Quantum Mechanics for Engineers |
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© Leon van Dommelen |
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D.1 Generic vector identities
The rules of engagement are as follows:
- The Cartesian axes are numbered using an index , with
1, 2, and 3 for , , and
respectively.
- Also, indicates the coordinate in the direction,
, , or .
- Derivatives with respect to a coordinate are indicated by
a simple subscript .
- If the quantity being differentiated is a vector, a comma is
used to separate the vector index from differentiation ones.
- Index is the number immediately following in the
cyclic sequence ...123123...and is the number
immediately preceding .
The first identity to be derived involves the “vectorial triple
product:”
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(D.1) |
To do so, first note that the -th component of
is given by
Repeating the rule, the -th component of
is
That writes out to
since the first and fourth terms cancel each other. The first three
terms can be recognized as the -th component of
and the last three as the -th
component of .
A second identity to be derived involves the “scalar triple
product:”
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(D.2) |
This is easiest derived from simply writing it out. The left hand
side is
while the right hand side is
Inspection shows it to be the same terms in a different order. Note
that since no order changes occur, the three vectors may be
noncommuting operators.