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```Physics 123
Final Exam
May 8, 2007, Tuesday
8:30 to 10:30 am
Constants and equations for final exam. You may detach this page if you wish.
Speed of light in vacuum c = 3.00 x 108 m/s
Planck's constant h = 6.63 x 10-34 Js = 4.14 x 10-15 eV s
Electron mass me = 9.11 x 10-31 kg=0.511 MeV/c2
Electron charge e = -1.60 x 10-19 C
1 nm = 10-9 m
1 eV = 1.60 x 10-19 J
1 MeV = 106 eV
Ionization energy of hydrogen= 13.6 eV
hc = 1240 eV  nm
h  h / 2  1.055  10 34 J  s  0.66  10 15 eV  s
Geometric Optics Sign convention
Positive
concave
converging
always
always
real
upright
upright
Focal distance for mirrors, f
Focal distance for lenses, f
Distance to object, do
Object height, ho
Distance to image, di
Image height, hi
Magnification, m
Negative
convex
diverging
never
never
virtual
inverted
inverted
 Mirror equation, lens equation
Magnification:
1
1
1


d0 di
f
m
 Focal distance for mirrors: f=r/2,
Waves v=f;
D=D0sin(kx-t+),
hi
d
 i
h0
do
Lens power: P=1/f, f – in m, P in diopter (D= m-1)
k=,
f,
f=1/T
For a wave traveling from a source in 3D: Intensity I1 /I2 =r22/r1 2, Amplitude D1/D2 =r2/r1
Sound level in decibels =10log(I/I0), I0=10-12W/m2
Doppler effect: f’=f/(1-vs/v)
Light as EM wave:
Law of reflection: i= r
c=f
Index of refraction: n=c/v
Total internal reflection: n1 sin1>n2
Snell’s law: n1 sin1=n2 sin 2
Wave reflection: shift in phase by  if n1<n2, no shift if n1>n2
Wave in medium:
Phase shift over extra path =2l/,
n=/n
Young’s experiment: extra path l=dsin
Constructive interference if relative phase =2m
m=0,1,2,3,4…
Rayleigh criterion for angular resolution =1.22/D
Half-width of the first diffractive maximum sin=/D
1
destructive if =2 (m+1/2)
Special relativity:
t0
t 
,
1 v2 / c2
m0c 2
E
,
1 v2 / c2
ΔL  L0 1  v 2 / c 2
m0v
p
KE  E  m0c 2 ,
1 v2 / c2
E 2  p 2c 2  m0 c 4
2
Particles and waves
Photons: E= hf = hc/, p= h/,
Photoelectric effect: E = W0+ KE
DeBroglie waves of matter (electrons, tennis balls, ...) = h/(mv)=h/p
Electrons (non-relativistic case): KEe = mev2/2
Compton effect (relativistic electron!): E  E ' K e ,  '   

 
h
(1  cos  ) , p  p ' p e
mc
Quantum mechanics
xp x  h
h 2 d 2
Heisenberg uncertainty principle

U
(
x
)


E

tE  h
2m dx 2
2
2
h k n
n2h2
2
En 

k n  p / h  n / L ;
Particle in a box:  n 
sin k n x ;
2m
8mL2
L
2m(U 0  E )
Tunneling probability T=exp(-2GL),
G
h 2
Wave function: probability to find in dV: |  | 2 dV , normalization – integral over all space  |  | 2 dV  1

Schrödinger equation:
Hydrogen-like ion (nucleus charge Z):
 Z 2e4m 1 
E n   2 2 2 
 8 0 h n 
2
Hydrogen atom: En = -(13.6 eV)/n ,
Bohr’s radius: r0 
 100 
1
r
3
0
h 2 0
1
 0.529  10 10 m
2
Z
Zme
e r / r0 ;  200 
1
32r
3
0
(2  r / r0 )e r / 2 r0
Pr  4r 2 |  | 2
Complex atoms:
n=1,2,3,…
l  0,1,2,3,...(n  1)
ml  l;(l  1);(l  2),...0
Electron shells
s: l=0
p: l=1
3s2 means “2 electrons on n=3 level with l=0 ”
Maximum number of states: 2n2 electrons on nth level;
Some trigonometry:
Cos(2x)=Cos2x-Sin2x
2Sin2x= 1-Cos(2x)
Sin(2x)=2Cos(x)Sin(x)
2Cos2x=1+Cos(2x)
d: l=2
2
f: l=3
2(2l+1) electrons on nth level on lth shell
d(Cos(x))/dx= - Sin(x)
d(Sin(x))/dx= Cos(x)
Sin(a+b)=Sin(a)Cos(b)+Cos(a)Sin(b)
ms  1 / 2
```
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