PHYSICS 2D
PROF. HIRSCH
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QUIZ 6
Formulas:
Time dilation; Length contraction : "t = #"t'$ # "t p ;
L = Lp /#
; c = 3 %10 8 m /s
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 '=
; uy '=
; inverse : v $ -v
2
1" ux v /c
# (1" ux v /c 2 )
Relativistic Doppler shift : f obs = f source 1+ v /c / 1" v /c (approaching)
r
r
Momentum : p = " mu ; Energy : E = " mc 2 ; Kinetic energy : K = (" #1)mc 2
Rest energy : E 0 = mc 2
;
Electron : me = 0.511 MeV /c 2
E=
p 2c 2 + m 2c 4
Proton : mp = 938.26 MeV /c 2
Neutron : mn = 939.55 MeV /c 2
Atomic mass unit : 1 u = 931.5 MeV /c 2
; electron volt : 1eV = 1.6 "10 -19 J
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/(kB T )
Boltzmann distribution : P(E) = Ce
8$
hc / "
8$f 2
Planck's law : u" ( ",T) = N " ( ") # E ( ",T) = 4 # hc / "kB T
;
N( f ) = 3
"
e
%1
c
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; Bragg equation : n% = 2d sin &
h
h
(1# cos $ );
= 0.0243A ; Coulomb constant : ke 2 = 14.4 eV A
mec
mec
r
r
r r
Force in electric and magnetic fields (Lorentz force) : F = qE + qv " B ;
Drag force : D = 6#a$v
Compton scattering : "'- " =
C
;
hc = 1,973 eV A
sin (# /2)
1
1
1
1
Hydrogen spectrum :
= R( 2 # 2 )
;
R = 1.097 $10 7 m#1 =
"mn
m
n
911.3A
kq q
kq q
mv 2
Electrostatic force, energy : F = 12 2 ; U = 1 2 . Centripetal force : Fc =
r
r
r
ke 2 Z
Z 2E0
ke 2
mev 2
ke 2 Z
Bohr atom : E n = "
= " 2 ; E0 =
= 13.6eV ; K =
; U ="
2rn
n
2a0
2
r
Rutherford scattering : "n =
!
hf = E i " E f ; rn = r0 n 2 ; r0 =
!
de Broglie : " =
!
FALL QUARTER 2009
DECEMBER 4th, 2009
h
E
;f =
p
h
4
a0
Z
; a0 =
h2
= 0.529A ; L = me vr = nh angular momentum
me ke 2
; # = 2$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
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d"
"
; vp =
; Heisenberg : #x#p ~ h ; #t#E ~ h
dk
k
E
-i t
h2 " 2#
"#
h
Schrodinger equation : +
U(x)#(x,t)
=
ih
;
#(x,t)
=
$
(x)e
2
2m "x
"t
group and phase velocity : v g =
PHYSICS 2D
PROF. HIRSCH
QUIZ 6
FALL QUARTER 2009
DECEMBER 4th, 2009
%
h 2 # 2$
+
U(x)
$
(x)
=
E
$
(x)
;
dx $ *$ = 1
&
2
2m #x
-%
2 2 2
2
2
n$x
$ hn
h
sin(
) ; En =
;
= 3.81eVA 2 (electron)
2
L
L
2mL
2me
Time " independent Schrodinger equation : -
" square well : # n (x) =
m$ 2
x
2h
1
p2 1
1
; E n = (n + )h$ ; E =
+ m$ 2 x 2 = m$ 2 A 2 ; %n = ±1
2
2m 2
2
!
Harmonic oscillator : "n (x) = H n (x)e
!
Expectation value of[Q] :< Q >= # " * (x)[Q]" (x) dx ; Momentum operator : p =
!
!
#
Eigenvalues and eigenfunctions : [Q] " = q " (q is a constant) ; uncertainty :
Step potential : reflection coef : R =
!
(k1 " k 2 ) 2
, T = 1" R ;
(k1 + k 2 ) 2
k=
#Q = < Q2 > $ < Q > 2
2m
(E " U)
h2
x2
%
-2 # (x )dx
Tunneling :
" (x) ~ e -# x
;
T = e -2#$x ;
T =e
;
x1
2m[U(x) - E]
h2
# (x) =
!
r
r
r
r -i E t
h2 2
$#
" # + 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 ] = ml h
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
Schrodinger equation in 3D : -
!
!
l
!
!
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!
h $
i $x
Justify all your answers to all problems
l
PHYSICS 2D
PROF. HIRSCH
QUIZ 6
FALL QUARTER 2009
DECEMBER 4th, 2009
Problem 1 (10 points)
An electron moves in a three-dimensional box of edge lengths L1=1A, L2=1A, L3=2A.
(a) Find the quantum numbers and the energies (in eV) of the ground state and the first
excited state.
(b) Find the quantum numbers and the energies (in eV) of the second excited state and of
the third excited state. Are any of these states degenerate?
(c) Which is the lowest state that is not the ground state for which the wavefunction is not
zero at the center of the box? Give its quantum numbers and justify your answer.
Hint: Use h 2 /(2me ) = 3.81 eVA 2
!
Problem 2 (10 points)
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Consider an electron in the state of hydrogen described by the wavefunction
" (r,# , $ ) = Cr 2e%r / 3a 0 sin 2 # e%i$
where C is a constant.
(a) Give the quantum numbers n, l and ml for this state, and the values for the
magnitude of its angular momentum and the z component of the angular momentum,
expressed in terms of h .
(b) In the absence of a magnetic field and ignoring spin, how many other states are there
!
! Give their quantum numbers.
with the same energy as this state?
(c) Ignoring spin, by how much (in eV) does the energy of this electron change when a
!
magnetic field of magnitude 3T pointing in the -z direction is turned on? Does the energy
increase or decrease? Justify your answer.
Problem 3 (10 points)
!
The radial wave function for the n=2, l =0 state of an electron in hydrogen is
1
r
R(r) =
(2 " )e"r / 2a 0
3/2
(2a0 )
a0
1
(a) Find
in this state. !
r
(b) Find the average potential energy for an electron in this state, expressed in terms of
ke2 and a0 (k=Coulomb constant, a0=Bohr radius).
(c) Find the average kinetic energy for an electron in this state, expressed in terms of ke2
!and a . Hint: remember that kin. energy + pot. energy = ...
0
(d) Compare the results in (b) and (c) with the results for the Bohr model for the n=2
orbit.
$
n!
Hint: Use that % dx x n e" #x = n +1
#
0
Justify all your answers to all problems
!