PHYSICS 2D
PROF. HIRSCH
QUIZ 6
WINTER QUARTER 2009
MARCH 13, 2009
Formulas:
Time dilation; Length contraction : "t = #"t'$ # "t p ;
; c = 3 %10 8 m /s
L = Lp /#
1
!
!
Lorentz transformation : x'= " (x # vt) ; y' = y ; z' = z ; t'= " (t # vx /c 2 ) ; inverse : v $ -v
Spacetime interval : ("s) 2 = (c"t) 2 - ["x 2 + "y 2 + "z 2 ]
uy
ux " v
Velocity transformation : ux '=
;
u
'=
; inverse : v $ -v
y
1" ux v /c 2
# (1" ux v /c 2 )
Relativistic Doppler shift : f obs = f source 1+ v /c / 1" v /c (approaching)
!
Momentum, energy (total, kinetic, rest) : p = # m u; E = # mc 2 ; K = (# $1)mc 2 ;E 0 = mc 2 ;E = p 2c 2 + m 2c 4
Electron : me = 0.511 MeV /c 2
Proton : mp = 938.26 MeV /c 2
Neutron : mn = 939.55 MeV /c 2
!
!
!
!
!
"
"
2
; electron volt : 1eV = 1.6 "10 -19 J
! Atomic mass unit : 1 u = 931.5 MeV /c
4
Stefan's law : etot = "T , etot = power/unit area ; " = 5.67 #10$8 W /m 2K 4
#
hc
etot = cU /4 , U = energy density = $ u( ",T)d" ;
Wien's law : "m T =
4.96kB
0
#E n / kB T
Planck : E n = nhf ; probability P(E n ) " e
; E ( $,T) = % E n P(E n ) / % P(E n )
n
!
!
!
!
!
!
!
!
!
Compton scattering : "'- " =
Electrostatics : F =
h
h
(1# cos $ );
= 0.0243A ; Coulomb constant : ke 2 = 14.4 eV A
mec
mec
kq1q2
kq q
(force) ; U = 1 2 (potential energy)
2
r
r
r
r
r
r
Force in electric and magnetic fields (Lorentz force) : F = qE + qv " B ;
Drag force : D = 6#a$v
C
Rutherford scattering : "n = 4
;
hc = 1,973 eV A
sin (# /2)
1
1
1
1
Hydrogen spectrum :
= R( 2 # 2 )
;
R = 1.097 $10 7 m#1 =
"mn
m
n
911.3A
2
2
2
2
ke Z
Z E
ke
mv
ke 2 Z
ke 2 Z
Bohr atom : E n = "
= " 2 0 ; E0 =
= 13.6eV ; K = e =
; U ="
2rn
n
2a0
2
2r
r
!
hf = E i " E f ; rn = r0 n 2 ; r0 =
!
de Broglie : " =
!
n
8$
hc / "
Planck's law : u( ",T) = N( ") # E ( ",T) = 4 # hc / "kB T
"
e
%1
Photons : E = hf = pc ; f = c / " ; hc = 12,400 eV A ; k B = (1/11,600)eV /K
Photoelectric effect : eVs = K max = hf " # , # $ work function
h
E
;f =
p
h
a0
Z
; a0 =
h2
= 0.529A ; L = me vr = nh angular momentum
me ke 2
; # = 2$f ; k =
2$
;
"
Wave packets : y(x,t) = $ a j cos(k j x " # j t), or y(x,t) =
E = h# ; p = hk ;
% dk a(k) e
i(kx -# (k )t )
E=
p2
2m
; &k&x ~ 1 ; &#&t ~ 1
j
!
!
group and phase velocity : v g =
d"
"
; vp =
;
dk
k
Heisenberg : #x#p ~ h ; #t#E ~ h
b
Born interpretation : P(x)dx =| " (x,t) |2 ; P(a # x # b) =
a
!
!
&
2
$ | " (x,t) |
dx ;
< f (x) >= $ | " (x,t) |2 f (x)
%&
PHYSICS 2D
PROF. HIRSCH
QUIZ 6
WINTER QUARTER 2009
MARCH 13, 2009
E
!
!
!
-i t
h2 " 2#
"#
Schrodinger equation : + U(x)#(x,t) = ih
;
#(x,t) = $ (x)e h
2
2m "x
"t
%
h 2 # 2$
Time " independent Schrodinger equation : +
U(x)
$
(x)
=
E
$
(x)
;
& dx $ *$ = 1
2m #x 2
-%
2 2 2
2
2
n$x
$ hn
h
" square well : # n (x) =
sin(
) ; En =
;
= 3.81eVA 2 (electron)
2
L
L
2mL
2me
Eigenvalues and eigenfunctions : [Q] " = q " (q is a constant) ; uncertainty : #Q = < Q2 > $ < Q > 2
m$ 2
#
x
1
p2 1
1
Harmonic oscillator : "n (x) = H n (x)e 2h ; E n = (n + )h$ ; E =
+ m$ 2 x 2 = m$ 2 A 2 ; %n = ±1
2
2m 2
2
x2
!
Tunneling :
!
!
!
!
%
-2 # (x )dx
" (x) ~ e -# x
;
T = e -2#$x ;
T =e
;
x1
!
!
!
!
!
!
!
!
2m[U(x) - E]
h2
(k1 " k 2 ) 2
2m
Step potential : reflection coef : R =
, T = 1" R ; k =
(E " U)
2
(k1 + k 2 )
h2
r
r
r
r -i E t
h2 2
$#
Schrodinger equation in 3D : " # + U( r )#( r ,t) = ih
;
#( r ,t) = % ( r )e h
2m
$t
2 2
# h n12 n 22 n 32
3D square well : "(x,y,z) = "1 (x)"2 (y)"3 (z) ; E =
( +
+ )
2m L12 L22 L23
Spherically symmetric potential: "n,l,m l (r,#, $ ) = Rnl (r)Ylm l (#, $ ) ; Ylm l (#, $ ) = Plm l (# )e im l$
r r r
h #
Angular momentum : L = r " p ; [Lz ] =
; [L2 ]Ylm = l(l + 1)h 2Ylm ; [L z ] = mh
i #$
ke 2 Z 2
Radial probability density : P(r) = r 2 | Rnl (r) |2 ;
Energy : E n = "
2a0 n 2
1 Z
Ground state of hydrogen and hydrogen - like ions : "1,0,0 = 1/ 2 ( ) 3 / 2 e$Zr / a 0
#
a0
"
"
#e
eh
Orbital magnetic moment : µ =
L ; µz = #µB ml ; µB =
= 5.79 $10#5 eV /T
2me
2me
r
1
"e r
Spin 1/2 : s = , | S |= s(s + 1)h ; Sz = msh ; ms = ±1/2 ;
µs =
gS
2
2me
r
r "e r
r r
Orbital + spin mag moment :
µ=
( L + gS ) ; Energy in mag. field : U = "µ # B
2m
r r
r r
Two particles : "(r1, r2 ) = + /# "( r2 , r1 ) ; symmetric/antisymmetric
Screening in multielectron atoms : Z " Z eff , 1 < Z eff < Z
Orbital ordering:
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 6d ~ 5f
l
!
!
# (x) =
Justify all your answers to all (3) problems
l
PHYSICS 2D
PROF. HIRSCH
QUIZ 6
WINTER QUARTER 2009
MARCH 13, 2009
Problem 1 (10 points)
An electron moves in a two-dimensional rectangular box of side lengths L and 10L, with
L=1A.
(a) Find the ground state energy, in eV, and the difference in energy between the ground
state and the first excited state.
(b) Give the quantum numbers of the 15 lowest energy states. Justify your answer.
(c) Find the quantum numbers of 2 states that are degenerate.
!
Problem 2 (10 points)
The electron in a hydrogen atom (Z=1) is in the state with wavefunction
" (r,# , $ ) = Cr 3e%r / 4 a 0 sin 3 #e i$
where C is a constant.
(a) Give the value of the quantum numbers n,l,ml .
(b) Find the most probable r in terms of a0.
(c) How much more likely is it to find this electron within 0.01A of r=4a0 than it is to find
it within 0.01A of r=2a0?
!
Problem 3 ( 10 points)
An electron in hydrogen is in the presence of a magnetic field of magnitude 10T.
Assuming it is in a state with quantum numbers n = 3, l = 2 ,
(a) Find the difference (in eV) between the highest and lowest energy that this electron
can have. Take into account electron spin but ignore spin-orbit coupling.
(b) Give all the quantum numbers of these two states.
!
(c) If you now included the effect of spin-orbit
coupling, would you expect the difference
in energy between those two states to increase, decrease or stay the same? Explain.
Justify all your answers to all problems