1. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Oct 4, 2012
Karine Chesnel

2. HW # 10 extended until Friday Oct 5 by 7pm
Quantum mechanics
Announcements
Homework
HW # 10 extended until Friday Oct 5 by 7pm
Pb 2.33, 2.34, 2.35

3. The finite square well Quantum mechanics V(x) Scattering states -a a x
Bound states
-V0

4. The finite square well Quantum mechanics V(x) x -V0 Ch 2.6
Continuity at boundaries
V(x) x
-V0
is continuous
X=+a
X=-a

5. The finite square well Quantum mechanics Ch 2.6 Scattering state For
Outside the well
For
For

6. The finite square well Quantum mechanics Ch 2.6 Scattering state For
General solution

7. The finite square well Quantum mechanics x -V0 Scattering state +a -a
(2)
(1)
V(x)
(3) -a +a x A B F
C,D
(1)
(3)
(2)
-V0

8. The finite square well Quantum mechanics x -V0
Continuity at boundaries
V(x) x
C,D A F B
-V0
at x = +a
at x = -a
Continuity of
Continuity of

9. The finite square well Quantum mechanics x -V0 Finally
Continuity at boundaries
V(x) x
C,D A F B
-V0
Finally

10. The finite square well Quantum mechanics x -V0 A F B Scattering state
V(x) x A F B
-V0
Coefficient of transmission

11. The finite square well Quantum mechanics x
The well becomes transparent (T=1)
when
V(x) x A F B
-V0
Coefficient of transmission

12. Quiz 14 True B. False Quantum mechanics
We have seen that the coefficient of transmission oscillates with energy,
and that the well becomes ‘transparent” for a particle in a scattering state
when its energy equals specific values En.
Similarly, we can show that the coefficient of reflection oscillates and
the well becomes like a perfect wall, so the particle is totally reflected
for some other specific values of energy En’.
True
B. False

13. Transmission versus energy
Ch 2.5
Quantum mechanics
Transmission versus energy
Transmission coefficient
Delta function well

14. Square wells and delta potentials
Quantum mechanics
Square wells and delta potentials
V(x) x
Scattering
States E > 0
V(x) x x
V(x)
-V0 -a +a
Bound states
E < 0

15. Square wells and delta potentials
Quantum mechanics
Square wells and delta potentials
V(x)
Physical considerations
Scattering
States E > 0 x
Symmetry considerations
Bound states
E < 0

16. ( ) Square wells and delta potentials 2 y a h m dx d - = ÷ ø ö ç è æ D
Ch 2.6
Quantum mechanics
Square wells and delta potentials
Continuity at boundaries
is continuous
is continuous except where V is infinite ( ) 2 y a h m dx d - = ÷ ø ö ç è æ D
Delta functions
Square well, steps, cliffs…
is continuous

17. Square wells and delta potentials
Ch 2.6
Quantum mechanics
Square wells and delta potentials
Finding a solution
Scattering states:
Find the relationship between transmitted wave
and incident wave
Transmission coefficient
Tunneling effect
Bound states
Find the specific values of the energy

18. Ch 2.6
Quantum mechanics
Square barrier
V(x) V0 x -a +a
Pb. 2.33

19. The finite square barrier
Phys 451
The finite square barrier
Scattering states
V(x) x
-V0 A B F
Pb. 2.33
for
Coefficient of transmission
for
for

20. transmission coefficient
Ch 2.6
Quantum mechanics
“Step” potential and “cliff” V0 x
V(x)
-V0 x
V(x)
Pb. 2.35
Pb. 2.34
With a different
definition for the
transmission coefficient
Analogy to
physical potentials