Hi, everybody.
In the previous videos, we talked about the rules and ...
the math that govern quantum mechanics.
I want to just spend one more short video talking about ...
some of the strange implications of quantum mechanics.
This is unique to quantum mechanics.
When we first learn classical mechanics,
the reality that we have to accept is more or less in line ...
with our everyday experience of watching objects move through space.
But if we're going to accept the rules of quantum ...
mechanics, we have to accept a pretty ...
strange reality.
In fact, even the scientist who ...
developed quantum mechanics didn't really want ...
to accept what it implied.
This is why, for example,
we have Schrodinger's cat or Einstein's famous quote,
"God does not play dice".
These are both examples of scientists being very unhappy ...
with the reality the quantum mechanics requires us to accept.
And I want to spend a few minutes talking about this.
One surprising implication of quantum mechanics is that ...
when we measure something, we change it.
Normally, we think of measurement in ...
science as something that happens externally.
A system goes about its business,
and we watch it, but in watching it,
we're not actually changing it, we're not disturbing it.
But quantum mechanics says that it's impossible not to ...
disturb a system when we watch it.
For example, if a particle starts off in a ...
superposition of different energy states,
and then we measure its energy,
now we know its energy.
So it's actually in an energy eigenstate after we take the ...
energy measurement.
To repeat that, the object was in a ...
superposition before we measured it,
and it's in an eigenstate after we measure it.
There are different ways of interpreting this.
In the traditional interpretation of quantum mechanics,
we would say that the measurement caused the ...
object to collapse to an eigenstate.
In another interpretation called the "many worlds ...
interpretation", we would say that measuring ...
it caused the universe to split into different realities ...
according to each possible measurement.
Which of these interpretations you accept is really up to you.
But either way, you have to accept the idea ...
that by measuring an object, you actually changed the ...
reality of that object.
This idea results and some famous paradoxes,
including the double-slit experiment and Schrodinger's cat,
both of which I've included some YouTube links to.
In the double-slit experiment, the idea is that if you shoot a ...
beam of electrons through two parallel slits,
if you don't watch which slit those electrons go through,
they actually go through both slits simultaneously like a ...
wave, but if you do watch which slit ...
those electrons go through, the electrons behave like ...
particles, going through one slit or the other.
And Schrodinger's cat is a thought experiment that ...
illustrates that some strange things happen when you ...
translate quantum mechanics to the macroscopic scale.
The idea of Schrodinger's cat is that a cat can be in a ...
superposition of alive and dead until you look at it,
in which case it decides on one or the other.
And this brings up some interesting questions of,
well, what counts as an observer?
Can it be an animal? Can it be a camera?
And I don't have clear answers to these questions.
They just illustrate that if we accept quantum mechanics,
we have to accept a pretty strange reality.
The other implication of quantum mechanics that I ...
want to mention is that certain properties can't be exactly ...
known simultaneously.
The most famous example of this is the Heisenberg ...
uncertainty principle, which is usually stated as:
It's impossible to know both the position and the ...
momentum of an object at the same time.
We've already seen the mathematical reason of why this is.
If we know the position of an object,
that means the object is in an eigenfunction of the position operator.
And if we know the momentum of an object,
that means the object is in an eigenfunction of the ...
momentum operator.
And those two sets of eigenfunctions are not the same.
So if we know the position exactly,
then it's actually in a superposition of different ...
momentum states. And if we know the ...
momentum exactly, then it's in a superposition of ...
different position states.
Related to this, I want to introduce one ...
mathematical concept in this video,
the idea of a "commutator".
If we know that measuring an object changes an object,
and that different properties correspond to different ...
eigenfunctions,
then by extension, the order in which we take ...
measurements matters.
A commutator is basically a test of whether it makes a ...
difference if we measure, for example,
position before momentum, or momentum before position.
Mathematically, a commutator is defined like this.
For two operators, which we'll call A1 and A2,
the commutator is defined as A1 applied to A2 minus A2 applied to A1.
If the commutator of A1 and A2 is equal to zero,
then that means it doesn't make a difference in which ...
order we measure those two properties.
In other words, it's possible for us to know the ...
values of those two properties A1 and A2 simultaneously.
But if the commutator isn't equal to zero,
then two properties can't be simultaneously measured.
They can't be simultaneously known.
Often when we calculate a commutator,
we'll want to apply that commutator to some test ...
function f, as we'll see in just a second.
So as an example, let's confirm the Heisenberg ...
uncertainty principle. Let's confirm that position and ...
momentum can't be simultaneously measured.
As we've seen before, the position and momentum ...
operators in one dimension look like this.
In order for us to find a commutator,
we're going to apply it to a test function that we'll call f.
By definition, the commutator of position ...
and momentum is equal to position applied to momentum ...
minus momentum applied to position.
So the commutator applied to a function f looks like this.
In the next step, let's actually write out those ...
position and momentum operators.
From this point on, the math is pretty ...
straightforward, as long as we remember that ...
the derivative with respect to x of x times a function ...
requires that we use the product rule.
So, our commutator applied to the ...
function f looks like this.
And from this point, we can combine some terms ...
and find out that it's equal to i times hbar times our test function f.
Since the commutator applied to f equals i hbar f,
then that means
that the commentator itself is equal to i times hbar.
I realize that derivation came pretty quickly,
so I would encourage you to go back and hit "pause"
as needed until you can follow every step of that math.
The important thing to notice about this is that the ...
commutator of position and momentum is not equal to zero.
So that means position and momentum cannot be ...
simultaneously known,
which is a statement of the Heisenberg uncertainty principle.
That's all for now, and in the next video,
we'll actually start to solve the Schrodinger equation in ...
certain simple situations.
