3-1
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 N-dimensional space.
(You're used to 3-D!)
•
We have (or can choose)
basis
vectors
:
ˆ
e
i
(N of them in N-dim 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 N-Dim 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
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 N-dim "vector space".
(adding or scaling vectors gives another vector.)