4. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Sep 21, 2010
Karine Chesnel

5. Friday 21: Review - Monday 24: Practice test 1
Quantum mechanics
Announcements
Homework this week:
Thursday Sep 20 by 7pm:
HW # 7 pb 2.19, 2.20, 2.21, 2.22
Friday 21: Review - Monday 24: Practice test 1
Plan to work on your selected problem
with your group and prepare the solution
to be presented in class (~ 5 to 7 min)
Test 1: Mon Sep 24 – Th Sep 27

6. Note from the TA about homework
Answer the problems completely. A lot of the problems
have multiple parts. For example, they first ask you to do
the derivation, and then ask for a qualitative description,
and finally let you give an analog to something.
Don't just do the math and forget everything else.
2. Use precise terminology to describe phenomena.
For example, in problem 2.2 of Homework 4, you are
supposed to comment on the concavity/divergence of the
function. Those are the terms I am looking for.
  Don't write something like "the function dies at infinity".
That is a vague expression and it is also unprofessional Muxue Liu

7. Quiz 9a True False Pb 2.13 Quantum mechanics
Since the operators a+ and a- are shifting the stationary states
from one level to another,
and since the stationary states are all orthogonal,
the expectations values for x and p on any state will ALWAYS be zero!
True
False
Pb 2.13

8. Solving the Schrödinger equation the direct way!
Quantum mechanics Ch 2.3
Harmonic oscillator x
V(x)
Solving the Schrödinger equation the direct way!
(analytic method)

9. Solving the Schrödinger equation the direct way!
Quantum mechanics Ch 2.3
Harmonic oscillator x
V(x)
Solving the Schrödinger equation the direct way!
(analytic method)
General solution
Expanding h in power series

10. Solving the Schrödinger equation the direct way!
Quantum mechanics Ch 2.3
Harmonic oscillator x
V(x)
Solving the Schrödinger equation the direct way!
(analytic method)
Is equivalent to:
Recursion formula

11. Solving the Schrödinger equation the direct way!
Quantum mechanics Ch 2.3
Harmonic oscillator x
V(x)
Solving the Schrödinger equation the direct way!
(analytic method)
Final solution:
Hermite polynomials

12. Quantum mechanics Ch 2.3
Harmonic oscillator

13. Quantum mechanics Ch 2.3
Harmonic oscillator
n=100

14. the energy of a particle is always quantized”
Quantum mechanics
Quiz 9b
“In quantum mechanics,
the energy of a particle is always quantized”
True
False

15. Quantum mechanics Ch 2.4
Free particle
V = 0 everywhere

16. Free particle Quantum mechanics Ch 2.4 with General Solution
Complete wave function

17. Free particle Quantum mechanics Ch 2.4 with
Wave function represents a physical wave:
wave travelling
in the (-x) direction
with speed v
wave travelling
in the (+x) direction
with speed v
with
Velocity of the phase

18. Free particle Quantum mechanics Ch 2.4 Talking about velocity
Velocity of the phase
Analogy with classical velocity
(using the de Broglie formulae)

19. Free particle Quantum mechanics Ch 2.4 Wave packet Normalization
A single wave for a given E is NOT a physical solution!
A superposition of waves IS normalizable!
Wave
packet
Individual
waves
dispersion
function
superposition
(summation)

20. Free particle Pb 2.20 Quantum mechanics Ch 2.4 Wave packet Fourier
transform
Inverse Fourier
transform
Plancherel’s
theorem
Pb 2.20
Extension from discrete sum to continuous integration

21. Free particle Pb 2.21, 2.22 Quantum mechanics Ch 2.4 Method:
1. Identify the initial wave function
2. Calculate the Fourier transform
3. Estimate the wave function at later times
Pb 2.21, 2.22

22. Quiz 9c B. A. C. Quantum mechanicsx -a a
A particle is in a given initial state Y(x,0)
what will be the shape of the Fourier transform F(k)? A. B. C. k
-p/a
p/a k k
-p/a
p/a

23. Free particle Quantum mechanics Ch 2.4 Dispersion relation
where
Dispersion relation
here
Physical interpretation:
velocity of the each wave at given k:
velocity of the wave packet: