Physics 451
Quantum mechanics I
Fall 2012
Oct 19, 2012
Karine Chesnel
Phys 451
Next homework assignments:
•HW # 14 due Friday Oct 19 by 7pm
Pb 3.7, 3.9, 3.10, 3.11, A26
• HW #15 due Tuesday Oct 23
Pb 3.13, 3.14, 3.17, 3.18, 3.22, 3.23
Practice test 2 M Oct 22
Work with your group!
Test 2 : Tu Oct 23 – Fri Oct 26
Phys 451
Generalized statistical interpretation

• Particle in a given state
• We measure an observable
• Operator’s eigenstates:
Q
(Hermitian operator)
Q  n  qn  n
eigenvalue
eigenvector
Eigenvectors are complete:
Discrete spectrum

   cn n
n 1
Continuous spectrum
 

 c(q)

q
( x) dq
Phys 451
Generalized statistical interpretation

Particle in a given state
   cn n
n 1
Fourier’s trick to find Cn
cn   n 

• Normalization:
    cn
2
n 1

• Expectation value
2
Q   Q   cn qn
n 1
Phys 451
Quiz 18
If you measure an observable Q
on a particle in a certain state    cn  n ,
n
what result will you get?
A. the expectation value
Q
B. one of the eigenvalues of Q
C. the average of all eigenvalues
D. A combination of eigenvalues
with their respective probabilities

q
n 1
n

c
n 1
n
2
qn
Phys 451
Generalized statistical interpretation
Operator ‘position’:
ˆ y  x   yf y  x 
xf
 ( x) 

c( y ) 

 c( y ) f
y
( x)dx

  ( x  y) ( x, t )dx   ( y, t )

c( y )   ( y, t )
2
Probability of finding the particle at x=y:
2
Phys 451
Generalized statistical interpretation
Operator ‘momentum’:
d
f p  x   pf p  x 
i dx
  x 

 c  p  f  x  dp
p

1
c( p) 
2

 ipx /
e
 ( x, t )dx    p, t 


c ( p )   ( p, t )
2
Probability of measuring momentum p:
Pb 3.11: probability of measuring p in a given range
2
Phys 451
The Dirac notation
Different notations to express the wave function:
• Projection on the position eigenstates
• Projection on the momentum eigenstates
• Projection on the energy eigenstates
  x, t     y, t    x  y  dy
eipx /
    p, t 
dp
2
  cn n ( x)eiEnt /
n
Quantum mechanics
The uncertainty principle
Finding a relationship between standard deviations
for a pair of observables
 A2 B 2
  A, B 

 2i





2
Uncertainty applies only for incompatible observables
Quantum mechanics
The uncertainty principle
Position - momentum
x p 
2
Quantum mechanics
The uncertainty principle
Position - Energy
x E 
Pb 3.14
2m
x   x
E   H
 xˆ , Hˆ  
ˆ

 im p
p
Quantum mechanics
The uncertainty principle
Energy - time
E t 
2
Special meaning of t
Quantum mechanics
Quiz 19
An excited particle emits a certain radiation of energy E
with a band width E. What can we say
about the characteristic lifetime of excited state?
A. Lifetime is a least
E
E 2
B. Lifetime is a most
E
E 2
C. Lifetime is a least
D. Lifetime is a most
2 E
2 E
Quantum mechanics
Heisenberg equation of motion
d Q
dt
Pb 3.17

i
Q
 H , Q 
t
Q  Hˆ
Ĥ
Q  xˆ
d x
dt
Q  pˆ
constant
 p /m
Ehrenfest’s theorem
Quantum mechanics
Heisenberg equation of motion
d Q
dt
Definition for t:
To evaluate t:
t 

Q
 H , Q 
t
Q
when
d Q dt
Q
0
t
• choose an appropriate operator
• calculate
Pb 3.18
i
Q
and
d Q / dt
Application: use Q = x, in the case of the infinite square well
Quantum mechanics
The Dirac notation
“bra” “ket”
ab
Inner product:
Operator:
a
b
Projection operator:
e
e

for orthonormal basis
en
en  1
ex
ex dx  1
n
or

Pb 3.22
Pb 3.23