SJP QM 3220
Formalism 1
Page F1
M. Dubson, (typeset by J. Anderson) Mods by S. Pollock
Fall 200
The
Formalism
of
Quantum
Mechanics
:
Our story so far …
State of physical system:
normalizable
Ψ
( x, t )
Observables:
operators
ˆ
x
,
ˆ
p
=
i
∂
∂
x
,
ˆ
H
Dynamics of
Ψ
:
TDSE
i
∂Ψ
∂
t
=
ˆ
H
Ψ
To solve, 1
st
solve TISE:
ˆ
H
ψ
=
E
Solutions are stationary states
ψ
n
(x), E
n
═►
special solutions of TDSE:
Ψ
n
(
x
,
t
)
=
n
(
x
)
e
−
iE
n
t
TDSE linear
═►
any linear combo. of solutions is also a solution.
Discrete case:
Ψ
(
x
,
t
)
=
c
n
e
−
iE
n
t
/
n
(
x
)
n
∑
(
n
=
1,2,3,
)
Continuum case:
Ψ
(
x
,
t
)
=
dk
φ
(
k
)
e
−
i
ω
(
k
)
t
∫
k
(x)
(
k any real number)
e
+
ikx
π
ψ
n
's ,
ψ
k
's
form complete, orthonormal sets:
dx
∫
m
*
n
=
δ
mn
dx
∫
k
'
*
k
=
1
2
dx
∫
e
i
(
k
−
k
')
x
=
(
k
−
k
Discrete
Discrete
Continuum
Discrete +
Continuum
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View Full DocumentSJP QM 3220
Formalism 1
Page F2
M. Dubson, (typeset by J. Anderson) Mods by S. Pollock
Fall 200
Notice similarity of
Ψ
's
to vectors;
Vector
V
//
complex function
Ψ
( x, t )
Scalar
real number a
//
complex number c
Any linear combination of vectors is a vector
C
=
a
A
+
b
B
//
Ψ
=
α
Ψ
1
+
β
Ψ
2
Orthonormal basis vectors
ˆ
x
⋅
ˆ
x
=
1
,
ˆ
x
⋅
ˆ
y
=
0
//
ψ
m
*
n
dx
∫
=
δ
mn
V
=
V
x
ˆ
x
+
V
y
ˆ
y
+
V
z
ˆ
z
//
Ψ
=
c
n
n
n
∑
V
x
=
ˆ
x
⋅
V
//
c
n
=
dx
n
*
Ψ
∫
Inner product
A
⋅
B
=
A
i
B
i
i
=
x
,
y
,
z
∑
//
∫
dx
Ψ
*
Φ
=
dx
∫
d
m
m
m
∑
*
c
n
n
n
∑
=
d
m
*
m
.
n
∑
c
n
m
*
n
dx
∫
mn
=
d
n
*
n
∑
c
n
The space of all complex, squareintegrable functions
Ψ
(x) is called
Hilbert Space.
Norm
V
⋅
V
=
V
2
//
Ψ
*
Ψ
dx
∫
<
∞
Hilbert Space is an infinitedimensional vector space with complex scalars and
normalizable vectors.
SJP QM 3220
Formalism 1
Page F3
M. Dubson, (typeset by J. Anderson) Mods by S. Pollock
Fall 200
Postulate 1:
Every possible physical state of a system corresponds to a normed vector
in Hilbert Space.
The correspondence is 1to1 except that vectors that differ by a phase
factor (scalar of modulus 1) corresponds to the same state
Ψ
( x, t )
◄═►
e
i
θ
Ψ
( x, t )
Dirac Notation
:
dx
−∞
+
∞
∫
f
*
(
x
)
g
(
x
)
=
f g
=
complex number
=>
f
g
=
g f
*
f
f
is real, nonnegative
c any complex number:
f c
⋅
g
=
c f g
c
⋅
f g
=
c
*
f g
Postulate 2:
(to be stated shortly!)
associates with every observable a linear,
hermitean operator. But first, a little background:
Definition:
An operator
ˆ
Q
is hermitean (or hermitian, both spellings are common) if
f
ˆ
Q g
=
ˆ
Q f g
for all f, g in Hilbert space (Hspace).
Which can be written (in position representation) as
dx
∫
f
*
(
ˆ
Q g
)
=
dx
∫
(
ˆ
Q f
)
*
g
Question: Is the operator
ˆ
Q
=
d
dx
(
)
Hermitian?
(The answer will be no.)
Let’s see why!
dx
∫
f
*
dg
dx
parts
?
=
dx
∫
df
dx
*
⋅
g
f
*
(
x
)
g
(
x
)

−∞
+
∞
0
−
d
dx
∫
(
f
*
)
⋅
g
(
x
)
dx
=
−
df
dx
∫
*
⋅
g
(
x
dx
So the answer is NO, there’s an extra unwanted minus sign that cropped up. It is NOT the
case that for this particular operator, that
f
ˆ
Q g
=
ˆ
Q f g
.
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 Fall '08
 STEVEPOLLOCK
 mechanics, S. Pollock, J. Anderson, M. Dubson, SJP QM

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