Download 5. Quantum Theory

5. Quantum Theory
5.0.
Wave Mechanics
5.1.
The Hilbert Space of State Vectors
5.2.
Operators and Observable Quantities
5.3.
Spacetime Translations and Properties of Operators
5.4.
Quantization of a Classical System
5.5.
An Example: the One-Dimensional Harmonic Oscillator
5.0. Wave Mechanics
Cornerstones of quantum theory:
•
Particle-wave duality
•
Principle of uncertainty
E  h

h
2
 
h
p
h

6.6256  1034 J s
History:
Planck:
Empirical fix for black body radiation.
Einstein:
Photo-electric effect → particle-like aspect of “waves”.
de Broglie: Particle-wave duality.
Thomson & Davisson: Diffraction of electrons by a crystal lattice.
Schrodinger: Wave mechanics.
 k
Planck’s
constant
Wave mechanics
State of a “particle” is represented by a (complex) wave function Ψ( x, t ).
Probability of finding the particle in d
P = probability density if
3x
about x at t = P  x, t  d x  c   x, t 
3
 P  x, t  d x  c    x, t 
3
2
d 3x  1
Ψ( x, t ) is called normalized if c = 1.
P d 3x = relative probability if Ψ cannot be normalized.
E.g., free particle:
  x, t   exp i  k  x   t  
(1st) quantization:
A  A  x, p
r-representation:
→
 xˆ , pˆ   i
Aˆ  A  xˆ , pˆ 
xˆ  x
pˆ 
i

2
d 3x
Hamiltonian :
1 2
H
p  V x
2m
Time-dependent Schrodinger equation :
i
Hˆ  
→
2
2m
2  V  x 
c.f. Hamilton-Jacobi equation

  x, t   Hˆ   x, t 
t
 2 2


i
  x, t   
  V  x    x, t 
t
2
m


Time-independent Schrodinger equation :
Hˆ   x, t   E   x, t 
 2 2




V
x
    x, t   E   x, t 
 2m


Ĥ  i

t
5.1.
The Hilbert Space of State Vectors
Specification of a physical state: Maximal set of observables M = { A, B, C, …}.
→ Pure quantum state specified by values { a, b, c, … } assumed by S.
Assumption:
Every possible instantaneous state of a system can be represented by a ray
(direction) in a Hilbert space. (see Appendix A.3)
Hilbert spaces are complex linear vector spaces with possibly infinite dimensions.
 φ | = 1-form dual to | φ  = | φ †



†
 * 
Inner product is sesquilinear :  α, β  C,

→
 

 
  
 

†

    
  *    *  
Norm / length / magnitude of | φ  is
 
   
*
  1
| φ  is normalized if
Maximal set M = { A, B, C, …} → pure states are given by | a, b, c, … .
Probability of measuring values { a, b, c, … } from a state | Ψ  is
P  a, b, c,
| 
a, b, c,
→
P  a, b, c,
  a, b, c,
a, b, c,

 P  a, b, c,
a, b, c,
|   1
→
a ,b , c ,
 

abc

2
|     aa bb  cc
  aa bb  cc

orthonormality
 a, b, c,
 1
a, b, c,
a ,b , c ,
 abc  a, b, c,
a, b, c,

a ,b , c ,


a, b, c,
a , b, c ,


a, b, c,
a, b, c,
1
a ,b , c ,
→
Completeness
a ,b , c ,

  1
else
P  a, b, c,
| 
a, b, c,
 

2
 a a    a  a
If a takes on continuous values,

a

da
Example: 1-particle system with x as maximal set.
x x    x  x
3
d
 x x
   d 3x x
orthonormality
x 1
x 
completeness
  d 3x x  x
P  x |      x
2
Ψ (x) = wave function
5.2.
Operator:
Operators and Observable Quantities

Ô


Iˆ
E.g., identity operator:
Ô   

  
Linear operators:
Oˆ      
   Oˆ
   Oˆ 
 α, β  C
Observables are represented by linear Hermitian operators.
Let the maximal set of observables be M = { A, B, C, …}.
If we choose | a,b,c,…  as basis vectors, then the operators Mˆ  Aˆ , Bˆ , Cˆ ,

are defined as
Aˆ a, b, c,
 a a, b, c,
Bˆ a, b, c,
 b a, b, c,
Cˆ a, b, c,
 c a, b, c,

Eigen-equations
eigenvalues
eigenvectors
…
P a   


a, b, c,
2
b, c,
Expectation value of A:
A  a P a 



a  a, b, c,
a, b, c,
a, b, c,
a


 Aˆ a, b, c,

a, b, c,
a , b, c,
  Â 
Adjoint
→
A†
of A is defined as:

 Aˆ †   Aˆ 
A is self-adjoint / hermitian if

 Aˆ †  Aˆ 

†

Aˆ †  Aˆ

†
  Â 
*

Consider 2 eigenstates, | a1  and | a2 , of A.
 a1  a1 a1
→
 a2  a2 a2
a2 Aˆ a1  a1 a2
If A is hermitian,
Hence,
a2  a1
a  a 
1
→
a2 Aˆ † a1  a2* a2
a1
*
2
a1  0
a2
a1  a1* is real
Eigenvalues of a hermitian operator are all real.
a1  a2
→
a2
a1  0
Eigenstates belonging to different eigenvalues
of a hermitian operator are orthogonal.
a1
Algebraic Operations between Operators
Addition:
 Aˆ  Bˆ 
Product:
Aˆ Bˆ 
  Aˆ   Bˆ 


 Aˆ Bˆ 
 Aˆ , Bˆ   Aˆ Bˆ  Bˆ Aˆ


Commutator:
Analytic functions of A are defined by Taylor series.


Aˆ  e
 Bˆ
n ˆn
1
B  1   Bˆ   2 Bˆ 2 
n!
n 0
Caution:
ˆ
ˆ
ˆ
ˆ
eB eC  e B  C
If B is hermitian, then
2
 Bˆ , Cˆ   0


unless
ˆ
Aˆ  ei B
→
ˆ
1
Aˆ †  e i B  Â
Ex. 5.7
( A is unitary )
5.3. Spacetime Translations and Properties of Operators
Time Evolution: Schrodinger Picture
Each vector in Hilbert space represents an instantaneous state of system.
  t   Uˆ  t    0 
U is the time evolution operator
Normalization remains unchanged :
 t   t     0   0
→
Setting
Uˆ †  t  Uˆ  t   Iˆ
Uˆ  0   Iˆ
   0  Uˆ †  t  Uˆ  t    0 
U is unitary
we can write
ˆ
Uˆ  t   ei H t
where
Hˆ †  Hˆ
If H is time-independent,
i
d
d
  t   i Uˆ  t    0 
dt
dt
1
→
Hˆ  Ĥ
ˆ
 Hˆ ei H t   0
H = Hamiltonian
 Hˆ Uˆ  t    0 
 Hˆ   t 
c.f. Liouville eq.
| Ψ (t)  is not observable.
Heisenberg Picture
A  t     t  Aˆ   t 
Observable:
   0  Aˆ  t 
ˆ ˆ t 
Aˆ  t   Uˆ †  t  AU
e
→
i
i ˆ
Ht
Aˆ e

ˆ ˆ t 
   0  Uˆ †  t  AU
  0
i ˆ
Ht
for time independent H
d ˆ
A  t    Aˆ  t  , Hˆ 
dt
 A is conserved if it commutes with H.
Classical mechanics:
Possible rule:
  0
d
A t    A , H
dt
 Aˆ , Bˆ  ~ i


 A, B P
H is conserved.
P
( Not always correct )
Example: Canonical commutation relations
 xˆ , pˆ    i  a 
 xˆ , xˆ    pˆ  , pˆ    0
Alternative derivation:
P i P,
Classical translational generator:
1
1
Pˆ  Pˆ   pˆ j
→
P
H i H ,
P
T ˆ  a   ei a  P
ˆ
j
All components of x or p should be simultaneously measurable →
 xˆ , xˆ    pˆ  , pˆ    0
T ˆ  a T ˆ  b   T ˆ  a  b 
e i a  P e i b  P  e  i  a  b   P
ˆ
→
Consider
A  xˆ   xˆ
ˆ
ˆ
Aˆ  A  xˆ 
→
ˆ
 xˆ , pˆ    i  a 
 Hˆ , Pˆ   0


ˆ
ˆ
xˆ  a  ei a  P xˆ ei a  P
 xˆ 
→
A  xˆ  a   ei a  P A  xˆ  e  i a  P
→

i

 Iˆ  i a  Pˆ 
ˆ
 xˆ  Iˆ  i a  Pˆ  
  a  pˆ  xˆ  xˆ  a  pˆ   
See Ex.5.3 for higher order terms
H independent of x (translational invariant)
Classical mechanics:
Conserved quantity ~ L invariant under corresponding symmetry transformation
Quantum mechanics:
Conserved operator = generator of symmetry transformation
J i   i j k x j pk
Example: Angular Momentum
J ˆ  xˆ  Pˆ 
1
xˆ  pˆ 
 Jˆi , Jˆ j   i  i j k Jˆk


1 ˆ
J
 i
J
i
, J j
P
5.4. Quantization of a Classical System
Canonical quantization scheme:
Classical
Quantum
L  qi , qi 
H  qˆi , pˆ i 
pi 
H   qi
i
L
 qi
L
L
 qi
 qˆi , pˆ j   i  i j
Schrodinger
 qˆi  t  , pˆ j  t    i  i j
Heisenberg
Difficulties:
• Generalized coordinates may not work.
Remedy: Stick with Cartesian coordinates.
• Ambiguity.
ˆ ˆ
E.g., AB when  A , B   0
• Constrainted or EM systems with
Possible remedy: use
L
0
 qk
Reminder: velocity is ill-defined in QM.

1 ˆ ˆ ˆˆ
AB  BA
2
Remedy: use pi .

Wave functions:
 x  x
x xˆ   x x 
  x  a   exp  a    x 
 x T
→
†
x pˆ  
a
i

x-representation
(Taylor series)
i

 x exp  a pˆ  


 x 

i
 x


x A  xˆ , pˆ    A  xˆ ,    x 
 i 


x A  xˆ , pˆ    t   A  xˆ ,    x, t 
 i 
 A  xˆ , pˆ     d 3 x  x
Many bodies:
  x1 , x2 ,



3
*
x A  xˆ , pˆ     d x   x  A  xˆ ,    x 
 i 
x1 , x2 ,

5.5. An Example: the One-Dimensional Harmonic Oscillator
L
1
1
mx 2  m 2 x 2
2
2
H
 xˆ , pˆ   i
1 2 1
p  m 2 x 2
2m
2
Some common choices of maximal sets of observables:
{ x } : coordinate (x-) representation
{ p } : momentum (p-) representation
{ E } : energy (E-) representation
{ n } : number (n-) representation
n-representation
Basis vectors are eigenstates of number operator:
n̂ n  n n
Lowering (annihilation) operator:
aˆ n  cn n 1
Raising (creation) operator:
aˆ † n  bn n  1
Canonical quantization:
 aˆ , aˆ †   1
n n   nn
n aˆ †aˆ n  cn n aˆ † n 1  cn cn* n 1 n 1
 cn
2
→
 cn bn1 n n  cn bn 1
 bn
n aˆ aˆ † n  bn n aˆ n  1  bn bn* n  1 n  1
2
→
 bn cn1 n n  bn cn 1
n aˆ , aˆ †  n  bn  cn
2
aˆ 0  0
→
2
 aˆ , nˆ   aˆ , aˆ †aˆ 
bn*  cn 1
cn 1  cn  1
2
→
2
c2  2
2
 cn cn* n  n n
Setting cn and bn real gives
a, bc  abc  bca
1
c1  1
c0  0
aˆ †aˆ n  cn bn1 n
2
cn*  bn 1
…
→
aˆ n  n n  1
cn  n
2
bn  n  1
2
nˆ  aˆ † aˆ
aˆ † n  n  1 n  1
 abc  bac  bac  bca   a, b c  b a, c
  aˆ , aˆ †  aˆ  aˆ †  aˆ , aˆ   â
 aˆ † , nˆ    aˆ † , aˆ † aˆ    aˆ † , aˆ †  aˆ  aˆ †  aˆ † , aˆ    â †
n 
1
† n
ˆ
a
0


n!
Exercise: Show that if n is restricted to the values 0 and 1,
then the commutator relation
 aˆ , aˆ †   1
must be replaced by the anti-commutator relation
ˆˆ
 aˆ , aˆ   aa
†
†
 aˆ † aˆ  1
For the harmonic oscillator, if we set
aˆ 
m 
i
ˆ
x


2 
m

pˆ 

aˆ † 

m 
i
ˆ
ˆ
x

p


2 
m 
then
 xˆ , pˆ   i
1
1


Hˆ   aˆ † aˆ      nˆ   
2
2


 aˆ , aˆ †   1
→
Hence, basis { | n  } is also basis of the E-representation.
1

Hˆ n   n n   n    n
2

1

n   n   
2

The nth excited state contains n vibrons, each of energy  ω.
aˆ , nˆ   aˆ
 aˆ † , nˆ   aˆ †
→
 aˆ , Hˆ    aˆ


→
 aˆ † , Hˆ     aˆ †


n t   e

i
n t
n
x-representation:
1
† n
ˆ
n 
a
0


n!
a 0 0
where

→
 n  x 
→
 m 


 2 
 m 2 
 0  x   N 0 exp  
x 
 2

Using
x
with
1  m 


n!  2 
n /2
n /2
n

d 
x


  0  x
m

d
x



d 
x


 0  x   0
m d x 

 
N0  

m



1/4

so
that

dx  0  x 

d
1
 d
 1

  exp   x 2 
exp    x 2 
d x
2
d x
 2

n
one can show
where
 m 2   d 
 m 2 
 n  x   N n exp 
x  
exp
x 


 2
 d x 



  2m  
Nn  n!


m




1/2

and


dx  n  x 
2
1
2
1
The x- and p- representations are Fourier transforms of each other.

x 

dp
p
p


p 
 dx
x
x

1
x 
2
p


dp
p e

1

2
i
 dx
x e
n 
p

 dp
p

p n

where
px
where
px
x
1
p 
2


Similarly
i

x x    x  x
so that

p    p  p
 dp
p n  p


n  p  p n

 dx
p
x
x n

1

2


dx e

i
 n  x
px
 n  x  x n

In practice, ψ (x) is easier to obtain by solving the Schrodinger eq.
2


2



V
x


 2m
  n  x   n  n  x


with appropriate B.C.s
i
e
px
Let V(x) → 0 as |x| → .
For E < 0, ψn(x) is a bound state with discrete eigen-energies εn.
For E > 0, ψk(x) is a scattering state with continuous energy spectrum ε(k).
However, scattering problems are better described in terms of the S matrix,
scattering cross sections, or phase-shifts.