Dept. of Materials Science & Engineering
Bill Knowlton
Extra Credit
Mathematical operators:
An operator is a symbol that tells you to do something to whatever follows the symbol.
Typically the operator will have a hat or caret symbol,^, over it. An example of an
operator is the following:
d
Aˆ =
dx
The operation it performs is the derivative with respect to x. If we us it to operate on
the function:
f(x) = x,
then:
ˆ ( x) = d f ( x) = d x = 1
Af
dx
dx
Pretty simple. Operators can also be complex. For instance, we can have the following:
∂
Pˆx = −i=
∂x
For the function:
f ( x) = e − ikx
then:
∂
Pˆx f ( x) = −i=
f ( x) = −ii=ke − ikx = =ke − ikx
∂x
1. Your assignment is the following. I will give you a list of operators and a list of
functions. I want you to use each operator and operate on each and every function.
You will do so using Mathematica. Please use the Style Sheet ArticleModern and use
various Styles to allow ease of reading your work.
List of Operators
List of Functions
∂
1. Pˆx = −i=
∂x
2
=
=2 ∂ 2
2. Hˆ x =
∇ 2x =
2m
2m ∂x 2
3. ∇ 2 =
1. ψ ( x) = Beikx
2. φ ( x) = Ax (a − x)
⎛πx ⎞
3. Φ ( x) = A sin ⎜
⎟
⎝ a ⎠
∂2
∂2
∂2
+
+
∂x 2 ∂y 2 ∂z 2
⎛ ∂
∂ ⎞
4. Lˆx = −i= ⎜ y − z ⎟
∂y ⎠
⎝ ∂z
4. ϕ ( x ) = axe
5. Ĉ Take complex conjugate C =
5. Ψ ( x, y, z ) = x 2 + y 2 + z 2
−
ax 2
2
6. Γ( x, y, z ) = x 2 yz + xy 2 z + xyz 2
For help, see the file: Operators Complex Conjugates & Rules.pdf
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Dept. of Materials Science & Engineering
Bill Knowlton
2. Given the function: φ ( x ) = Ax( a − x)
a
and the following integral: b ∫ φ ( x)φ ( x)dx = 1
0
use Mathematica and solve for the constant b.
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