's a nice start.