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Chem120A Notes 8/29/08
Schrödinger Equation →
)
(
)
(
ˆ
x
E
x
H
Ψ
=
Ψ
(note: “^” signifies an operator)
→ This is an eigenvalue equation
→
H
ˆ
is the Hamiltonian operator
→
Ψ
(x) is an eigenfunction
→ E is the eigenvalue
→ Not all functions are eigenfunctions, only those which give you a number
→ Operator operating on a function gives you that function multiplied by a number
(Operator ∙ Function = Number ∙ Function → Eigenvalue Equation)
Examples of Operators
D
ˆ
→ Differentiation operator with respect to x
f
(x)
→ differentiable equation
)
(
)
(
)
(
ˆ
x
f
x
f
dx
d
x
f
D
′
=
≡
→
x
x
e
x
e
x
D
3
2
)
3
(
ˆ
2
+
=
+
Sum of two operators
A
ˆ
and
B
ˆ
)
(
ˆ
)
(
ˆ
)
(
)
ˆ
ˆ
(
x
f
B
x
f
A
x
f
B
A
+
≡
+
Product of two operators
A
ˆ
and
B
ˆ
[
]
)
(
ˆ
ˆ
)
(
ˆ
ˆ
x
f
B
A
x
f
B
A
≡
→
A
ˆ
operates on the result of
B
ˆ
operating on
f
(x)
Operators obey associative law of multiplication
i.e.
C
B
A
C
B
A
ˆ
)
ˆ
ˆ
(
)
ˆ
ˆ
(
ˆ
=
However, order of operations is important
→ For numbers,
ba
ab
=
→ For operators,
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This note was uploaded on 09/28/2009 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at University of California, Berkeley.
 Spring '07
 Whaley
 Physical chemistry, pH

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