Physics 451
Quantum mechanics I
Fall 2012
Sep 26, 2012
Karine Chesnel
Quantum mechanics
Announcements
Test 1
Until tomorrow Thursday night!
• Homework this week: HW # 8
due Friday Sep 28 by 7pm
Ch 2.5
Quantum mechanics
Bound and Scattering states
In solid state physics
Electron, X-ray scattering:
Free particle
‘Scattering state”
Crystal
Atom
Bound state
Bound states
Ch 2.5
Quantum mechanics
Bound and Scattering states
A mechanical analogy
V(x)
E2
E1
x
Ch 2.5
Quantum mechanics
Bound and Scattering states
E  [V    orV   ]
E  [V    andV   ]
Scattering state
Bound state
Harmonic oscillator
Infinite square well
Free particle
V(x)
0
a
Bound state
x
x
Bound state
Scattering state
Ch 2.5
Quantum mechanics
Bound and Scattering states
V(x)
x
In physical reality:
V ()  0
E 0
E0
Scattering state
Bound state
Ch 2.5
Quantum mechanics
The Dirac delta function
 x  

  x dx  1

0
for
x0

for
x0

 f  x    x  a dx  f  a 

Ch 2.5
Phys 451
Quiz 11
3
What is
 f  x    x  5dx
2
A. 1
B. f(-5)
C. f(5)
D. f(x)
E. zero
equal to?
Ch 2.5
Quantum mechanics
The delta function potential
V x   x
2

2
i

  ( x)
2
t
2m x
For
x0
d 2
2mE
 2 
2
dx
Ch 2.5
Quantum mechanics
The delta function potential
Continuity at boundaries

is continuous
d
dx
is continuous except where V is infinite
d 2

  ( x)  E
2
2m dx
2
2m
 d 

   2  0

 dx 