1
PHYSICS 13
FINAL EXAM
FALL 2003
NAME___________________________________________
Useful data
and equations
:
Schrödinger equation: (
η
=h/2
π
)
−
h
´
2
2m
d
2
dx
2
ψ
(x)
+
V(x)
ψ
(x)
=
E
ψ
(x)
For two interacting bodies, the m is replaced by reduced mass
µ
and x is the separation.
Quantum mechanical angular momentum |
L
|= [
l(l
+1)]
1/2
η
and
L
z
= m
l
η
Note the
h-bar
or more generally |
J
| = [j(j+1)]
1/2
η
and J
z
= m
j
η
with m
j
= j, j-1, ..., -j
Addition of orbital (
l
) and spin (1/2) angular momenta yields j=
l
-1/2,
l
+1/2.
Addition of
J
1
and
J
2
yields sum
J
with quantum number j = |j
1
-j
2
|,
|j
1
-j
2
|+1, ..., j
1
+j
2
-1, j
1
+j
2
Atomic electron spectroscopic notation: n
l
1 or 2 or…
, where n is principal quantum number,
l
is orbital angular
momentum quantum number,
l
=0,1,2,...,n-1, with notation
l
=0->s, 1->p, 2->d, 3->f,...
Avogadro's Number: N
A
= 6.0 x 10
23
number of atoms, molecules or nuclei in a MOLE (or gram molecular wt)
Non-relativistic Kinetic Energy =mv
2
/2
Magnetic dipole interaction energy =
µ
Z
B
Z
Lorentz Transformation frame S´ moves in the +x direction with V as seen in S:
x=
γ
(x´+Vt´), y=y´, z=z´, t=
γ
(t´+Vx´/c
2
)
where
γ
=1
/
√
(1-v
2
/c
2
)
Doppler effect
λ
´
± =
λ
√
(1+
v/c
)
/
√
(1−
v/c
)
Time dilation t
´
=
γ
t
Relativistic Energy E= m
0
c
2
/
√
(1-v
2
/c
2
) ; Rest energy = m
0
c
2
;
Relativistic Kinetic Energy = E - m
0
c
2
Photon: E=hf ;
c=
λ
f
; k=2
π
/
λ
;
ω
=2
π
f
Electric dipole moment p
E1
= Q r (Q=charge, r=separation)
Average Thermal Energy at Temperature T(Kelvin) = k(Boltzmann)T/2 per degree of freedom
Decay N(t) = N(0)exp(-
λ
t),
λ
=1/(mean lifetime) = ln2/T
1/2
where T
1/2
is the half-life
Nuclear Binding Energy
E
b
= (Zm(p) + Nm(n) - M
A
) c
2
Coulomb Potential k
EM
Q
1
Q
2
/r
Newton Law of Gravitation : potential energy =
−
Gm
1
m
2
/r
Constants
:
Velocity of light: c = 3.00x 10
8
meter/sec
electron charge: e=1.60x10
-19
Coul
Newton gravitational constant G = 6.67x10
-11
J*m/Kg
2
k(Boltzmann) = 1.38x10
-23
J/°K = 8.62x10
-5
eV/°K
Planck’s constant: h = 6.63 x 10
-34
J sec= 4.14 x 10
-15
eV sec
η
= h/2
π
= 1.05 x 10
-34
J sec = 6.58 x 10
-16
eV sec
k
EM
=
1/4
πε
0
= 8.99x10
9
Nm
2
/Coul
2
hc = 1.24 x 10
-6
eV m
Rest energy of electron = m
e
c
2
= 511 keV = 0.511 MeV
m
e
=9.1x10
-31
Kg
Unified mass units
1 u = 1.66x10
-27
Kg
(1u)c
2
= 931.5 MeV
Units:
Energy: 1 J(oule) = 1 Kg m
2
/sec
2
;
1 eV(electron Volt) = 1.6x10
-19
J
CHOOSE 2 OUT OF LAST 3 PROBLEMS (V, VI, VII) i.e drop one
**EXTRA CREDIT Prob.VIII