De Broglie's postulate - wave-like properties of particles
Wave-particle duality
Electromagnetic radiation can exhibit both particle and wave properties. It appears dicult to reconcile
these facts.
The full paradox of particle-wave duality is illustrated
Models of an atom and old quantum theory
Classical models of atoms
Thompson's model
Chemical elements dier by the number Z of electrons in their atoms. Atoms are electrically neutral,
so that the charge Ze of all electrons in an atom must be compensated
Mid-term Exam Solutions
1. (a) The explorer travels in two stages, the trip to the star and the trip back home. Each trip lasts
t = 40 years as observed on earth, but it corresponds to a shorter time experienced by the explorer:
t=
1
v2
c2
The total time
Homework 3: Quantization of electromagnetic radiation
1. A typical chemical bond in a biological molecule has a strength of a few electron-volts, say 4 eV to be
specic.
(a) Can low-intensity microwave radiation with a wavelength 1 cm break a bond, thus ca
Homework 6: Schrodinger equation
1. Important properties of complex numbers:
(a) Use Euler's formula ei = cos + i sin to express a complex number z = x + iy as z = rei . Write
x and y in terms of r and , and vice versa.
(b) Show for any complex number z =
Homework 4: Matter waves
1. Consider a head-on elastic collision between a photon of momentum p0 and a stationary free electron.
(a) Calculate the momentum p of the photon after collision assuming that it bounces directly back (in
the direction opposite t
Special theory of relativity
Electrodynamics and Michelson-Morley experiment
At the end of the 19th century physics was solidly shaped by two great theories: Newton's mechanics
and Maxwell's electrodynamics. Maxwell's equation provided a complete descript
Relativistic dynamics
Lorentz transformations also aect the accelerated motion of objects under the inuence of forces. In
Newtonian physics a constant force
that the velocity
v = at
F
accelerates an abject at a constant rate
a = dv/dt = F/m,
of the object
Solutions to homework 4: Matter waves
1. (a) The incoming photon with momentum
pe
p0 kicks the electron and bounces back, giving a momentum
p0 . Projecting all momenta onto the positive x axis, the
to the electron in the same direction as
momentum conserv
Solutions to homework 3: Quantization of electromagnetic radiation
1. (a) No. Energy of a photon with = 1 cm is 0.12 meV, which is not enough to break a bond.
(b) hc/ > Ebond < hc/Ebond = 310 nm
(c) This is an ultra-violet wavelength.
2. (a) E = hc/ = Tou
Solutions to homework 5: Atomic spectra
1. The Rydberg formula for the Hydrogen atom emission spectrum is
1
=R
1
1
2
m2
n
,
n>m1
Each series of emission wavelengths, obtained by xing m and varying n, is bounded from both above
and below:
R
1
1
R
2
m
1
1
z = rei = r(cos + i sin ) = x + iy
x = r cos y = r sin
x2 + y 2 = r2 (cos2 + sin2 ) = r2 r = x2 + y 2 y/x = tan
= arctan(y/x) f (z) = arctan(z)
2
z (/2, /2)
arctan 2
x y
(, )
+ arctan
y
arctan x
= arg(x + iy) =
+ arctan
y
x
, x<
2m = 2
E = 2mc2 M
h
E = M gh g 9.8 2
M=
2mc2
1.83 1010
gh
E = mc2 E0 = mc2
E
1
=
2
E0
1 (v/c)
=
v2
=
c2
1
v=c
t=
l
=
v
2
E0
E
E0
E
1
2
l
l
c
1
E0 2
E
tl
=
E0 2
E
1
l = ctl t
tl = 4 l t 4.052 E0
Schrodinger equation in three dimensions
The stationary Schrodinger equation in three dimensions is a partial dierential equation involving three
coordinates per particle.
The mathematical complexity behind such an equation can be intractable
by analytica
Quantum Mechanics
Dynamics of matter waves
We learned that the discoveries such as blackbody radiation and photoelectric eect led to a general
conclusion that energy of electromagnetic waves is transferred and transmitted in discrete quanta.
The quantizat