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This video is the Schrodinger equation part 2, so if you haven’t seen it, you might
want to watch part 1 here first. On the other hand, if you’re just here for me to explain
the derivation of the equation and how to use it, stick around.
OK, If it were your task to build a new theory of physics, you’d need to decide what things
were so fundamental you should build them into your theory. Conservation of energy is
probably high up there on your list, as it clearly was on Schrodinger’s, because that
principle is baked right into quantum mechanics via the Schrodinger equation.
Suppose we have a particle with this wavefunction at the start. Remember we can write our state
in any basis we want, so a perfectly valid one is the energy eigenbasis. This means we
can always write a state as a superposition of different energies. And remember if we
measure the energy of the particle, it will be one of these with this probability. Now
say this state evolves in time, in other words we apply the time evolution U. What condition
do we want to impose on the new energies of the state? In other words, how do we want
conservation of energy to look in quantum mechanics?
First let’s look at the special case were a particle had just one energy E when we start-
in other words, it was an energy eigenstate. Then we evolve it forward in time, and look
at the energy of the new state. We definitely want that energy to be E -because otherwise
energy is not conserved. So that’s the first thing we’ll require. But we’ll require
more. Going back to the general case, if we looked at the average energy the particle
has, that also shouldn’t change after some time. Because if on average the energy increased,
I don’t think we’d be willing to say the energy of this particle was conserved. In
fact, let’s ask for even more from conservation. If you were to measure the particle’s energy
initially, you have this probability of measuring this energy. Well let’s require that if
instead you measured it afterward, you have the same probability of getting that energy.
You can’t tell the difference between the two states by measuring energy. This requirement
is strong enough to pretty much give us the Schrodinger equation, as we’ll see.
We can always write the final state as a superposition of energies- just because we can do that with
any state. But we couldn’t have another energy in this superposition that wasn’t
in the original. That would mean there was a 0 chance of measuring that energy originally,
but later that chance is not zero. So we’ve got the same energies in both, we just need
to know about how the coefficients have changed in the new one. We want the probablities to
be the same, but that probability is just the length of this complex number squared.
So this number here better have the same length. Let me draw this complex number as an arrow
for a second. Then this complex number is the same length, but it might be rotated.
Let’s call that angle theta. There’s a convenient way to represent rotations, which
we’ve seen before: this equation says the new coefficient is the old one rotated by
angle theta. Ok so let’s put that in for all the energies.
Great, almost done. 2 little things though. First, these can’t all have rotated the
same amount. Why? cos then you could bring this rotation to the front, and now both the
future and present state are essentially the same. That’s because an overall rotation
doesn’t affect any measurement outcomes. But that would mean that no matter what crazy
situation you put a particle in, it does nothing. So that’s not right. So the only solution
is to make sure the angles here is different for every energy. Also the amount of rotation
should depend on how much you’ve gone forward in time- so that if it’s only a little you
only rotate a bit. That suggests that the right amount of angle to rotate is Energy
times time. Oh and throw in the some constant to deal with units and scaling and all that.
Andddd we’re done. That’s what the schrodinger equation tells you happens to the state. But
let’s talk a little about using it.
Say that we start with this state where this energy is twice as big as the lowest energy,
and the next is 3 times as big, and these are their coefficients. As we evolve it in
time, the second one’s angle changes twice as fast, and the third one’s three times
as fast. But regardless, you might be thinking, but these states hardly look different. That’s
right, in this basis. But if you look in another basis, that difference can be much more obvious.
For example, suppose that, in the position basis, each of these had wavefunctions that
look like this. Then at the very start when you add them up, the probability of finding
the particle at various points looks like this. But see how that evolves in time to
give a probability distribution that changes dramatically. The reason is this complex numbers.
At the start they all just added like normal to give you this. But later in time, when
these complex numbers are pointing different ways, they don’t just add up anymore. As
an extreme example, if these two arrows pointed in opposite directions, this they would subtract
from one another. This is why the overall wavefunction in the position basis can vary
so much, so that -depending on the time you measur- the particle might have a very big
probability of ending up here versus here. Yet if you measure energy, you always get
one of these three energies with these probabilities. This is ingeneral true- things look pretty
calm and unchanging for measurements of energy, but they can look very very different when
measuring something else.
And that’s it! Everything you need to know about the schrodinger equation. Ok just kidding.
There’s one last bit to it and it’s a big deal. How do we know what this energy
eigenbasis is? If we take an energy eigenvector, how to we find what it looks like in the position
basis? Well thankfully Schrodinger also gave us an equation that defines what an energy
eigenstate is, and it’s what you’d expect as well. The issue is, this equation is spectacularly
hard to solve. By that, I mean it’s probably impossible to solve it mathematically for
just about anything bigger than 2 particles- you have to resort to doing it by computer,
preferably the super type. That’s why physical chemists do so much hard work on the Schrodinger
equation, and it’s something I feel completely under qualified to talk about, since as physicists
we tend to put these problems into the too hard basket. Maybe when if I learn it one
day, we can come back.
But let’s move along to homework time.
First. Throughout these 2 videos, I kept talking
about predicting the future, and that if you know the present state, you can predict the
future. Does this mean quantum mechanics is deterministic? If you don’t think so, comment
on where the determinism ends and the randomness starts in this theory.
Second. Show that, for the Schrodinger equation, this
is true: That it’s the same thing to go forward in time t1 plus t2 as go forward t1
and then go forward t2. Seems obvious, but explain what philosophical consequences that
might have.
Third. I said that linearity follows from the shrodinger
equation. Can you see why? Try and prove it. Please type your answer in the comments but
afterward you can click this link for my short proof. Yours will likely be a little different
though, so I think it will be really interesting and helpful if you write your answer down
right.
Fourth. This is for the people who’ve done Quantum
mechanics before and know that in this theory time evolution is unitary- yet I just went
on about linearity so much in the previous video. Show that linearity plus the assumption
that time evolution maps valid states to other valid states is equivalent to saying that
the evolution is unitary. I’ll put the proof of that in the video here as well.
Sorry these videos and my replying to commenting has been so haphazard lately. As you might
know I started my PhD recently, but I’m finally settling into it, so I’ll try be
more consistent now. But more about that soon.