Chem. 540
Instructor: Nancy Makri
Postulates of Quantum Mechanics
1.
Wavefunction
The state of a system is described by a vector in a Hilbert space.
The state of a system
can be fully specified by its wavefunction in position space,
( ; )
t
Ψ
r
, or by its wavefunction in
momentum space,
( ; )
t
Ψ
p
%
etc.
The probability of finding the particle within a volume
d
r
around point
r
is equal to
2
( )
d
Ψ
r
r
.
In order for the probability of finding the particle anywhere to be equal to unity, we
require wavefunctions to be normalized:
2
( )
1
d
Ψ
=
∫
r
r
.
Hilbert space:
A vector space (over complexvalued vectors) of up to infinite dimension, which is complete,
and in which the inner product
*
×
u
v
is finite.
2.
Operators
States are transformed by linear operators:
1
2
1
2
ˆ
ˆ
ˆ
(
)
A
A
A
α
β
α
β
Ψ +
Ψ
=
Ψ +
Ψ
.
The commutator of two operators
ˆ
ˆ
,
A B
is defined as
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
[
,
]
A B
AB
BA
≡

.
If the commutator of two operators is equal to zero we say that the operators commute.
This is
not always the case.
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 Fall '08
 Mccall
 Hilbert space, Kronecker delta, Hermitian Operator, Hermitian operator form, dr Ψ

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