Transcript Sep 12

Physics 451
Quantum mechanics I
Fall 2012
Sep 12, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework remaining this week:
• Extended Friday Sep 14 by 7pm:
HW # 5 Pb 2.4, 2.5, 2.7, 2.8
Note:
Penalty on late homework: - 2pts per day
Credit for group presentations:
Homework 2: 20 points
Quiz 5: 5 points
Quantum mechanics
Announcements
No student assigned to the following transmitters:
2214B68
17A79020
1E5C6E2C
1E71A9C6
Please register your i-clicker at the class website!
Quantum mechanics
Ch 2.1
Time-independent
Schrödinger equation
• Space dependent part:
2
2


d
 V ( x) y  Ey

2
 2m dx

Solution y(x) depends on the potential function V(x).
( x, t )  y ( x)e
iEt /
Stationary state
Associated to energy E
Quantum mechanics
Ch 2.1
Stationary states
Properties:
• Expectation values are not changing in time (“stationary”):

Q    Q ( x,
)dx
i x
*
with
( x, t )  y ( x)eiEt /

Q  y Q ( x ,
)y dx
i x
*
Q
is independent of time
p m v m
d x
dt
0
The expectation value
for the momentum is always zero
In a stationary state!
(Side note: does not mean that
 x and  p are zero!)
Quantum mechanics
Ch 2.1
Stationary states
Properties:
• Hamiltonian operator - energy
2


d2
 V ( x) y  Ey

2
 2m dx

^
H
^
^
H  y H y dx  E y *y dx  E
^
*
^
H 2  y * H 2 y dx  E 2 y *y dx  E 2
H  0
Quantum mechanics
Ch 2.1
Stationary states
• General solution

 ( x, t )   cn  n ( x, t )
n 1
where
 n ( x, t )  y n ( x)e  iEnt /
• Associated expectation value for energy

H   cn2 En
n 1
Quantum mechanics
Quiz 6a
A particle, is in a combination of stationary states:

 ( x, t )   cn  n ( x, t )
n 1
What will we get if we measure its energy?
A.
H
B.
c E
n
n
n
C.
one of the values
D.

n
En
En
Quantum mechanics
Quiz 6b
A particle, is in a combination of stationary states:

 ( x, t )   cn  n ( x, t )
n 1
What is the probability of measuring the energy En?
A. 0
B. cn
C.
cn
 cn
D. cn 2
E. 1 2
cn
Quantum mechanics
Ch 2.2
Time-independent potential
Expectation value for the energy:

*

  

H     cm  m  H   cn  n 
  n1

  m 1
^
^



H   c c
m 1 n 1
^
*
m n
H y n  Eny n


^
 H  n dx
*
m

^


H   cm* cn Ene
m1 n 1
^

H   cn En
n 1
2
i ( En  Em )
t
 nm
Quantum mechanics
Ch 2.2
Infinite square well
V(x)=0 for 0<x<a
V=∞
0
a
The particle can only exist
in this region
else
x
Shape of the
wave function?
Quantum mechanics
Ch 2.2
Infinite square well
Solutions to Schrödinger equation:
d y


E
y
2
2m dx
2
2
d 2y
2


k
y
2
dx
Simple harmonic oscillator
differential equation
with
k
2mE
Quantum mechanics
Ch 2.2
Infinite square well
Solutions to Schrödinger equation:
y ( x)  A sin kx  B cos kx
Boundary conditions:
At x=0:
y (0)  0
At x=a:
y (a)  0
y ( x)  A sin kx
with
n
kn 
a
Quantum mechanics
Ch 2.2
Infinite square well
Possible states and energy values:
2
 n
yn 
sin 
a
 a

x

n2 2 2
En 
2
2ma
Quantization of the energy
Each state yn is associated to an energy En
^
H y n  Eny n
Quantum mechanics
Ch 2.2
Infinite square well
Properties of the wave functions yn:
y 3 , E3
1.They are alternatively
even and odd
around the center
y 2 , E2
2. Each successive state
has one more node
y 1 , E1
3. They are orthonormal
Excited states
Ground state
0
a
*
y
 my n   nm
x
4. Each state evolves in time with the factor
e
 iEn t /
Quantum mechanics
Ch 2.2
Infinite square well
Pb 2.4
x
Pb 2.5
Particle in one stationary state
x
2
p
 x p 
p2
2
Particle in a combination of two stationary states
  x, 0   A(y 1 y 2 )
  x, t    ( x, t )
oscillates in time
2
x
H
p
evolution in time?
expressed in terms of E1 and E2
Quantum mechanics
Ch 2.2
Infinite square well
Expectation value for the energy:
^

H   cn En
2
n 1
The probability that a measurement
2
yields to the value En is
cn
Normalization

c
n 1
n
2
1