This preview shows pages 1–4. Sign up to view the full content.
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
This preview has intentionally blurred sections. Sign up to view the full version.
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
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 02/27/2012 for the course PHYSICS 3220 taught by Professor Stevepollock during the Fall '08 term at Colorado.
 Fall '08
 STEVEPOLLOCK
 mechanics

Click to edit the document details