Physics 451 Quantum mechanics I Fall 2012 Sep 10, 2012 Karine Chesnel
Feb 25, 2016
Physics 451Quantum mechanics I
Fall 2012
Sep 10, 2012
Karine Chesnel
Announcements
Quantum mechanics
• 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
Homework
No student assigned to the following transmitters:
Quantum mechanics
Please register your i-clicker at the class website!
2214B6817A790201E5C6E2C1E71A9C6
Uncertainty principle Quiz 4a
Which statement is accuratefor these electronic wave functions?
A. Both the position x and the momentum 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.
p
Quantum mechanics Ch 1.6
Uncertainty principle
Position
x
2x p
Heisenberg’s uncertaintyPrinciple 1927
2p m E V
particle
De Broglie formula
1924
2 hp
wave
Momentum
p ix
Quantum mechanics Ch 1.6
Uncertainty principle
Quantum mechanics Ch 1.6
Pb 1.9
How to check the uncertainty principle?
• Calculate and x 2xx
• Calculate and p 2pp
• Estimate the product x p
• Compare to 2
Quantum mechanics Ch 1
Probability current
Pb 1.14
Density of probability2( , ) ( , ) *x t x t
Probability between two points ( , )b
aba
P x t dx
, ( , )abdPJ a t J b t
dt
where *, *2iJ a tm x x
Quantum mechanics Ch 2.1
Time-independentSchrödinger equation
2 2
22i V
t m x
In general ( , )V x t Here function of x only The potential is independent of time
( )V x
General solution: ( , ) ( ) ( )x t x t “Stationary state”
Quantum mechanics Ch 2.1
Time-independentSchrödinger equation
2 2
2
1 12
d di Vdt m dx
Plugging the general solution: ( , ) ( ) ( )x t x t in the Schrödinger
equation
Function of time only
Function of space only
E
Quantum mechanics Ch 2.1
Time-independentSchrödinger equation
1 di Edt
d iEdt
• Time dependent part:
General solution: ( )iE t
t e
Quantum mechanics Ch 2.1
Time-independentSchrödinger equation
Solution (x) depends on the potential function V(x).
• Space dependent part: 2 2
2
12
d V Em dx
2 2
2 ( )2d V x E
m dx
Global solution:/( , ) ( ) iEtx t x e Stationary state
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
Quantum mechanics Ch 2.1
Stationary states
Properties:
• Expectation values are not changing in time (“stationary”):
/( , ) ( ) iEtx t x e * ( , )Q Q x dxi x
with
* ( , )Q Q x dxi x
Q is independent of time
0d x
p m v mdt
The expectation value for the momentum is always zero
In a stationary state!
p(Side note: does not mean that and x are zero!)
Quantum mechanics Ch 2.1
Stationary states
Properties:
2 2
2 ( )2
d V x Em dx
• Hamiltonian operator - energy
^ ^* *H H dx E dx E
^ ^2 * 2 2 * 2H H dx E dx E
^H
0H
Quantum mechanics Ch 2.1
Stationary states
• General solution
1
( , ) ( , )n nn
x t c x t
/( , ) ( ) niE tn nx t x e
where
• Associated expectation value for energy 2
1n n
n
H c E
Quantum mechanics Ch 2.1
Stationary states
Pb 2.1
/( , ) ( ) niE tn nx t x e
a) En must be realb) n(x) can always be real
c) n(x) is either real or odd, when V(x) is even
Pb 2.2 minnE VClassical analogy: The kinetic energy is always positive!
However, in QM, it is possible that at some locations x( )nE V x