4. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Review 2
Karine Chesnel

5. EXAM II When: Tu Oct 23 – Fri Oct 26 Where: testing center
Quantum mechanics
EXAM II
When: Tu Oct 23 – Fri Oct 26
Where: testing center
Time limited: 3 hours
Closed book
Closed notes
Useful formulae provided

6. EXAM II 1. The delta function potential 2. The finite square potential
Quantum mechanics
EXAM II
1. The delta function potential
2. The finite square potential
(Transmission, Reflection)
3. Hermitian operator, bras and kets
4. Eigenvalues and eigenvectors
5. Uncertainty principle

7. 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

8. ( ) Square wells and delta potentials 2 y a h m dx d - = ÷ ø ö ç è æ D
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

9. The delta function potential
Quantum mechanics
The delta function potential
For

10. The delta function well
Quantum mechanics
The delta function well
Bound state

11. The delta function well/ barrier
Ch 2.5
Quantum mechanics
The delta function well/ barrier
Scattering state A F B x
“Tunneling”
Reflection coefficient
Transmission coefficient

12. The finite square well Quantum mechanics x -V0 Bound state
Symmetry considerations
The potential is even function about x=0
V(x)
The solutions are either even or odd! x
-V0

13. Quantum mechanics
The finite square well
Bound states
where

14. 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

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

16. Linear transformation
Quantum mechanics
Formalism
Wave function
Vector
Linear transformation
(matrix)
Operators
Observables are Hermitian operators

17. Eigenvectors & eigenvalues
Quantum mechanics
Eigenvectors & eigenvalues
For a given transformation T, there are “special” vectors for which:
is an eigenvector of T
l is an eigenvalue of T

18. Eigenvectors & eigenvalues
Quantum mechanics
Eigenvectors & eigenvalues
To find the eigenvalues:
We get a Nth polynomial in l: characteristic equation
Find the N roots
Spectrum
Find the eigenvectors

19. Hilbert space Quantum mechanics Infinite- dimensional space
N-dimensional space
Hilbert space: functions f(x) such as
Inner product
Orthonormality
Schwarz inequality

20. The uncertainty principle
Quantum mechanics
The uncertainty principle
Finding a relationship between standard deviations
for a pair of observables
Uncertainty applies only for incompatible observables
Position - momentum

21. The uncertainty principle
Quantum mechanics
The uncertainty principle
Energy - time
Derived from the
Heisenberg’s equation
of motion
Special meaning of Dt

22. Different notations to express the wave function:
Quantum mechanics
The Dirac notation
Different notations to express the wave function:
Projection on the energy eigenstates
Projection on the position eigenstates
Projection on the momentum eigenstates

23. The Dirac notation Bras, kets Operators, projectors Quantum mechanics
= inner product
= matrix (operator)
Operators, projectors
projector on vector en