Transcript 2B22 Revision Lectures - University College London

PHAS2222 Revision Lecture
• The plan:
– First hour:
• Summary of main points and equations of course
• Opportunity to request particular topics
– Second hour:
• Specially requested topics
• The 2007 exam paper (mainly Section B)
• Note: model answers to exams not available on
website, but I am happy to give feedback on
attempts at past examination papers
PHAS2222 Revision Lecture 2008
Photo-electric effect, Compton
scattering
E  h
p
Particle nature of light in
quantum mechanics
Davisson-Germer experiment,
double-slit experiment
h

Wave nature of matter in
quantum mechanics
Wave-particle duality
Postulates:
Time-dependent Schrödinger
Operators,eigenvalues and
equation, Born interpretation
eigenfunctions, expansions
2246 Maths
Separation of
in complete sets,
Methods III
variables
Time-independent Schrödinger
commutators, expectation
Frobenius
equation
values, time evolution
method
Quantum simple
Legendre
harmonic oscillator
Hydrogenic atom
1D problems
equation 2246
En  (n  12 ) 0
Radial solution
Rnl , E  
2
1Z
2 n2
Angular solution
Yl m ( ,  )
Angular momentum
operators
Lˆz , Lˆ2
Section 1 – Failure of classical
mechanics
The photoelectric effect:
E  h (1.1)
Kmax  h  W (1.2)
De Broglie’s relationship for matter
waves:
h
p

Compton scattering: when photon deflected through
angle θ, new wavelength is
 '  
h
(1  cos  ) (1.4)
me c
Diffraction from crystal surfaces and double-slit experiments:
Maximum scattering when path difference = nλ
Verifies
Section 2 – A wave equation for
matter waves
Time-dependent Schrödinger equation:
2

2
i

(2.2)
t
2m x 2
(for matter waves in free space)
i
Hamiltonian operator (represents energy of
particle):


d
2
2

2

 V ( x, t ) (2.3)
t
2m x 2
(generally)
2
ˆ
  2m dx 2  V ( x, t )    H  (2.4)


Born interpretation: probability of finding particle in a small length δx at
position x and time t is equal to
2
 ( x, t )  x (2.6)
If Hamiltonian is independent of time,
can try solution
( x, t )   ( x)T (t ) (2.9)
(timeindependent
Find
ˆ
T (t )  exp(iEt / )
H  E (2.13)
SE)
Uncertainty principle:
xp 
2
(strict version)
Section 3 – Examples of the timeindependent SE in 1 dimension

Free particle
V 0
3. Have a finite normalization
integral.
Rectangular barrier
V 
V(x)
I
-a
I
2. Have a continuous first derivative
(unless the potential goes to infinity)
Infinite square well
V(
x)
Finite square well
The wavefunction must:
1. Be a continuous and singlevalued function of both x and t
Travelling waves,
arbitrary value of
energy
V 
d 2
 V ( x)  E (2.12)
2m dx 2
2
II
a
Quantization of energy
V0
III
II
V(x
)
III
V0
Matching of solutions: a
travelling waves (sines or
cosines) in well, exponentials
in barriers
Potential step
V(x)
V0
Tunnelling
Transmission
and reflection
x
Section 3 – contd
Particle flux at position x
Particle flux (flow of
probability):
i  * 
* 
( x ) 



2m 
x
x 
2
Simple harmonic oscillator:
 m0 
y 



1/ 2
x
2
d
2 2
1
Hamiltonian Hˆ  

m

0x
2
2
2m dx
Substitute  ( y)  H ( y)exp( y 2 / 2)
Series solution for H(y) must
terminate, so H is a finite power
series (polynomial) – called a Hermite
polynomial.
Termination condition
1
Energy E  (n  ) 0
2
Section 4 – Postulates of quantum
mechanics
Postulate 4.1: Existence of wavefunction, related to probability
density by Born interpretation.
Postulate 4.2: to each observable quantity is associated a linear, Hermitian
operator (LHO). The eigenvalues of the operator represent the possible results
of carrying out a measurement of the corresponding quantity. Immediately
after making a measurement, the wavefunction is identical to an eigenfunction
of the operator corresponding to the eigenvalue just obtained as the
measurement result.
Eigenfunction: Oˆn ( x)  onn ( x)
Hermitian operator:



Postulate 4.3: the operators representing the position
and momentum of a particle are
xˆ  x
pˆ x  i

x
 * ˆ

* ˆ
f (Og )dx    g (Of )dx 
 

*
(4.2)
Section 4 - contd
The eigenfunctions of a Hermitian operator belonging to
different eigenvalues are orthogonal.
If
Qˆn  qnn ; Qˆm  qmm with qn  qm

then
The eigenfunctions φn of a Hermitian operator
form a complete set, meaning that any other
function satisfying the same boundary
conditions can be expanded as
Note that
 an  1
2

 ( x)   ann ( x)
Postulate 4.4: suppose a measurement of the quantity Q is made,
and that the (normalized) wavefunction can be expanded in terms of
the (normalized) eigenfunctions φn of the corresponding operator as
 ( x)   ann ( x).
Then the probability of obtaining the eigenvalue qn as the
measurement result is a
2
n
n
if ψ normalized, φ orthonormal.
n
n
*

 nmdx  0
Section 4 - contd
ˆ ˆ  RQ
ˆˆ
Qˆ , Rˆ   QR


Commutator:
Commuting operators have same eigenfunctions,
can have well-defined values simultaneously
(‘compatible’)
Expectation value:
Q   an qn .
2
n
Postulate 4.5: Between measurements (i.e. when it is not disturbed by
external influences) the wave-function evolves with time according to
the time-dependent Schrődinger equation.
Time development in terms of eigenfunctions of Hamiltonian:
If
Hˆ  n  En n
and
( x,0)   an n ( x)
n
then
( x, t )   an exp(iEnt / ) n ( x)
n
Section 5 – Angular momentum
Different components do not commute:
[ Lˆx , Lˆ y ]  i Lˆz but
 Lˆ2 , Lˆz   0


In spherical polar coordinates:
Lz  i


L2  
2
 1  
 
1 2 
sin



 sin   
  sin 2   2 


Their simultaneous eigenfunctions
are spherical harmonics:
Lz
Ly
LˆzYl m ( , )  m Yl m ( , ),  l  m  l
Lˆ2 Yl m ( , )  l (l 1) 2Yl m ( , )
Conserved for problems with
spherical symmetry
Lx
Section 6 – The hydrogen atom
2
2

Ze
Hˆ 
2 
2me
4 0 r
Now look for solutions in the form
 (r, ,  )  R(r )Y ( ,  )
Angular parts are spherical harmonics
Radial part:
R (r ) 
 (r )
r
Y ( , )  Yl m ( , )
 2 d2 
2me dr 2
with
Atomic units:
 Veff (r )   E 
Ze2 l (l  1) 2
Veff (r ) 

4 0 r
2me r 2
Planck constant  1 (dimensions [ ML2T 1 ])
Electron mass me  1 (dimensions [ M ])
Constant apearing in Coulomb's law
e2
4 0
 1 (dimensions [ ML3T 2 ])
Section 6 - contd
Put
 (r )  F (r ) exp( r )
with E  
2
2
Series solution for F must terminate:
possible only if
Z
 n where n is an integer  l : n  l  1, l  2

Z2
En   2 (in atomic units)
2n

Z2 
E  in units
Eh 
2


0
n is principal quantum number
l=0,1,2,…,n-1
m  l , (l  1),
0,
(l  1), l
-1
l=0 l=1 l=2 l=3
Section 7 - Spin
Interaction with magnetic field:
B 
e
(the Bohr magneton).
2me
Stern-Gerlach experiment: atoms
with single outer electron divide into
two groups with opposite magnetic
moments.

ˆ  gSˆ )
Hˆ  Hˆ 0  B B  (L
g  2 (Dirac's relativistic theory)
g  2.00231930437 (Quantum Electrodynamics)
Coupling of spin to orbital
angular momentum:
Lz
|L-S|
S
Ly
Quantum numbers describing spin:
L
1
(for electron); like l in ordinary angular momentum
2
1
L+S
ms   s s   (for electron); like m in ordinary angular momentum
2
s
Lx
Section 7 - contd
Total angular momentum (orbital +
spin)
S
J  LS
L
Described by two quantum numbers:
•j (determining quantity of total angular momentum
present); ranges from |l-s| to l+s in integer steps
•mj (determining projection of total angular
momentum along z), ranges from –j to +j in integer
steps