Quantum Mechanics Concepts: 5 The EPR Paradox explanation - Subtítulos bilingües
Hello, today we are continuing in our series on quantum mechanics concepts and
we're going to be looking firstly at the implications of Heisenberg's uncertainty
principle which we derived last time and then we're going to look at the Einstein Podolsky-Rosen experiment.
and deal with questions which I have been asked on many occasions about my actual video that I did on this subject.
Now last time we showed that if you want to do two simultaneous measurements,
measurements of course are represented by operators,
We showed that the commutator of two measureables,
those are operators, must equal zero, if you are going to measure them both at the same time.
And the significance of this, which the commutator is simply m1 m2 minus m2 m1, that is what this shorthand means.
It simply means the order of the operators is reversed.
The significance of this being equal to zero is that both those operators have the same eigenvector,
so they can both have the same eigenvectors, and so they can both operate on the same state at the same time.
If they do not have the same common eigenvectors,
then since this is the key,
that the operator acting on a state is equal to the eigenvalue which is the result you want times that state.
If m2 cannot also operate on that state,
Then you cannot have those two measurements at the same time,
because they both need to measure the state that the system is in in order to give you two different eigenvalues, two different measurements.
And the significance of this is that we showed last time that therefore the commutator needed to be equal to zero.
We also showed last time that the commutator of the position operator and the momentum operator is equal to ih bar,
which significantly we said does not equal zero, and that's the reason that you can't measure position and momentum at the same time.
Now, when we did spin, you'll remember that we came up with three measureables, three matrices, sigma x, sigma y and sigma z.
These were the measureables, they were operators, so let's give them hats, which measured spin along the x, the y and the z axis.
So now I want to give them a and Can we measure any two of those things at the same time?
Can we measure along the x-axis and along the y-axis at the same time?
Before I do that, I just want to show one property of a commutator.
If you have a commutator A,
B, that of course is equal to A, B, minus B, A, by And if you have the commutator B, A,
then by definition that is bA minus ab and you can see that this is simply the
minus of this right because minus this would be minus ba plus ab which is this
so in other words what I've just shown is that the commutator of ab is equal to minus the commutator of the A.
So if you reverse the operators you simply have to put in a minus sign.
So let's kick off by asking whether we can measure along the z-axis and the x-axis
Which means that the commutator of the measurements along the x-axis and the z-axis must be zero.
Otherwise, you can't do it.
And that, don't forget, is sigma z, sigma x, minus sigma x, sigma z.
So let's do it.
Let's actually do the matrix multiplication.
Sigma was 1 0 0 minus 1 times sigma x which is 0 1 1 0 minus sigma x which is
0 1 1 0 times sigma z which is 1 0 0 minus 1.
And that is going to give us 2 by 2 matrices, 1 minus the other.
So let's do it.
1 times nought plus nought times 1 is nought.
1 times 1 plus nought times nought is 1.
Nought times nought minus 1 times 1 is minus 1.
and then nought times one plus minus one times nought is nought and over this side nought times one plus one times nought is nought,
nought times nought plus one times minus one is minus one,
one times one plus nought times nought is one 1, and 1 times naught plus naught times minus 1 is naught.
You'll notice that these two are not the same.
But if I multiply inside and outside by minus 1, effectively I'm just multiplying by plus 1, 2 lots of minus 1, I get that.
And so now I've got,
the two matrices are the same, and that is now 2 times naught 1 minus 1 naught, which I don't immediately recognize.
I multiply by minus i on the inside and divide by minus i on the outside,
I'm going to get 2 divided by minus i,
and then inside I multiply by minus i,
I'm going to get naught minus i plus i, because i times minus 1 is plus i, naught.
you should recognize that.
That, of course, is simply the sigma y matrix.
And if I multiply top and bottom by i, then i times minus i is plus 1, so that just goes.
So, I've shown that the commutator of sigma z sigma x is equal to 2i times the spin operator sigma y, which significantly does an odd equation.
So you cannot measure along the z and the x components or coordinates at the same time.
What about z and y?
So we now need to look at sigma z, sigma y and ask is that equal to zero?
Well that of course is going to be equal to sigma x,
sigma y minus sigma z, actually sigma z, sigma y minus sigma y, sigma z.
Sigma z is 1 naught,
naught minus 1, times sigma y, which is naught minus i i naught, minus sigma y, naught minus i i naught.
times sigma z one nought nought minus one.
So that's going to give us two matrices,
one minus the other,
one times nought plus nought times i is nought,
one times minus i plus nought times nought is minus i,
nought times nought minus 1 times i is minus i, and then naught times minus i plus minus 1 times naught is naught.
Over here I've got naught times 1 minus i times naught is naught, naught times naught.
plus minus i times minus 1 is plus i,
i times 1 plus naught times naught is i, and then i times naught plus naught times minus 1 is naught.
These two are not the same,
but again if I simply change signs inside and outside,
I change the plus to a minus,
I change the plus to a minus and then outside,
I change the minus to a plus, you'll notice that I've now got two that are the same.
So this is two times 0, minus i, minus i, 0.
If I multiply by i inside and divide outside,
so outside I divide by i, and inside I multiply by i, minus i squared is just plus 1.
minus i times i plus 1.
And of course,
if I multiply the outside top and bottom by minus i, then I get minus i squared on the bottom, which is simply 0.
That's sorry, 1.
And so this is equal to minus 2i times this here, but what is this here?
That is simply sigma x, and again that does not equal zero.
So the commutator of sigma z with sigma y is minus 2i sigma y.
And remember,
if you swap these two around,
if this was sigma y with sigma z, if the order were reversed, then you simply, that is simply minus sigma z, sigma y.
So if you did sigma y, sigma z, you would get plus 2i sigma x.
But significantly, it's not zero, so you can't measure these two at the same time.
And for completeness,
though I think we can guess what's going to happen,
let's look at sigma x, sigma y, which of course is sigma x, sigma y minus sigma y, sigma x.
And that's got to equal zero if we can measure along the x and the y coordinates at the same
So we now have to write down what these are,
that's a 0,
1, 1, 0, that's sigma x times sigma y, which is 0, minus i, i, 0, minus sigma y, 0, minus i, 0, times sigma x, 0, 1, 1, 0.
And once again,
we're going to get two matrices,
one minus the other,
nought times nought plus one times i is i,
nought times minus i plus one times nought is nought, one times nought plus nought times i is nought.
1 times minus i plus naught times naught is minus i,
and over here we're going to get naught times naught minus i times 1 is minus i,
naught times 1 plus minus i times naught is naught, i times naught plus naught times 1 is naught.
i times one plus naught times naught is i.
Once again these two matrices are not the same,
but they can be made the same just by simply multiplying by minus one inside and outside.
So if you multiply by minus one inside and outside you get that.
And this is now simply i, 0, 0, minus i.
Now let me multiply inside by minus i and divide outside by minus i.
So I divide that by minus i, and then inside I'm going to multiply by minus i.
Minus i times i is minus i squared, which is plus 1.
Minus i times minus i is i squared, which is minus 1.
And you might record, oh, and we better just multiply top and bottom by i.
If you multiply the bottom by i, you minus i squared, which is minus So this gives you 2i times this term here.
Guess what?
That is sigma z, sigma z, which does not equal 0.
So in each case,
what we've shown is,
well, in this case, we've shown that sigma x, the commutator of sigma x and sigma y gives you 2i sigma z.
which significantly doesn't equal zero, so you can't measure the spin along the x and the y-axis at the same time.
But you'll recall that in each case,
if you take the commutator,
of one spin with another,
you get either plus or minus,
depending on which way you do it,
plus or minus 2i times the spin operator of the third,
whichever one is not in the square brackets, whichever one is not in the commutator, goes into the result.
So now I want to deal with the questions that I'm often asked about my video on the Einstein-Padolsky Rosen paradox.
And before I answer the question,
let me just explain briefly what that paradox was,
just to remind you that quantum mechanics had been born in about 1925,
and Einstein didn't like it,
and he didn't like it because the whole essence of quantum mechanics,
as we have and Einstein was convinced that the only reason that you had probabilities was because you didn't know all the information.
It's a bit like tossing a coin.
You toss a coin, it might come down heads or tails.
We use that as a random number generator because you don't know which way it will fall,
but you only don't know because all the information.
such as the torque applied to the coin,
the interference with air resistance, the coefficient of elasticity as it hits the ground, all those things are unknown, so you can't work it out.
It's too complicated.
But if you knew it all, you could work it out.
Quantum says you can never work it out.
Einstein didn't like that and he together with Podolskin Rosen about 10
years after quantum mechanics was born posed this question suppose you have a
particle which has a spin zero we won't worry too much about what that But the important thing is that the particle decays,
and in decaying, it two other particles.
One of them is an electron, the other is a positron.
You obviously have to be very careful with the positron because if it hits an electron, it will annihilate.
And typically, we send these particles into the laboratories of Alice and Bob, always called Alice and Bob.
So the particle which originated the electron and the positron has now decayed,
so it's not there anymore, and all you've got are an electron and a positron.
But you'll recall that the original particle had spin zero.
And since conservation of angular momentum is important, this means that whatever spin we don't know what spin the electron will have.
But whatever spin it has, the positron will have to have the opposite spin because the total angular momentum must equal zero.
And so the two must cancel out.
So, although we don't know what direction this spin will be in, and for each pair production there'll be a different spin.
We do know that the other particle will have to have the opposite spin.
So what we're going to do is to get Alice and Bob to measure the spin along the Z
axis using that piece of equipment that we developed in our earlier videos, and in particular on the video on electron spin.
And each of them is going to measure the spin of their particle along the Z axis,
and obviously they're going to get one of two results, either they're going to get an up result or a down result.
Now, you know enough from the video we did on electron spin to know that if you take an
electron with a spin like this,
and you put it through our device,
then there is a probability that you will get an up, but there's also a probability that you'll get a down.
Similarly, similarly if similarly if if if you you take take an
this particle and put it through the self-same measuring device, there's a probability of getting up and a probability of getting down.
Which means that in theory Alice could measure up for this and Bob could measure up for this.
and there's a possibility that Alice could measure down for this and Bob could measure down for this.
But in fact what is found is that no matter what happens if Alice measures up,
Bob will measure down and if Alice measures down, Bob will measure up.
There is never an occasion where they both measure up or down for those particles.
And the explanation for that is that instead of these particles being two separate particles each with their own wave function,
each with their own linear superposition of up and down states,
These two particles are somehow conjoined with a common wave function,
and that common wave function dictates that whatever particle, whatever spin Alice measures, Bob will always measure the other.
Now this is something that Einstein did not like.
He wanted to know how this particle could know what this particle had been measured.
As soon as this particle is measured up, this will be measured down.
There were only two ways Einstein could think of.
The was that somehow this particle sends a message to this particle instantaneously to say,
I have been measured up so you had better measured down.
But that would require instantaneous communication, and Einstein himself had already ruled that out in his theory of special relativity.
So, the only other explanation he could think of was that buried inside each of these particles was a kind of DNA.
They're called hidden variables.
And there was information stored in each of these particles created at birth, as it were, when they were first formed here.
Hidden variables stuck inside the electron and the positron that told the electron and
what to do in every conceivable set of circumstances they might encounter.
So this electron would have in its DNA if you are measured along the z-axis.
You must give an honor.
And this positron would have buried in its hidden variables.
If you are measured along these z-axis, you must give an answer down.
That way, you could be sure that they would always be opposite.
But it was John Stuart Bell in 1964, and I've done a video on this separately, who showed that that could not be possible.
that you can't have hidden variables because they do not accord with experimental results.
So that goes out of the window as well.
So we're now really left stuck.
We can't explain it by instantaneous communication.
We can't explain it by hidden variables.
And so the idea that there is a common wave function which somehow determines the
result and the fate of these two particles remains with us, but not entirely understood or explained.
Now that's the EPR, the Einstein-Padolsky Rose in paradox, but here was the question that I got asked.
You see, I put forward this idea.
That since Bob knows that whatever Alice measures in the z direction, he will always get the opposite.
Let's not bother to measure in the z direction.
Let's now measure in the x direction.
So Bob turns his equipment around.
and measures the spin of his positron in the x direction.
So this is the way it's supposed to work.
Alice measures the spin of the electron in the z direction, and let's say she gets an up.
She shouts across the laboratory, I got an up.
Which means Bob knows, if I had measured in the Z direction, I would definitely have got a down, because I always get a down.
So I know that the spin in the Z direction of my positron is down.
But I am going to measure in the X direction.
Now we just showed earlier in this video that you cannot simultaneously know the Z spin and the X spin and yet it appears that Bob is going to thwart Heisenberg's uncertainty
principle and so he makes his measurement in the X direction.
direction, and he will get either an up in the x-direction or a down.
He's bound to get one, and of course up in the x-direction is right and down in the x-direction is left.
So let's suppose he gets a right.
He'll definitely get one or the other.
Well, he's thwarted Heisenberg because now he knows that the positron has a spin down
in the z direction and a spin right in the x direction.
Or has he?
Unfortunately, no he hasn't and this is the reason.
You see, you may remember that when we did the spin video, I said that something happens when you do a measurement.
What happens is,
you know,
that the electron which is in this spin state here goes through the measuring
device and if it is measured as up that's fine but I also said something else
happens the electrons is actually left in an up position.
The state is actually changed from being a combination of up and down to wholly up,
because we've picked out just one of those states of linear superposition.
position.
Now because these two particles are conjoined by a common eigenvalue,
or wave function, this means that as soon as this electron is measured and its state changes to entirely up.
This positron will instantly change to entirely down because they are in the same wave function.
So when Bob measures the rightness or the leftness of his particle, he is no longer measuring this spin state.
He is simply measuring this spin state.
And we know that if you put a particle whose spin is entirely in the z direction,
through a measuring device which is in the x direction,
you've got a 50% chance of getting a right and a 50% chance of getting a left.
So Bob may well measure right, he may well measure left.
but he has significantly not measured the Z and the X components of this spin.
He has deduced the Z dimension of this spin simply because Alice got an up, therefore he knows he would have got a down.
But when he thinks that he has measured this spin,
as far as the right or leftness is concerned, he's wrong, that relates to the collapsed state of the result of Alice doing her measurement.
What I meant when I said in the Einstein-Podolsky-Rosen experiment that Bob won't get a result,
it doesn't mean that the lights won't come on, it doesn't mean that he won't get a right or a left measurement.
What it means is that that doesn't tell him anything because all he has got is the Z component of the original spin,
But when he makes his measurement along the x-axis,
all he is getting is a 50%
probability of getting a right or a 50%
probability of getting a left because he is measuring the collapsed state and that is not the same as measuring the original state.
Must be zero.
Otherwise can't do it.
And that don't forget is sigma z, sigma x.
minus sigma x, sigma z.
So let's do it.
That's actually do the matrix multiplication.
Sigma z was 1 0 0 minus 1 times sigma x which is 0 1 1 0.
minus sigma x, which is 0, 1, 1, 0, times sigma z, which is 1, 0, 0, minus 1.
And that is going to give us 2 by 2 matrices, 1 minus the other.
So let's do it.
1 times 0 plus 0 times 1 is 0.
1 times 1 plus 0 times 0 is 1.
nought times nought minus one times one is minus one and then nought times one plus minus one times nought is nought.
This shorthand means, it simply means the order of the operators is reversed.
The significance of this being equal to zero is that both those operators have the same eigenvector,
so they can both have the same eigenvectors, and so they can both operate on the same state at the same time.
If they do not have the same common eigenvectors,
then since this is the key formula that the operator acting on a state is
equal to the eigenvalue which is the result you want times that state.
If m2 cannot also operate on that state,
Then you cannot have those two measurements at the same time,
because they both need to measure the state that the system is in, in order to give you two different eigenvalues, two different measurements.
And this is going to show one property of a If you have a commutator a,
b, that of course is equal to a, b, minus b, a, by definition.
And if you have the commutator b, a, then by definition that is b, a, minus a, b.
And you can see that this is simply the minus of this, right?
Because minus this would be minus b, a plus a, b.
So in other words, what I've just shown is that the commutator of AB is equal to minus the commutator of Ba.
So if you reverse the operators, you simply have to put in a minus.
So let's kick off by asking whether we can measure along the z-axis and the x-axis at the same time,
which means that the commutator of the measurements along the x-axis and the z-axis up.
Today we are continuing in our series on quantum mechanics concepts,
and going to be looking firstly at the implications of Heisenberg's uncertainty principle,
which we derived last time, and then we're going to look at the Einstein-Podolsky-Rosen experiment.
and deal with questions which I have been asked on many occasions about my actual video that I did on this subject.
Now last time we showed that if you want to do two simultaneous measurements,
measurements of course are represented by operators,
we showed that the commutator of two Measurables,
those are operators, must equals zero, if you are going to measure them both at the same time.
And the significance of this,
which the commutator is simply M1,
M2, minus M2, M1, that is what the significance of this is that we showed last time
that therefore the commutator needed to be equal to zero.
also showed last time that the commutator of the position operator and the momentum operator is equal to i h bar,
which significantly we said does not equal zero.
And that's the reason that you can't measure position and momentum at the same time.
Now when we did spin, you'll remember that we came up with three measurables, three matrices, sigma x, sigma y and sigma z.
These were the measurables, they operators, so measure the spin along the x, the y and the z.
So now I want to ask the question, can we measure any two of those things at the same time?
Can we measure along the x-axis and along the y-axis at the same time?
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