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Heisenberg's Uncertainty Principle
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From the course by University of Manchester
Introduction to Physical Chemistry
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From the lesson
Quantum Chemistry I
This module explores Planck's quantum of energy, particle nature of light, wave nature of matter, Heissenberg's uncertainty principle, the Schrödinger equation, free particle & the particle in a box, Born's interpretation of the wavefunction, and normalisation of the wavefunction.
Meet the Instructors
Patrick J O'Malley, D.ScReader
Chemistry
Michael W. Anderson, FRSCProfessor of Materials Chemistry
School of Chemistry
Jonathan Agger, MRSCLecturer and Deputy Director of Outreach
Chemistry
[BLANK_AUDIO].
Hi, there.
So, the next topic we're going to
move on to is the Heisenberg's Uncertainty Principle.
And here we have Heisenberg, Vernon Heisenberg on the right,
and here we have a statement of the Heisenberg uncertainty principle.
So it is impossible to know both position
and the momentum of a particle with certainty.
Now this is usually written as delta x, which is the uncertainty and the position,
and again we're talking about the x direction, multiplied by delta p sub x.
Which should be the uncertainty in the, in the
x direction, uncertainty of momentum in the x direction.
And the formal definition is at that, greater than or equal h
over 4 pi or if you want to h bar it's h bar over 2;
where we know that h bar is find h divided by pi 2 pi.
This principle is a fundamental law of nature.
And it's, it's you can trace it back to what we've been talking about, about
the, the wave particle duality of light and, and particles in motion.
One way to think about this, it's, it's a difficult
concept and most students and teachers have problems with it.
But one way of thinking about it, is if you take a, a very simple system so if you
have some particle and that's moving, let's say our, our x direction.
Now, to observe this particle, we have to shine some light on it, okay.
That's the way we observe its position.
And so let's have a photon of light coming in here, h nu.
And we know also from our, previous work that
you can write that, that's equal to hc over lunde.
C is the speed of light and, lamda is the wave length of that light.
And we also know that, that photon is going to in the particle,
The, the particle aspect of it is going to have a momentum, and that
momentum which we define as smaller p is going to be h over, over lambda.
So the precision that we're going to be able
to measure particle to here along the x direction say.
And we'll call our precision delta x, is ging to be
proportional to the wavelength of our observing light.
So if we if we want to decrease this, we want to make this smaller and smaller;
then we're going to have to use smaller and
smaller wavelength light, to observe that, that, that particle.
Now as we go along and we are trying to increase the precision that we measured
this delta x here, we are going to have to go to smaller and smaller wavelengths.
So as we try to reduce delta x, or
increase the precision on which we measured the precision
of the particle as much as possible, we have
to decrease lambda but of course when we decrease lambda.
We're also increasing the momentum of the particle.
So now if you treat this photon as coming in as a
particle as we go to land the momentum is going to be higher.
So what that is going to do of course is, basically its going to, if
you can think of it as a part, its going to strike against this particle here.
And, of course, like any two particles colliding,
we have a large particle here coming in.
Then, you're going to change the direction and
the momentum, if you like, of this particle here.
So by trying to increase the precision of our measurement,
we're actually going to have a less and less precise measure of the momentum.
And this really is the basis of the, the Heisenberg's uncertainty principle.
What you call, you call the momentum.
P sub x in this case, and delta x.
We call these our complimentary observables.
So let's write that down.
So these are complimentary,
observables.
And momentum and position are not the only complimentary observables.
Say some other familiar complimentary observables will be
the position which you measure energy and time.
So again taking our analogy up here, a simple system up here a bit
further, we had, we could write say that let's put a different color in.
Delta x is approximately equal to, to lambda.
And similarly you could write that delta p
sub x, approximately equal to h over lamed.
So if you then do the product delta x times delta p sub x.
[BLANK_AUDIO]
what you find is that, you can say delta x by delta p sub x is equal to h [INAUDIBLE]
constant and this in essence is, Is the Heisenberg relationship, its
formally given as greater sum equal to H over, over 4 pi, that's what were
saying is that the message which we can
measure position and momentum in the same directions.
Its important to remember its in the same direction.
If you are at delta x.
Delta p y set here, then this doesn't govern, this that product.
Only governance these two products when we're talking about the same direction.
But it's approximately equal to h.
And this is a fundamental right before the middle law of nature.
It's got nothing to do with the measuring
device we're using, how precise we can measure it.
If we had a, what we're saying is if we had
a measuring device that could measure exactly the momentum and the position.
You're always limited by this value, this value here.
So let's now look at a few examples to try
to engrain this in, in, in, in, in your thinking.
So we're just going to take two, two examples here.
And see what it means for, for different objects.
So let's talk about about, we talked when we
were in approximation, we talked about a tennis ball.
So let's bring that up again.
It's an object or particle that we're all familiar with.
So let's suppose now this case, we could measure the velocity of the tennis ball.
So this is the precision which we measure the velocity of the tennis ball, and let's
suppose that that's equal to 1.0 by 10 to the minus 6 meters, seconds minus 1.
We could ask another question.
What is position?
Or what is the position in which we could measure the position of that tennis ball.
Or what is delta, delta x, and again, like
the previous example, we're going to assume
that the tennis ball has a mass of 57 grams.
So the first thing we have got to do is work out the momentum.
We know of the uncertainty of the momentum, we know
the uncertainty of the velocity, and we know the mass.
So we know that the mass so we just know that
the mass times that uncertainty in the velocity is going to
give us the uncertainty in momentum so we can write down
that Delta p is going to be the mass times delta v.
There for we are going to have 57 by 10 to the minus 3.
And again we're converting everything to si units kilograms.
And the velocity is already given in meters per second.
So that turns to minus six.
And that's meter seconds minus one.
Now we right down the Heisenberg uncertainty relationship.
So that says delta p times delta x, and now it says greater than or
equal to so the minimum uncertainty is going to be h over 4 pi,
so that's from the equation as written, as written up here.
So if we go back down, back down here, now we
can rearrange that equation because we are looking for delta x, remember?
So delta x is going to be equal to h over 4 pi.
And that's going to be multiplied by p, by delta p.
So that should be delta p there.
So you can plug in then the values in to that, and if you
do that you get 6.626 by ten to the minus 34, that's Blake's constant.
And it's in joule second.
And you divide that by 4 times pi.
And it's 57 by 10, to the minus 3, and that's kilograms.
And that's all been divided by the uncertainty and the
velocity, which is 1.0 by 10 to the minus 6.
And that's meters, second minus 1.
And if you plug these values in you should get a value of 9.25
by ten to the minus 28 meters.
So what you can see here is this value.
It's incredibly small for the tennis ball so what it means
is there is an uncertainty in, in the position though
it's so small that there is now way we're ever
going to be able to measure this for the tennis ball.
So, like we had in the previous occasion when we
measured the the wavelength of the, of the moving tennis ball
in this case here again, it has no, no effect on our, our observation.
But let's now move on to a much smaller object like an electron.
So you can ask the same question again.
So what is now for a, for an electron.
[BLANK_AUDIO]
electron, and we'll assume same velocity.
So it's the same velocity as our tennis ball up there.
So again, if you plug our values into our, our equation, so we get delta x now's
going to be at Planck's Constant again, 6.626 by 10 to the minus 34 joule-second.
Again it's going to be 4 times pi and then we're going to have to put in the mass
of the electron, 9.11 by 10 to the minus 31.
That's kilograms, and the velocity is the same that we have for the tennis ball, so
that's 1.0 by 10 to the minus six, and that's meters seconds minus one.
And when you work that out, you should come with a value of 58 meters.
So we can see that for the electron, at 58 meters, is huge.
So in essence we really have no idea or we
can have no idea of where the actual electron is.
So this is a big difference if you like from looking
at the classical few of particle movement and the quantum mechanical view.
Classical view says, we can measure the momentum and the position exactly so
therefore, at any time in the future, we can predict position of that particle.
So we can if you'd like work out a trajectory for that particle.
Just like we can work out a trajectories
of the planets as they circle the, the sun.
But for small tiny objects, like electrons, and
is also look just like protons and neutrons.
We can't do this, other words we can't trace
out trajectories for the electrons circling around the, the nuclei.
And all we can do as we show when we go on to move about
on to the equation, is we can work our away functions and all we can do.
And then we're left to do is work
out the probabilities of where the electrons will be.
[NOISE]
[BLANK_AUDIO]
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