31
Lecture notes
(these are from ny earlier version of the course  we may follow these at a
slightly different order, but they should still be relevant!)
Physics 3220,
Steve Pollock.
Basic Principles of Quantum Mechanics
The first part of Griffith's Ch 3 is in many ways a
review
of what we've been talking about, just
stated a little formally, introducing some new notation and a few new twists. We will keep coming
come back to it 
First, a quick review of ordinary
vectors
. Think carefully about each of these  nothing here is
unfamiliar (though the notation may feel a little abstract), but if you get comfortable with each idea
for regular vectors, you'll find it much easier to generalize them to more abstract vectors!
•
Vectors live in Ndimensional space.
(You're used to 3D!)
•
We have (or can choose)
basis
vectors
:
ˆ
e
i
(N of them in Ndim space.)
(Example in an "older
Phys 1110 notation" of these would be the old familiar unit vectors:
ˆ
i
,
ˆ
j
,
ˆ
k
•
They are
orthonormal
:
ˆ
e
i
⋅
ˆ
e
j
=
δ
ij
(This is the
scalar
, or
inner
, or
dot
product.)
•
They are
complete
:
This means
any
vector
v
=
v
i
ˆ
e
i
i
∑
is a unique linear combo of basis vectors.
•
The basis set
spans
ordinary space. This is like
completeness,
but backwards  every linear
combination of basis vectors is again an NDim vector, and all such linear combos generate all
possible vectors in the space.
•
We can choose a specific
representation
of
v
, namely
{v
1
,v
2
,v
3
,
…
v
n
}
, but it is
not
unique, it
depends on the choice of basis. (e.g. polar vs. rectangular, and even which particular rotation of
the rectangular axes.)
•
Each number
v
i
is the
projection
of
v
in the
ˆ
e
i
direction, and can be obtained by the formula
v
i
=
v
⋅
ˆ
e
i
.
(This involves the same scalar product, again, as we used above in the statement of
orthonormality.
)
You can
prove
the last formula by using orthogonality and completeness.
•
Addition, or multiplication by a scalar (number), keeps you in the same Ndim "vector space".
(adding or scaling vectors gives another vector.)
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32
We can make
very
powerful analogies to
all
of the above in the world of square integrable
functions: Note the one to one correspondence between each of the following statements about
functions, with the preceding ones about vectors.
•
Square integrable functions live in
Hilbert
space. (Never mind what this means for now!)
•
We have (or can choose)
basis
functions
:
u
n
(
x
)
(
Infinitely
many of them.)
(This infinity might be
countable
(discrete), or it might be uncountable, in which case you can't use
integers as labels, but need real numbers.)
We have already met some examples of both such types of u
n
's, as eigenfunctions of operators.
•
They are
orthonormal
:
dx u
n
(
x
)
∫
u
m
(
x
)
=
δ
n m
.
This is apparently our new way of writing
the
inner
product.
(If the labeling is continuous, the right side will be a Dirac delta function!)
•
They are
complete
:
Any
function
ψ
(
x
)
=
c
n
u
n
(
x
)
n
∑
is a unique linear combo of the basis
vectors.
(If the labeling is continuous, then
ψ
(
x
)
=
dE c
(
E
)
u
E
(
x
)
∫
)
•
The basis
spans
Hilbert space. This is similar to
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 Fall '08
 STEVEPOLLOCK
 mechanics, Light, Hilbert space, Hermitian, Hermitian Operators, Dirac notation

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