WEBVTT
00:00:01.730 --> 00:00:03.770
BARTON ZWIEBACH: We
were faced last time
00:00:03.770 --> 00:00:05.660
with a question
of interpretation
00:00:05.660 --> 00:00:08.760
of the Schrodinger
wave function.
00:00:08.760 --> 00:00:15.470
And so to recap the main
ideas that we were looking at,
00:00:15.470 --> 00:00:19.820
we derive this
Schrodinger equation,
00:00:19.820 --> 00:00:25.550
basically derived it
from simple ideas--
00:00:25.550 --> 00:00:31.010
having operators, energy
operator, momentum operator,
00:00:31.010 --> 00:00:36.680
and exploring how the de
Broglie wavelength associated
00:00:36.680 --> 00:00:40.970
to a particle would be a wave
that would solve the equation.
00:00:40.970 --> 00:00:43.455
And the equation was the
Schrodinger equation,
00:00:43.455 --> 00:00:46.250
a free Schrodinger
equation, and then
00:00:46.250 --> 00:00:48.110
we added the
potential to make it
00:00:48.110 --> 00:00:53.720
interacting and that way,
we motivated the Schrodinger
00:00:53.720 --> 00:01:10.250
equation and took this form
of x and t psi of x and t.
00:01:10.250 --> 00:01:15.400
And this is a dynamical equation
that governs the wave function.
00:01:15.400 --> 00:01:18.030
But the interpretation
that we've
00:01:18.030 --> 00:01:22.840
had for the wave function,
we discussed what Born said,
00:01:22.840 --> 00:01:26.670
was that it's related
to probabilities
00:01:26.670 --> 00:01:31.230
and psi squared
multiplied by a little dx
00:01:31.230 --> 00:01:33.690
would give you the
probability to find
00:01:33.690 --> 00:01:39.220
the particle in that little
dx at some particular time.
00:01:39.220 --> 00:01:50.430
So psi of x and t squared
dx would be the probability
00:01:50.430 --> 00:01:56.700
to find the particle at
that interval dx around x.
00:01:56.700 --> 00:02:00.000
And if you're describing the
physics of your Schrodinger
00:02:00.000 --> 00:02:02.940
equation is that of
a single particle,
00:02:02.940 --> 00:02:04.650
which is the case here--
00:02:04.650 --> 00:02:12.525
one coordinate, the coordinate
of the particle, this integral,
00:02:12.525 --> 00:02:17.550
if you integrate
this all over space,
00:02:17.550 --> 00:02:22.930
must be 1 for the
probability to make sense.
00:02:22.930 --> 00:02:28.420
So the total probability of
finding the particle must be 1,
00:02:28.420 --> 00:02:30.040
must be somewhere.
00:02:30.040 --> 00:02:32.200
If it's in one part
of another part
00:02:32.200 --> 00:02:35.230
or another part, this
probabilities-- for this
00:02:35.230 --> 00:02:37.075
to be a probability
distribution,
00:02:37.075 --> 00:02:41.470
it has to be well-normalized,
which means 1.
00:02:41.470 --> 00:02:50.350
And we said that this equation
was interesting but somewhat
00:02:50.350 --> 00:02:55.960
worrisome, because if the
normalization of the wave
00:02:55.960 --> 00:03:07.880
function satisfies, if this
holds for t equal t-nought
00:03:07.880 --> 00:03:13.490
then the Schrodinger equation,
if you know the wave function
00:03:13.490 --> 00:03:18.890
all over space for t equal
t-nought, which is what you
00:03:18.890 --> 00:03:23.450
would need to know in order
to check that this is working,
00:03:23.450 --> 00:03:28.550
a t equal t-nought, you take
the psi of x and t-nought,
00:03:28.550 --> 00:03:30.650
integrate it.
00:03:30.650 --> 00:03:36.900
But if you know psi of x
and t-nought for all x,
00:03:36.900 --> 00:03:39.200
then the Schrodinger
equation tells you
00:03:39.200 --> 00:03:43.290
what the wave function
is at a later time.
00:03:43.290 --> 00:03:46.320
Because it gives you the
time derivative of the wave
00:03:46.320 --> 00:03:49.580
function in terms of
data about the wave
00:03:49.580 --> 00:03:52.350
function all over space.
00:03:52.350 --> 00:03:55.470
So automatically, the
Schrodinger equation
00:03:55.470 --> 00:04:02.230
must make it true that this
will hold at later times.
00:04:02.230 --> 00:04:06.651
You cannot force the wave
function to satisfy this at all
00:04:06.651 --> 00:04:07.150
times.
00:04:07.150 --> 00:04:11.650
You can force it maybe
to satisfy at one time,
00:04:11.650 --> 00:04:16.290
but once it satisfies it at
this time, then it will evolve,
00:04:16.290 --> 00:04:20.470
and it better be that
at every time later,
00:04:20.470 --> 00:04:23.730
it still satisfies
this equation.
00:04:23.730 --> 00:04:28.640
So this is a very
important constraint.
00:04:28.640 --> 00:04:37.910
So we'll basically develop this
throughout the lecture today.
00:04:37.910 --> 00:04:43.610
We're going to make a big point
of this trying to explain why
00:04:43.610 --> 00:04:47.650
the conditions that we're going
to impose on the wave function
00:04:47.650 --> 00:04:51.770
are necessary; what it teaches
you about the Hamiltonian,
00:04:51.770 --> 00:04:55.970
we'll teach you that it's
a Hermitian operator;
00:04:55.970 --> 00:05:00.120
what do you learn
about probability--
00:05:00.120 --> 00:05:03.810
you will learn that there
is a probability current;
00:05:03.810 --> 00:05:09.720
and all kinds of things will
come out of taking seriously
00:05:09.720 --> 00:05:12.930
the interpretation
of this probability,
00:05:12.930 --> 00:05:18.210
the main point being that
we can be sure it behaves
00:05:18.210 --> 00:05:23.130
as a probability at one time,
but then for later times,
00:05:23.130 --> 00:05:26.670
the behaviors and probability
the Schrodinger equation must
00:05:26.670 --> 00:05:27.600
help--
00:05:27.600 --> 00:05:31.770
must somehow be part of
the reason this works out.
00:05:31.770 --> 00:05:35.750
So that's what we're
going to try to do.
00:05:35.750 --> 00:05:43.550
Now, when we write an equation
like this, and more explicitly,
00:05:43.550 --> 00:05:55.460
this means integral of psi star
of x and t, psi of x and t dx
00:05:55.460 --> 00:05:56.300
equal 1.
00:05:59.270 --> 00:06:02.690
You can imagine that not
all kind of functions
00:06:02.690 --> 00:06:05.660
will satisfy it.
00:06:05.660 --> 00:06:09.020
In particular, any wave
function, for example,
00:06:09.020 --> 00:06:13.250
that at infinity approaches
a constant will never
00:06:13.250 --> 00:06:16.700
satisfy this, because
if infinity, you
00:06:16.700 --> 00:06:20.300
approach a constant,
then the integral
00:06:20.300 --> 00:06:22.980
is going to be infinite.
00:06:22.980 --> 00:06:25.020
And it's just not
going to work out.
00:06:25.020 --> 00:06:31.920
So the wave function cannot
approach a finite number,
00:06:31.920 --> 00:06:35.350
a finite constant as
x goes to infinity.
00:06:35.350 --> 00:06:39.810
So in order for this to hold--
00:06:39.810 --> 00:07:04.190
order to guarantee this can
even hold, can conceivably hold,
00:07:04.190 --> 00:07:07.610
it will require a bit
of boundary conditions.
00:07:07.610 --> 00:07:12.290
And we'll say that
the limit as x
00:07:12.290 --> 00:07:16.770
goes to infinity
or minus infinity--
00:07:16.770 --> 00:07:22.790
plus/minus infinity of psi of
x and t will be equal to 0.
00:07:25.640 --> 00:07:28.210
It better be true.
00:07:28.210 --> 00:07:31.240
And we'll ask a little more.
00:07:31.240 --> 00:07:36.700
Now, you could say,
look, certainly
00:07:36.700 --> 00:07:39.610
the limit of this
function could not
00:07:39.610 --> 00:07:45.190
be in number, because it
would be non-zero number,
00:07:45.190 --> 00:07:47.270
the interval will diverge.
00:07:47.270 --> 00:07:49.840
But maybe there is no limit.
00:07:49.840 --> 00:07:54.160
The wave function is so crazy
that it can be integrated,
00:07:54.160 --> 00:07:57.250
but suddenly, it
has a little spike
00:07:57.250 --> 00:08:00.820
and it just doesn't
have a normal limit.
00:08:00.820 --> 00:08:05.170
That could conceivably
be the case.
00:08:05.170 --> 00:08:10.000
Nevertheless, it doesn't seem
to happen in any example that
00:08:10.000 --> 00:08:11.770
is of relevance.
00:08:11.770 --> 00:08:16.210
So we will assume that
the situations are not
00:08:16.210 --> 00:08:19.740
that crazy that this happened.
00:08:19.740 --> 00:08:23.860
So we'll take wave
functions that necessarily
00:08:23.860 --> 00:08:26.560
go to 0 at infinity.
00:08:26.560 --> 00:08:32.830
And that certainly is good.
00:08:32.830 --> 00:08:36.010
You cannot prove it's
a necessary condition,
00:08:36.010 --> 00:08:41.080
but if it holds, it
simplifies many, many things,
00:08:41.080 --> 00:08:46.300
and essentially, if the wave
function is good enough to have
00:08:46.300 --> 00:08:49.690
a limit, then the
limit must be 0.
00:08:49.690 --> 00:08:51.610
The other thing
that we will want
00:08:51.610 --> 00:09:04.710
is that d psi/dx, the limit as
x goes to plus/minus infinity
00:09:04.710 --> 00:09:05.380
is bounded.
00:09:09.220 --> 00:09:14.050
That is, yes, the limit may
exist and it may be a number,
00:09:14.050 --> 00:09:17.510
but it's not infinite.
00:09:17.510 --> 00:09:21.890
And In every example
that I know of--
00:09:21.890 --> 00:09:25.340
in fact, when this goes to
0, this goes to 0 as well--
00:09:25.340 --> 00:09:31.640
but this is basically all
you will ever need in order
00:09:31.640 --> 00:09:35.900
to make sense of the wave
functions and their integrals
00:09:35.900 --> 00:09:37.700
that we're going to be doing.
00:09:37.700 --> 00:09:40.670
Now you shouldn't
be too surprised
00:09:40.670 --> 00:09:45.080
that you need to say something
about this wave function
00:09:45.080 --> 00:09:46.760
in the analysis
that will follow,
00:09:46.760 --> 00:09:50.150
because the derivative--
00:09:50.150 --> 00:09:52.220
you have the function
and its derivative,
00:09:52.220 --> 00:09:56.190
because certainly, there
are two derivatives here.
00:09:56.190 --> 00:10:04.020
So when we manipulate
these quantities
00:10:04.020 --> 00:10:08.890
inside the integrals,
you will see very soon--
00:10:08.890 --> 00:10:15.280
single derivatives will show up
and we'll have to control them.
00:10:15.280 --> 00:10:16.870
So the only thing
that I'm saying
00:10:16.870 --> 00:10:20.920
is that when you see
a wave function that
00:10:20.920 --> 00:10:24.320
satisfies this
property, you know
00:10:24.320 --> 00:10:27.290
that unless the function
is extremely crazy,
00:10:27.290 --> 00:10:32.600
it's a function that goes
to 0 at plus/minus infinity.
00:10:32.600 --> 00:10:35.540
And it's the relative
pursuant it also goes to 0,
00:10:35.540 --> 00:10:42.570
but it will be enough to say
that it maybe goes to a number.
00:10:42.570 --> 00:10:47.820
Now there's another
possibility thing for confusion
00:10:47.820 --> 00:10:52.930
here with things that
we've been saying before.
00:10:52.930 --> 00:10:58.410
We've said before that the
physics of a wave function
00:10:58.410 --> 00:11:05.890
is not altered by multiplying
the wave function by a number.
00:11:05.890 --> 00:11:11.150
We said that psi added to
psi is the same state; psi
00:11:11.150 --> 00:11:14.820
is the same state as
square root of 2 psi--
00:11:14.820 --> 00:11:17.760
all this is the same
physics, but here it
00:11:17.760 --> 00:11:25.080
looks a little surprising if you
wish, because if I have a psi
00:11:25.080 --> 00:11:31.290
and I got this
already working out,
00:11:31.290 --> 00:11:37.540
if I multiply psi by square
root of 2, it will not hold.
00:11:37.540 --> 00:11:43.334
So there seems to be a little
maybe something with the words
00:11:43.334 --> 00:11:44.250
that we're been using.
00:11:44.250 --> 00:11:47.970
It's not exactly
right and I want
00:11:47.970 --> 00:11:52.050
to make sure there is no
room for confusion here,
00:11:52.050 --> 00:11:56.730
and it's the following fact.
00:11:56.730 --> 00:12:03.330
Here, this wave function
has been normalized.
00:12:03.330 --> 00:12:08.810
So there's two kinds of wave
functions that you can have--
00:12:08.810 --> 00:12:12.205
wave functions that can be
normalized and wave functions
00:12:12.205 --> 00:12:13.640
that cannot be normalized.
00:12:13.640 --> 00:12:18.940
Suppose somebody comes to you
and gives you a psi of x and t.
00:12:22.530 --> 00:12:25.240
Or let's assume that--
00:12:25.240 --> 00:12:27.320
I'll put x and t.
00:12:27.320 --> 00:12:29.270
No problem.
00:12:29.270 --> 00:12:36.270
Now suppose you go and
start doing this integral--
00:12:36.270 --> 00:12:42.230
integral of psi squared dx.
00:12:42.230 --> 00:12:44.990
And then you find that
it's not equal to 1
00:12:44.990 --> 00:12:49.840
but is equal to
some value N, which
00:12:49.840 --> 00:12:52.080
is different from 1 maybe.
00:12:55.690 --> 00:13:10.700
If this happens, we say that
psi is normalizable, which
00:13:10.700 --> 00:13:16.580
means it can be normalized.
00:13:16.580 --> 00:13:21.440
And using this idea that
changing the value--
00:13:21.440 --> 00:13:23.660
the coefficient
of the function--
00:13:23.660 --> 00:13:32.030
doesn't change too
much, we simply say, use
00:13:32.030 --> 00:13:42.280
instead psi prime, which
is equal to psi over
00:13:42.280 --> 00:13:50.860
square root of N. And look what
a nice property this psi prime
00:13:50.860 --> 00:13:52.180
has.
00:13:52.180 --> 00:14:04.430
If you integrate psi prime
squared, it would be equal--
00:14:04.430 --> 00:14:06.500
because you have psi
prime here is squared,
00:14:06.500 --> 00:14:13.300
it would be equal to the
integral of PSI squared divided
00:14:13.300 --> 00:14:15.730
by the number N--
00:14:15.730 --> 00:14:18.940
because there's two of them--
00:14:18.940 --> 00:14:25.680
dx, and the number goes out and
you have the integral of psi
00:14:25.680 --> 00:14:34.000
squared dx, but that integral
was exactly N, so that's 1.
00:14:34.000 --> 00:14:42.570
So if your wave function
has a finite integral
00:14:42.570 --> 00:14:51.610
in this sense, a number
that is less than infinity,
00:14:51.610 --> 00:14:56.430
then psi can be normalized.
00:14:56.430 --> 00:15:01.480
And if you're going to
work with probabilities,
00:15:01.480 --> 00:15:05.480
you should use instead
this wave function,
00:15:05.480 --> 00:15:11.050
which is the original wave
function divided by a number.
00:15:11.050 --> 00:15:15.280
So they realize
that, in some sense,
00:15:15.280 --> 00:15:18.390
you can delay all of
this and you can always
00:15:18.390 --> 00:15:23.010
work with wave functions
that are normalizable,
00:15:23.010 --> 00:15:27.000
but only when you're going to
calculate your probabilities.
00:15:27.000 --> 00:15:31.350
You can take the trouble
to actually normalize them
00:15:31.350 --> 00:15:35.280
and those are the ones
you use in these formulas.
00:15:35.280 --> 00:15:42.480
So the idea remains that we work
flexibly with wave functions
00:15:42.480 --> 00:15:48.300
and multiply them by numbers and
nothing changes as long as you
00:15:48.300 --> 00:15:52.860
realize that you cannot change
the fact that the wave function
00:15:52.860 --> 00:15:57.240
is normalizable by multiplying
it by any finite number,
00:15:57.240 --> 00:15:59.490
it will still be normalized.
00:15:59.490 --> 00:16:05.900
And if it's normalizable, it's
equivalent to a normalized wave
00:16:05.900 --> 00:16:06.730
function.
00:16:06.730 --> 00:16:09.940
So those two words
sound very similar,
00:16:09.940 --> 00:16:11.670
but they're a little different.
00:16:11.670 --> 00:16:17.250
One is normalizable, which
means it has an integral of psi
00:16:17.250 --> 00:16:20.730
squared finite, and
normalize is one
00:16:20.730 --> 00:16:25.450
that already has been
adjusted to do this
00:16:25.450 --> 00:16:31.490
and can be used to define
a probability distribution.
00:16:31.490 --> 00:16:32.030
OK.
00:16:32.030 --> 00:16:35.540
So that, in a way
of introduction
00:16:35.540 --> 00:16:39.160
to the problem
that we have to do,
00:16:39.160 --> 00:16:43.660
our serious problem
is indeed justifying
00:16:43.660 --> 00:16:48.820
that the time evolution doesn't
mess up the normalization
00:16:48.820 --> 00:16:51.480
and how does it do that?