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2 Experimental Methods for Determination of Nucleation Rates
Then, an inner product or a scalar product is a scalar that effectively “samples”
a component that is parallel to the two vectors in question:
< a|b >=< b|a >= scalar
(2.3.2)
where |a> and |b> are any vectors. The meaning of taking a scalar product is especially
clear for base vectors, such as |x>, |y>, and |z>, in our three dimensions. An inner
product of two perpendicuar base vectors is zero: e.g., = 0. And an inner
product of two like base vectors is one: e.g., = 1. For example, consider a
vector |a> that is defined as
|a >≡ (3, −1, 2) = 3|x > −1|y > +2|z >
(2.3.3)
We can take inner products of |a> with the three base vectors of the space, |x>,
|y>, and |z>:
< a|x >= (3, −1, 2) · (1, 0, 0) = 3 + 0 + 0 = 3
< a|y >= (3, −1, 2) · (0, 1, 0) = 0 − 1 + 0 = −1
< a|z >= (3, −1, 2) · (0, 0, 1) = 0 + 0 + 2 = 2
(2.3.4)
As can be seen, taking an inner product of a vector with one of the base vectors
“samples” the component of the original vector that is parallel to the direction of the
base vector.
2.3.2 The Fourier Transform
It turned out that each component of trigonometric functions is orthogonal to every
other. An interesting demonstration is presented by Feynman in his lecture series
(50) [20]. In short, any one component, say sin(x), cannot be expressed in a linear
combination of the others (sin(2x), etc.). That is, no matter how one linearly combines
sin(2x), sin(3x), sin(0.1x), sin(–x), etc., one can never obtain sin(x) or a multiple of
it. A few other examples are
| sin(x) > = α| sin(2x) > +β| sin(3x) >
| cos(−4x) > = α| cos(5x) > +β| cos(6x) >
|ex p(7i x) > = α| exp(8i x) > +β| exp(9i x) >
(2.3.5)
The above example shows that, unlike our three-dimensional world that are limited
to only three base vectors, infinite numbers of “directions” or base vectors are possible
for trigonometric functions. In mathematical terms, it can be said that trigonometric
functions span a Hilbert space of infinite dimensions. Then, what would be the
meaning of an inner product or a scalar product in this trigonometric space? Can
2 Experimental Methods for Determination of Nucleation Rates
Then, an inner product or a scalar product is a scalar that effectively “samples”
a component that is parallel to the two vectors in question:
< a|b >=< b|a >= scalar
(2.3.2)
where |a> and |b> are any vectors. The meaning of taking a scalar product is especially
clear for base vectors, such as |x>, |y>, and |z>, in our three dimensions. An inner
product of two perpendicuar base vectors is zero: e.g.,
product of two like base vectors is one: e.g.,
vector |a> that is defined as
|a >≡ (3, −1, 2) = 3|x > −1|y > +2|z >
(2.3.3)
We can take inner products of |a> with the three base vectors of the space, |x>,
|y>, and |z>:
< a|x >= (3, −1, 2) · (1, 0, 0) = 3 + 0 + 0 = 3
< a|y >= (3, −1, 2) · (0, 1, 0) = 0 − 1 + 0 = −1
< a|z >= (3, −1, 2) · (0, 0, 1) = 0 + 0 + 2 = 2
(2.3.4)
As can be seen, taking an inner product of a vector with one of the base vectors
“samples” the component of the original vector that is parallel to the direction of the
base vector.
2.3.2 The Fourier Transform
It turned out that each component of trigonometric functions is orthogonal to every
other. An interesting demonstration is presented by Feynman in his lecture series
(50) [20]. In short, any one component, say sin(x), cannot be expressed in a linear
combination of the others (sin(2x), etc.). That is, no matter how one linearly combines
sin(2x), sin(3x), sin(0.1x), sin(–x), etc., one can never obtain sin(x) or a multiple of
it. A few other examples are
| sin(x) > = α| sin(2x) > +β| sin(3x) >
| cos(−4x) > = α| cos(5x) > +β| cos(6x) >
|ex p(7i x) > = α| exp(8i x) > +β| exp(9i x) >
(2.3.5)
The above example shows that, unlike our three-dimensional world that are limited
to only three base vectors, infinite numbers of “directions” or base vectors are possible
for trigonometric functions. In mathematical terms, it can be said that trigonometric
functions span a Hilbert space of infinite dimensions. Then, what would be the
meaning of an inner product or a scalar product in this trigonometric space? Can