PROFESSOR: Very good. So it's time to start. So today, I want to talk about
general features of quantum mechanics. Quantum mechanics is something
that takes some time to learn, and we're going to be doing some
of that learning this semester. But I want to give you a
perspective of where we're going, what are the
basic features, how quantum mechanics looks,
what's surprising about it, and introduce some
ideas that will be relevant throughout
this semester and some that will be relevant for
later courses as well. So it's an overview
of quantum mechanics. So quantum mechanics,
at this moment, is almost 100 years old. Officially-- and we will hear-- this year, in 2016, we're
celebrating the centenary of general relativity. And when will the centenary
of quantum mechanics be? I'm pretty sure it
will be in 2025. Because in 1925,
Schrodinger and Heisenberg pretty much wrote down the
equations of quantum mechanics. But quantum mechanics
really begins earlier. The routes that led to quantum
mechanics began in the late years of the 19th century
with work of Planck, and then at the
beginning of the century, with work of Einstein and
others,m as we will see today and in the next few lectures. So the thoughts, the
puzzles, the ideas that led to quantum
mechanics begin before 1925, and in 1925, it
suddenly happened. So what is quantum mechanics? Quantum mechanics is really
a framework to do physics, as we will understand. So quantum physics has
replaced classical physics as the correct description
of fundamental theory. So classical physics may
be a good approximation, but we know that at some
point, it's not quite right. It's not only not
perfectly accurate. It's conceptually very
different from the way things really work. So quantum physics has
replaced classical physics. And quantum physics
is the principles of quantum mechanics applied to
different physical phenomena. So you have, for example,
quantum electrodynamics, which is quantum mechanics
applied to electromagnetism. You have quantum
chromodynamics, which is quantum mechanics applied
to the strong interaction. You have quantum optics when
you apply quantum mechanics to photons. You have quantum
gravity when you try to apply quantum
mechanics to gravitation. Why the laughs? And that's what gives rise
to string theory, which is presumably a quantum
theory of gravity, and in fact, the quantum
theory of all interactions if it is correct. Because it not only
describes gravity, but it describes
all other forces. So quantum mechanics
is the framework, and we apply it to many things. So what are we going
to cover today? What are we going to review? Essentially five topics--
one, the linearity of quantum mechanics, two, the
necessity of complex numbers, three, the laws of
determinism, four, the unusual features
of superposition, and finally, what
is entanglement. So that's what we
aim to discuss today. So we'll begin with
number one, linearity. And that's a very
fundamental aspect of quantum mechanics,
something that we have to pay a lot of attention to. So whenever you
have a theory, you have some dynamical variables. These are the variables
you want to find their values because they are
connected with observation. If you have dynamical
variables, you can compare the values
of those variables, or some values deduced
from those variables, to the results of an experiment. So you have the equations
of motion, so linearity. We're talking linearity. You have some equations
of motion, EOM. And you have
dynamical variables. If you have a theory,
you have some equations, and you have to solve for
those dynamical variables. And the most famous example
of a theory that is linear is Maxwell's theory
of electromagnetism. Maxwell's theory
of electromagnetism is a linear theory. What does that mean? Well, first, practically,
what it means is that if you have a solution-- for example, a plane wave
propagating in this direction-- and you have another solution-- a plane wave propagating
towards you-- then you can form
a third solution, which is two plane waves
propagating simultaneously. And you don't have
to change anything. You can just put them together,
and you get a new solution. The two waves propagate
without touching each other, without affecting each other. And together, they
form a new solution. This is extraordinarily
useful in practice because the air
around us is filled with electromagnetic waves. All your cell phones send
electromagnetic waves up the sky to satellites
and radio stations and transmitting stations, and
the millions of phone calls go simultaneously without
affecting each other. A transatlantic cable can
conduct millions of phone calls at the same time, and as much
data and video and internet. It's all superposition. All these millions
of conversations go simultaneously
through the cable without interfering
with each other. Mathematically, we have
the following situation. In Maxwell's theory, you
have an electric field, a magnetic field, a charge
density, and a current density. That's charge per unit
area per unit of time. That's the current density. And this set of data
correspond to a solution if they satisfy
Maxwell's equations, which is a set of equations
for the electromagnetic field, charged densities,
and current density. So suppose this is a solution,
that you verify that it solves Maxwell's equation. Then linearity
implies the following. You multiply this by alpha,
alpha e, alpha b, alpha rho, and alpha j. And think of this as
the new electric field, the new magnetic field,
the new charged density, and the new current
is also a solution. If this is a solution,
linearity implies that you can
multiply those values by a number, a constant number,
a alpha being a real number. And this is still a solution. It also implies more. Linearity means
another thing as well. It means that if you have two
solutions, e1, b1, rho 1, j1, and e2, b2, rho 2, j2-- if these are two
solutions, then linearity implies that the sum e1 plus e2,
b1 plus b2, rho 1 plus rho 2, and j1 plus j2 is
also a solution. So that's the meaning, the
technical meaning of linearity. We have two solutions. We can add them. We have a single solution. You can scale it by a number. Now, I have not shown
you the equations and what makes them linear. But I can explain this
a little more as to what does it mean to
have a linear equation. Precisely what do we mean
by a linear equation? So a linear equation. And we write it schematically. We try to avoid details. We try to get
across the concept. A linear equation, we write
this l u equal 0 where u is your unknown and l is what
is called the linear operator, something that acts on u. And that thing, the equation,
is of the form l and u equal 0. Now, you might say,
OK, that already looks to me a little strange,
because you have just one unknown, and here we
have several unknowns. So this is not very general. And you could have
several equations. Well, that won't change much. We can have several
linear operators if you have several equations,
like l1 or something, l2 on something, all
these ones equal to 0 as you have several equations. So you can have several
u's or several unknowns, and you could say something
like you have l on u, v, w equals 0 where you
have several unknowns. But it's easier to just
think of this first. And once you understand this,
you can think about the case where you have many equations. So what is a linear equation? It's something in which this l-- the unknown can
be anything, but l must have important properties,
as being a linear operator will mean that l on a times
u, where a is a number, should be equal to alu and l
on u1 plus u2 on two unknowns is equal to lu 1 lu 2. This is what we mean by
the operator being linear. So if an operator
is linear, you also have l on alpha u1 plus beta u2. You apply first the second
property, l on the first plus l on the second. So this is l of alpha
u1 plus l of beta u2. And then using the
first property, this is alpha l of
u1 plus beta l of u2. And then you realize that
if u1 and u2 are solutions-- which means lu 1
equal lu 2 equals 0 if they solve the equation-- then alpha u1 plus
beta u2 is a solution. Because if lu1 is 0 and lu2 is
0, l of alpha u1 plus beta u2 is 0, and it is a solution. So this is how we write
a linear equation. Now, an example
probably would help. If I have the
differential equation du dt plus 1 over
tau u equals 0, I can write it as an
equation of the form lu equals 0 by taking
l on u to be defined to be vu vt plus 1 over tau u. Now, it's pretty much-- I haven't done much here. I've just said, look, let's
define l [? active ?] [? on ?] u to be this. And then certainly, this
equation is just lu equals 0. The question would be maybe
if somebody would tell you how do you write l alone-- well, l alone, probably
we should write it as d dt without anything
here plus 1 over tau. Now, that's a way
you would write it to try to understand
yourself what's going on. And you say, well, then when
l acts as the variable u, the first term takes
the derivative, and the second term, which is
a number, just multiplies it. So you could write
l as this thing. And now it is
straightforward to check that this is a linear operator. l is linear. And for that, you have to
check the two properties there. So for example, l on
au would be ddt of au plus 1 over tau au,
which is a times du d tau plus 1 over tau u, which is alu. And you can check. I asked you to check
the other property l on u1 plus u2 is
equal to lu 1 plus lu 2. Please do it.
