Quantum Mechanica 2
, Quantum Mechanics
Hoorcollege 1 - 7-2-22
Formalisms of QM
Schrödinger eg : MIX) =
-chVYIx = E
Heisenberg's uncertainty principle : AxAps
Hilbert space
The Cartesian space in 3 dimensions : #3 ; Fetth
↳ You can write
any vector in 3 dimensions as a sum /linear combination) : = an + Gratis
By using a matrix
you can transform a vector into another rector
The 3 vectors form a basis : [ ,
2 , s)
By choosing a ,
92 and as
you can recreate
any element in the
vector space
* Basisvectors orthonormal which that the scalar product (inproduct) Vi
Sij the scalar product is either 7 or 0
Uj
are means · = >
-
.
,
To describe the Hilbert space , we need state rectors > (x)
We call 1> a ket > Dirac notation
↑x E H (Hilbert spaces
↳> vector space
complex
All info about a quantumstate is incorporated into the State Vector (x)
State vectors are transformed by linear
operators
dimension of Hilbert space
al ↑
R
A2
Dirac notation &
ai /Xi]
(x :
: =
i = 1 ↳ basiselement
basis an
indepen -
cent
this expression makes it more explicit that
you're using a
choice of basis
Specific
J
n dit K
dimensional with
Ibasis dependent)
We've seen in linear that if elements of the Cartesian Vector space that multiplication by and
algebra two recoors are ,
some coefficients adding the results ,
the result will also be an element of the same vector space
En M
(a + bez) ERM a ,
De R
The holds in the Hilbert space
same
thing
(x) , 19)
(x) = a(x) + biB) -H a, be
↳ a and b are now complex numbers because the
Hilbert space is a
complex vector space .
analogue with the Schrodinger cat
Both the Hilbert
States are part of space , so :
1 alive) -H : Idead H
but also : lalive) + Idead) -H
Electron Spin
Spin up ↑ and spin down I
t -
We have 2 dimensions , therefore we need 2 basisvectors
The dimensions of the Hilbert the number of independent basisvectors
space are
a choice of basisrecorcould be : 1+ =
(b) ; 1- (i)
any (x)EH
element can be written as the linear superposition of the basiselements :
(x) =
c+ |+ 7 + c 1 -
> = c+ (b) + c (i) =
(c)
↳ This is the most general state that this
quantum system can have
J T measure the
-
spin of the
system , you will
only find 2 possibilities <
Spin up or spin down
↳ a of this will return two possible outcomes
measurement system
However, this does not mean states of 1) exist there is an infinite number of quantum States
only two >
-
↳ choice of coefficients
every different is a different quantum State
,Inner/Scalar product in Hilbert
Space
The scalar product reviewed :
E R
a =
(91 ,
92 , 93) : 5 = ( bi , ba , b3)
Scalar product : = Saibi = tâllcos
↳> parallel : /âlI
perpendicular : o
Scalar product in Hilbert Space :
# In-dim I .
=adi b
1) 13
The innerproduct is
given by <x1B) (t) :
↳ "Draket"
(7 = "Ket"
Bl =
"bra"
If Space (1x)tH) 1H * )
(x) is an element of the Hilbert
,
then each
corresponding 'bra-vector is not an element of the Hilbert space ,
but the dual Hilbert
space
the Hilbert Vector
Transforming from space to the dual Hilbert space
al
A2
(ai*, *,
*
IX) =
> (x1 =
aa an (
an
...
Ket > bra
So to ket to bra is to make the column a row and then take the
go from complex conjugate of each element
In different notation
(x)
Bai) with SIN] as basis of
(xl= Bi *
(il with Mil as basis of
Example of braket
(18) =
19. *, 92 *, ...,
an
*
)
(b)
Physical interpretation
(1B) is a measure of overlap in the Hilbert space
= (b) : Ec =
(9) >
-
plotting these vectors
you see
they are
perpendicular so
they have no overlap
= 16) : va =
(6) >
the recoors overlap :
So what this means is when the innerproduct is zero , there is no overlap ,
but if it's not zero there is some overlap
(x1B) = 0
↳> no overlap ; Vectors are
orthogonal
Using normalized State vectors
We know that 15(X)12 is the
a measure of the probability of finding particle in a
space
The idea is that when we take innerproducts of a state vector we can
assign them a ~
interpretation :
(x(x) = 17 H
Now, back to the electron and we saw that (x) is a linear combination of elements
going spin ,
Since the basis (xIB) of
is
orthogonal moet er
gelden dat 0 in dit
geval ( + 1 -7 0
= =
: :
,
We have two elements of this rector space :
1x
=
11 + > -
it -) = (1)
1p) = be 1 + + b21 -
3 =
(3)
What ?
are the values of bi , be such that (xIB) = 0
First , compute the innerproduct :
(x1B) = ( .
+ i)(b) = 0
↳
complex
↓ (b) conjuga
te
by = -
ibz
, the this value is , the smaller the
Using Kronecker delta
& larger overlap
S
(4j(4)
=
(4
j /ci) I
Sij
linearalgebra Sij
↳ Kronecker delta
* a basis should be orthonormal
Here ,
(j
= <
4j14)
In the limit of complete overlap :
1) Ci(i) then
Cj < and Ci = 0
forif
=
When can compute cj by taking the innerproduct of one of the basis rectors with the
big state vector
↳
(j (
4j(4)
=
N
IN: > <Mil =
1 is an
operator >
-
something that acts on a state rector and
gives me another state vector where both state vectors
belong do the same Hilbert space
i =1
↳ 814) =
153 Where 14) , 153 &H
Identity operator :
"14 :>< Mi =
1
i =1
3 =
C
Xi (4)
14
↳
dummy
incét
Hilbert space in finite dimensions vs infinite dimensions
In the Hilbert dimensions we need an infinite number of basis elements
space of infinite
(dx4(x)(x)
S
(4) =
continuous
"Wave' particle's
&
I
element of
function Position
Hilbert
1y) =
i
"Ciltis
=
1
3 discrete (finite dimensions of Hilbert Space)
So, for the wave function
the infinite dimensions plays the role of the coefficient ci
( (dxx2(x)(x) (dxdxy, (x(x2(x) .x(y S dxy, *(x)X2(x)] overlap integral
*
(4. 142) = ax'4, (x)(x) = =
an infinite dimensions we use S(X'-X) (Dirac delta)
Discrete <4: /Kronecker deltal
:
14j) =
Sij
Continuous :< X11x) =
SIX-X) (Dirac deltal
(4143 =
( *
a xy (X)X(x) =
( d x(4(x) = 1 (because the warefunction in normalized(
M
Finite Hilbert space : 141) =
= =,
jic 141)
Infinite Hilbert space : 141) =
( dx + (x) (x)
,
* Table in the lecture notes to determine if finite of infinite
Note : Spatial dimension o Hilbert dimension
State rectors and operators
= (a) : m
=InS
m == (b) : Ben
Operator :8 applied to a staterector to make another State vector
& 141) =
152) ; 14 · 7, 1427 H
Inner product : 14 . 3
,
122) > C
Operator : 14,7 - 1427
In the finite dimensional Hilbers space : 8141) =
142)
1413
ailtibil
814 3
=(
--
,
=
142) =
:
·
Operators are basis independent ,
but the representations do depend on the basis
Action of 8 on 14) -H is determined
by 81 asiselements
·
, Quantum Mechanics
Hoorcollege 1 - 7-2-22
Formalisms of QM
Schrödinger eg : MIX) =
-chVYIx = E
Heisenberg's uncertainty principle : AxAps
Hilbert space
The Cartesian space in 3 dimensions : #3 ; Fetth
↳ You can write
any vector in 3 dimensions as a sum /linear combination) : = an + Gratis
By using a matrix
you can transform a vector into another rector
The 3 vectors form a basis : [ ,
2 , s)
By choosing a ,
92 and as
you can recreate
any element in the
vector space
* Basisvectors orthonormal which that the scalar product (inproduct) Vi
Sij the scalar product is either 7 or 0
Uj
are means · = >
-
.
,
To describe the Hilbert space , we need state rectors > (x)
We call 1> a ket > Dirac notation
↑x E H (Hilbert spaces
↳> vector space
complex
All info about a quantumstate is incorporated into the State Vector (x)
State vectors are transformed by linear
operators
dimension of Hilbert space
al ↑
R
A2
Dirac notation &
ai /Xi]
(x :
: =
i = 1 ↳ basiselement
basis an
indepen -
cent
this expression makes it more explicit that
you're using a
choice of basis
Specific
J
n dit K
dimensional with
Ibasis dependent)
We've seen in linear that if elements of the Cartesian Vector space that multiplication by and
algebra two recoors are ,
some coefficients adding the results ,
the result will also be an element of the same vector space
En M
(a + bez) ERM a ,
De R
The holds in the Hilbert space
same
thing
(x) , 19)
(x) = a(x) + biB) -H a, be
↳ a and b are now complex numbers because the
Hilbert space is a
complex vector space .
analogue with the Schrodinger cat
Both the Hilbert
States are part of space , so :
1 alive) -H : Idead H
but also : lalive) + Idead) -H
Electron Spin
Spin up ↑ and spin down I
t -
We have 2 dimensions , therefore we need 2 basisvectors
The dimensions of the Hilbert the number of independent basisvectors
space are
a choice of basisrecorcould be : 1+ =
(b) ; 1- (i)
any (x)EH
element can be written as the linear superposition of the basiselements :
(x) =
c+ |+ 7 + c 1 -
> = c+ (b) + c (i) =
(c)
↳ This is the most general state that this
quantum system can have
J T measure the
-
spin of the
system , you will
only find 2 possibilities <
Spin up or spin down
↳ a of this will return two possible outcomes
measurement system
However, this does not mean states of 1) exist there is an infinite number of quantum States
only two >
-
↳ choice of coefficients
every different is a different quantum State
,Inner/Scalar product in Hilbert
Space
The scalar product reviewed :
E R
a =
(91 ,
92 , 93) : 5 = ( bi , ba , b3)
Scalar product : = Saibi = tâllcos
↳> parallel : /âlI
perpendicular : o
Scalar product in Hilbert Space :
# In-dim I .
=adi b
1) 13
The innerproduct is
given by <x1B) (t) :
↳ "Draket"
(7 = "Ket"
Bl =
"bra"
If Space (1x)tH) 1H * )
(x) is an element of the Hilbert
,
then each
corresponding 'bra-vector is not an element of the Hilbert space ,
but the dual Hilbert
space
the Hilbert Vector
Transforming from space to the dual Hilbert space
al
A2
(ai*, *,
*
IX) =
> (x1 =
aa an (
an
...
Ket > bra
So to ket to bra is to make the column a row and then take the
go from complex conjugate of each element
In different notation
(x)
Bai) with SIN] as basis of
(xl= Bi *
(il with Mil as basis of
Example of braket
(18) =
19. *, 92 *, ...,
an
*
)
(b)
Physical interpretation
(1B) is a measure of overlap in the Hilbert space
= (b) : Ec =
(9) >
-
plotting these vectors
you see
they are
perpendicular so
they have no overlap
= 16) : va =
(6) >
the recoors overlap :
So what this means is when the innerproduct is zero , there is no overlap ,
but if it's not zero there is some overlap
(x1B) = 0
↳> no overlap ; Vectors are
orthogonal
Using normalized State vectors
We know that 15(X)12 is the
a measure of the probability of finding particle in a
space
The idea is that when we take innerproducts of a state vector we can
assign them a ~
interpretation :
(x(x) = 17 H
Now, back to the electron and we saw that (x) is a linear combination of elements
going spin ,
Since the basis (xIB) of
is
orthogonal moet er
gelden dat 0 in dit
geval ( + 1 -7 0
= =
: :
,
We have two elements of this rector space :
1x
=
11 + > -
it -) = (1)
1p) = be 1 + + b21 -
3 =
(3)
What ?
are the values of bi , be such that (xIB) = 0
First , compute the innerproduct :
(x1B) = ( .
+ i)(b) = 0
↳
complex
↓ (b) conjuga
te
by = -
ibz
, the this value is , the smaller the
Using Kronecker delta
& larger overlap
S
(4j(4)
=
(4
j /ci) I
Sij
linearalgebra Sij
↳ Kronecker delta
* a basis should be orthonormal
Here ,
(j
= <
4j14)
In the limit of complete overlap :
1) Ci(i) then
Cj < and Ci = 0
forif
=
When can compute cj by taking the innerproduct of one of the basis rectors with the
big state vector
↳
(j (
4j(4)
=
N
IN: > <Mil =
1 is an
operator >
-
something that acts on a state rector and
gives me another state vector where both state vectors
belong do the same Hilbert space
i =1
↳ 814) =
153 Where 14) , 153 &H
Identity operator :
"14 :>< Mi =
1
i =1
3 =
C
Xi (4)
14
↳
dummy
incét
Hilbert space in finite dimensions vs infinite dimensions
In the Hilbert dimensions we need an infinite number of basis elements
space of infinite
(dx4(x)(x)
S
(4) =
continuous
"Wave' particle's
&
I
element of
function Position
Hilbert
1y) =
i
"Ciltis
=
1
3 discrete (finite dimensions of Hilbert Space)
So, for the wave function
the infinite dimensions plays the role of the coefficient ci
( (dxx2(x)(x) (dxdxy, (x(x2(x) .x(y S dxy, *(x)X2(x)] overlap integral
*
(4. 142) = ax'4, (x)(x) = =
an infinite dimensions we use S(X'-X) (Dirac delta)
Discrete <4: /Kronecker deltal
:
14j) =
Sij
Continuous :< X11x) =
SIX-X) (Dirac deltal
(4143 =
( *
a xy (X)X(x) =
( d x(4(x) = 1 (because the warefunction in normalized(
M
Finite Hilbert space : 141) =
= =,
jic 141)
Infinite Hilbert space : 141) =
( dx + (x) (x)
,
* Table in the lecture notes to determine if finite of infinite
Note : Spatial dimension o Hilbert dimension
State rectors and operators
= (a) : m
=InS
m == (b) : Ben
Operator :8 applied to a staterector to make another State vector
& 141) =
152) ; 14 · 7, 1427 H
Inner product : 14 . 3
,
122) > C
Operator : 14,7 - 1427
In the finite dimensional Hilbers space : 8141) =
142)
1413
ailtibil
814 3
=(
--
,
=
142) =
:
·
Operators are basis independent ,
but the representations do depend on the basis
Action of 8 on 14) -H is determined
by 81 asiselements
·
