0Introduction

IB Methods 0 Introduction
In the previous courses, the (partial) differential equations we have seen are
mostly linear. For example, we have Laplace’s equation:
2
φ
x
2
+
φ
y
2
= 0,
and the heat equation:
φ
t
= κ
2
φ
x
2
+
2
φ
y
2
.
The Schr¨odinger’ equation in quantum mechanics is also linear:
i~
Φ
t
=
~
2
2m
2
φ
x
2
+ V (x)Φ(x).
By being linear, these equations have the property that if
φ
1
, φ
2
are solutions,
then so are λ
1
φ
1
+ λ
2
φ
2
(for any constants λ
i
).
Why are all these linear? In general, if we just randomly write down a
differential equation, most likely it is not going to be linear. So where did the
linearity of equations of physics come from?
The answer is that the real world is not linear in general. However, often we
are not looking for a completely accurate and precise description of the universe.
When we have low energy/speed/whatever, we can often quite accurately approx-
imate reality by a linear equation. For example, the equation of general relativity
is very complicated and nowhere near being linear, but for small masses and
velocities, they reduce to Newton’s law of gravitation, which is linear.
The only exception to this seems to be Schr¨odinger’s equation. While there
are many theories and equations that superseded the Schr¨odinger equation, these
are all still linear in nature. It seems that linearity is the thing that underpins
quantum mechanics.
Due to the prevalence of linear equations, it is rather important that we
understand these equations well, and this is our primary objective of the course.