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This... is the Schrodinger equation.
I know this looks ugly but don’t run away!
I’m going to talk about what the Schrodinger equation actually means, and hopefully you’ll
see it doesn’t deserve its bad reputation.
Physics tries to predict the future.
What I mean is, if you have an object in some state that you know, and then you do something
to it- you let it go above the ground, or you give it a big push, or you put it next
to something electric, the job of physics is to tell you, what happens to the object
next?
Newtonian Mechanics did a good job of that.
If you know the initial position and velocity of an object, it will tell you exactly where
the particle is any time in the future.
In other words, you give it the initial state of the object, and based surroundings it give
you the state at time t.
What we want for quantum mechanics is something that will do the same- tell us what the future
state of the system is.
That’s what the Schrodinger equation is for.
In fact there’s a much nicer way to write the equation that puts it into this form.
Remember the state of a quantum particle is given by the wavefunction.
So if this is the wavefunction at the start, the Schrodinger equation tells you that this
thing acts on it to produce the wavefunction t seconds into the future.
Then the question is, what is this transformation, and why is it that way?
If you’re interested in the history of quantum mechanics then you may have heard Schrodinger
essentially just came down from the mountains with his fully formed equation, with no derivation.
This is kind of true- but that doesn’t mean there is no way to motivate the Schrodinger
equation these days.
The main thing it comes down to is conservation of energy.
It’s surprisingly straight forward, but I decided to put that derivation of the equation
to the next video.
See, the derivation of the equation, and how to precisely use the it are good to know but
kind of optional in terms of what we’ll do later.
However there is one fact about the equation that is so vital we will use it again and
again in future videos.
That one fact is so important, it’s all we’ll need to understand quantum computers,
and prove the no cloning theorem and quantum teleportation, and discuss the many worlds
interpretation of quantum mechanics, plus basically everything else we’re going to
do.
So here it is: time evolution is linear.
Let me explain that.
Suppose that I have an object and there are two different states I can start the object
in.
For example, I could just drop this object so it has no velocity to begin, or I could
throw it, so it has a big starting velocity.
We know what would happen to the object in each case, right?
We apply time evolution to each, and this is how they evolve.
But now suppose my object didn’t start in one of these 2 states -but actually a superposition
of both.
In this case, it both had no velocity and a big velocity to start.
What happens to it over time?
What do you expect happens?
Think about it.
Hopefully, what you expect is that it will be a superposition of the evolution of each
state, so at each point in time, the state looks like this.
The mathematical way to put this is, the time evolution of a superposition is the linear
combination of the time evolution of the each of the states.
This is what I mean.
If I have a state in any superposition like this, to get the overall future state, I just
look at the future state of the individual parts and add them back up.
It even holds if you write you that same state in a different basis.
For example if I changed to the momentum basis here, the same rule applies, and so these
two states are equal to each other.
So the general statement of linearity is this:
This does seem like a reasonable thing to request from quantum mechanics, but linearity
has some subtle philosophical statements about superposition hidden in it.
Look back at this example.
It says that, these separate parts of the superposition don’t act like they know the
other part is there.
This part of the object here doesn’t look over its shoulder and see that there’s another
part of the superposition and decide to do something else.
Regardless of whether there is another state in the superposition or not, this part of
the superposition does the same thing.
In other words, anything in this part of the superposition evolves completely unaware of
the other parts.
Well, that is until measurement happens.
When you measure, the branches of the superposition have the chance to interact which each other
again.
That’s because measurement doesn’t follow the Schrodinger equation.
This fact, that ‘measurement’ doesn’t obey the same rule that everything else in
the universe is subject to, is so bizarre, if not down right contradictory that it has
it’s own name: the measurement problem, and it’s the biggest foundational issue
in quantum mechanics.
Meanwhile, everything else evolves perfectly linearly in time, unaware of the other parts
of the superposition.
If you want to know where linearity comes from and what might have inspired the Schrodinger
equation in the first place, click this link to the next video.