1
PHYS 342
Fall Semester 2011
Homework No. 1: Math Review
Due Date:
August 29, 2011
1.
When we discuss the Schrödinger wave equation for
the H atom, we will make extensive use of the polar
co-ordinate system.
Anticipating this development, the
Figure
to
the
right
illustrates
a
point
P
in
both
rectangular and polar coordinates. If the coordinates of
a point P in rectangular coordinates is (x,y,z), what are
the equivalent coordinates (r,
θ
,
φ
) in spherical polar
coordinates?
Write
down
expressions
for
an
area
element dA and a volume element dV using both
coordinate systems.
2.
We will use complex notation regularly throughout the semester and it is a good idea to
make sure that you you have a good idea what the imaginary number i=√(-1) is all about.
When
dealing
with
complex
numbers
(ie
numbers that have a real and an imaginary
part), it is convenient to draw a new co-
ordinate system as shown. Note that we no
longer use (x,y) to label the axis, but instead
use
(real,
imaginary).
As
an
example,
the
position of a point specified by the co-ordinates
(3, 2 )
i
is shown. Using a complex number co-
ordinate system is quite useful and you must
understand
how
it
is
different
than
the
conventional (x,y) system that you use all the
time.
First, we need a definition. When a real number
is multiplied by i, what actually happens?
It is
useful to think of “multiplication by i” as an operation that just rotates
a “real” number
counter clockwise by 90
o
. Normally, we associate multiplication with an operation that
changes the original value of a number to a new value which is different. Not so when you
multiply by I - the magnitude of the number stays the same, but the location of the number
rotates.
z
y
P(x, y, z) or P(r,
ϑ
,
ϕ
)
x
ϑ
ϕ
r
0
1
2
3
4
Real
Imaginary
4
3
2
0
i
i
i
i
(3, 2 )
i