Download Sep 10 - BYU Physics and Astronomy

Physics 451
Quantum mechanics I
Fall 2012
Sep 10, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework
• Homework 4: T Sep 11 by 7pm
Pb 1.9, 1.14, 2.1, 2.2
• Homework 5: Th Sep 13 by 7pm
Pb 2.4, 2.5, 2.7, 2.8
Quantum mechanics
No student assigned to the following transmitters:
2214B68
17A79020
1E5C6E2C
1E71A9C6
Please register your i-clicker at the class website!
Quantum mechanics
Ch 1.6
Quiz 4a
Uncertainty principle
Which statement is accurate
for these electronic wave functions?
A. Both the position x and the momentum
p are fairly well defined
B. The position of the particle is fairly well defined but the momentum is poorly defined
C. The momentum of the particle is fairly well defined but the position is poorly defined
D. Both the position and the momentum are poorly defined.
Quantum mechanics
Ch 1.6
Uncertainty principle
Position
x
Momentum
p  i

x
p  2m  E  V 
particle
 x p 
p 
2

wave
2

h

De Broglie
formula
1924
Heisenberg’s uncertainty
Principle 1927
Quantum mechanics
Ch 1.6
Uncertainty principle
How to check the uncertainty principle?
• Calculate
x
x2
and
x
• Calculate
p
p 2 and
p
• Estimate the product
• Compare to
 x p
2
Pb 1.9
Quantum mechanics
Ch 1
Probability current
 ( x , t )   ( x, t )   * 
2
Density of probability
b
Probability between two points
Pab    ( x, t )dx
a
where
dPab
 J  a, t   J (b, t )
dt
i   *
 
J  a, t  
*


2m 
x
x 
Pb 1.14
Quantum mechanics
Ch 2.1
Time-independent
Schrödinger equation
2

2
i

V 
2
t
2m x
In general
V ( x, t )
Here
V ( x)
function of x only
The potential is independent of time
General solution:
 ( x, t )   ( x) (t )
“Stationary state”
Quantum mechanics
Ch 2.1
Time-independent
Schrödinger equation
Plugging the
general solution:
 ( x, t )   ( x) (t )
in the Schrödinger
equation
2
1 d
1 d 2
i

V
2
 dt
2m  dx
Function of
time only
Function of
space only
E
Quantum mechanics
Ch 2.1
Time-independent
Schrödinger equation
• Time dependent part:
1 d
i
E
 dt
d
iE
 
dt
General solution:
 (t )  e

iE
t
Quantum mechanics
Ch 2.1
Time-independent
Schrödinger equation
1 d 2

V  E
2
2m  dx
2
• Space dependent part:
d 2

 V ( x)  E
2
2m dx
2
Solution (x) depends on the potential function V(x).
Global solution:
( x, t )   ( x)eiEt /
Stationary state
Quantum mechanics
Quiz 3b
“If the particle is in one stationary state,
its expectation value for position is not changing in time.”
A. True
B. False
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 )   ( x)eiEt /

Q   Q ( x ,
) 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)   E

2
 2m dx

^
H
^
^
H   H  dx  E  * dx  E
^
*
^
H 2   * H 2  dx  E 2  * 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 )   n ( x)eiEnt /
• Associated expectation value for energy

H   cn2 En
n 1
Quantum mechanics
Ch 2.1
Stationary states
 n ( x, t )   n ( x)eiEnt /
Pb 2.1
a) En must be real
b) n(x) can always be real
c) n(x) is either real or odd, when V(x) is even
Pb 2.2
En  Vmin
Classical analogy:
The kinetic energy is always positive!
However, in QM, it is possible that
En  V ( x)
at some locations x