Time-independent Hamilton, Time-dependent Schrodinger's equation, Larmor Precession, Hamiltonian for the particle in E, B fields, Scalar and vector potentials, Rayleigh-Ritz Variational Principal, Identical & indistinguishable particles, Time-dependent Hamilton, Emission & absorption of radiation b...
Course Outline : I particle
Time
independent phenomena
&M many particles
lidentical particles/
Time dependent phenomena
1 . Time -
independent H
Revision for QM
Hilbert
Space
Definition Hibert is rector which is
·
space space
: a
ummu
with
product between its elements
complete an inner
Elements of the rector
space
are ware
functions .
·
Hibert space
is the home of OM states/worefunctions
wavefunctions are
representations of
&M states
Hibert
Example => :
space
·
1 Hibert of single particle warfunctions in ID :
space
& IX) , &2 IX), ...
(infinite dimensions)
2
Hibert
space of spin- states :
spin-up state
spin down states (2D)
147 IV
I'd 1il
17 197
·
Definition : Inner
product of infinite dimenension
H. S le g single-particle werefunctions ID/
↑
.
in
, f (x) , gIX) :
(dx +
*
[f(g) =
(x)g(x)
In Hibert we it rector
space ,
say is a
space
With inner
product we mean
Kflg > / < 8
fig HS
Definition :
·
Norm of f(x) =
Kflf > )E
· A
we is normalized if <
flf) =
1
finite It S spin-
Definition : For dimension .
leg
Statel
Es (f) =
f, (d) +
+2 (i) =
H
197 =
g2(d) +
ga (i) =
192
Inner
The product :
<
flg =
If* )() =
fig 1
+ * S
transpose of complex conjugate
of f
·
Definition :
ket rector :
If)
=
1e
*
bra rector (f)
:
=
If * t )
Hilbert space is
composed of ket rectors .
·
Definition
:
The set of Vectors (d , I , P2X/ ,
... ) is
complete if any
other rector f(x) can be
expanded
C
f(x) = &
, Linear
independence is when I4uhu are
orthogonal
i .
e <
dild; = & ,
when itj
The set and
that is
ulte muly
-
independent
wi forms a basis .
Von Neumann's axioms
·
physical quantities are called observable quantities
measured in experiment
·
Van Neumann's axioms :
1) In QM observable is
,
every ze n represented by a
Hermition operator
-
Always true for closed systems i.e
quantum system +
the environment -
Not true for systems quantum system interact
-
i
open .
e
with the environment
2) The mechanical state of is
quantum ~
a
system
represented by a state it or a
wifn of IX) in
H . S eigenfunction
-
HIX) and < 21X) with < constant e.g-4(x) , 10034(X)
represent the same the same
physical system
. just
different "length"
3) If we measure the physical quantity & of the
(i
"
times"
system many
. e
many copies or
replica of
the same system) , then the mean l
expectation
Q
value) of is
< Q7 = < 41947
(ax4
*
=
(x)4(X)
, Hermitian Operators
For
general operator &
·
a
*
< 41947 =
Q4147
Definition An Hermition if
operator
:
·
is
*
<
41947 =
(41947
i . e < 41047 = <
&4147
Notation
·
:
For Hermition
operators , the
expectation value is
written as -
< Q7 =
< 419147
=
< 41947 =
184147
(ax
*
4 Q4
=
Properties operators
=> Hermition have
·
:
1) t of mmmagonal
set eigenfunctions in H S .
i .
2
& Un (x) =
XnUn(X) Un-eigenfunction
H S An-eigenvalue
where & eigenfunction f(x) .
f(x) = Un (X)
and Un/Um>
< = 0
if n + m
2)
w
Eigenvalues Xu are red
wi
experimental measurements
=
Meaning eigenfunctions => :
*
-
A Q M system is
. described by eigenfunction 2 W =
Us (X)
of Q .
If we measure & , the result of
measurement will Xs
be
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