physics 451 quantum mechanics i fall 2012 review 2 karine chesnel
TRANSCRIPT
EXAM II
Quantum mechanics
• Time limited: 3 hours• Closed book• Closed notes• Useful formulae provided
When: Tu Oct 23 – Fri Oct 26Where: testing center
EXAM II
Quantum mechanics
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
Quantum mechanics
Square wells and delta potentials
V(x)
x
Bound statesE < 0
ScatteringStates E > 0
Symmetry considerations
even evenx x
odd oddx x
Physical considerations
ikxreflected x Be
ikxincident x Ae
ikxtransmitted x Fe
Quantum mechanics
Square wells and delta potentials
Continuity at boundaries
Delta functions
Square well, steps, cliffs…
dx
d
is continuous
is continuous except where V is infinite
022
m
dx
d
dx
d
is continuous
is continuous
Quantum mechanics
Scattering state 0E
Ch 2.5
0
ikx ikxleft x Ae Be ikx
right x Fe
A F
Bx
Reflection coefficient Transmission coefficient
2 2
1
1 2 /R
E m
2 2
1
1 / 2T
m E
The delta function well/ barrier
V x x
“Tunneling”
Quantum mechanics
The finite square well
V(x)
x
-V0
Symmetry considerations
The potential is even function about x=0
cos
kx
kx
Ae
D lx
Ae
even
The solutions are either even or odd!
Bound state
0E
Quantum mechanics
The finite square well
Scattering state 0E
V(x)
x
-V0
A
B
FC,D
-a +a
(1)
(2)
ikx ikx
ilx ilx
ikx
Ae Be
Ce De
Fe
(1)
(3)
(2)
(3)
Quantum mechanics
The finite square well
1
220
00
21 sin 24
V aT m E V
E E V
V(x)
x
-V0
A
B
F
Coefficient of transmission
2 22
0 22 (2 )nE V nm a
The well becomes transparent (T=1)
when
Formalism
Quantum mechanics
ˆijH H Linear transformation
(matrix)Operators
Wave function Vector
Observables are Hermitian operators †Q Q
Quantum mechanics
Eigenvectors & eigenvalues
For a given transformation T, there are “special” vectors for which:
T a a
a is an eigenvector of T
is an eigenvalue of T
Quantum mechanics
Eigenvectors & eigenvalues
0T I a
det 0T I
To find the eigenvalues:
We get a Nth polynomial in : characteristic equation
Find the N roots 1 2, ,... N Spectrum
Find the eigenvectors 1 2, ,... Ne e e
Quantum mechanics
Hilbert space
N-dimensional space
1 2 3, , ,... Ne e e e
Infinite- dimensional space
1 2 3, , ... ...n
Hilbert space: functions f(x) such as 2
( )b
a
f x dx
*( ) ( )f g f x g x dx
Inner product
Schwarz inequality f g f g
2 2*( ) ( ) ( ) ( )f x g x dx f x dx g x dx
Orthonormalitym n nmf f
Quantum mechanics
The uncertainty principle
2
2 2,
2A B
A B
i
Finding a relationship between standard deviations for a pair of observables
Uncertainty applies only for incompatible observables
Position - momentum 2x p
Quantum mechanics
The uncertainty principle
Energy - time
2E t
Special meaning of t
Qtd Q
dt
,d Q i Q
H Qdt t
Derived from the Heisenberg’s equation
of motion
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
/niE tn n
n
c e
( , ) ( )y t x y dy
/1( , )
2ipxp t e dp