PHYSICS 4E
PROF. HIRSCH
QUIZ 4
SPRING QUARTER 2010
MAY 21
Formulas and constants:
hc = 12,400 eV A ; k B = 1/11,600 eV/K ; ke 2 = 14.4eVA ; me c 2 = 0.511"10 6 eV ; m p /me = 1836
!
!
!
Relativistic energy - momentum relation E = m2 c 4 + p 2 c2
;
c = 3 ! 108 m / s
r
r
r r
Photons: E = hf ; p = E /c ; f = c / "
Lorentz force: F = qE + qv " B
1
Photoelectric effect : eV0 = ( mv 2 ) max = hf " # , # $ work function
2
%
%
2
dIn
1 '
1
x3
'4
!Integrals : In " & x n e# $x dx ;
! ; I1 =
= #In +2 ; I0 =
; & x dx =
d$
2 $
2$
15
0
0 e #1
_
_
8#
hc
1
Planck's law : u( ") = n( " ) E ( ") ; n( ") = 4
;
E ( ") =
"
" e hc / "kB T $1
Energy in a mode/oscillator : E f = nhf ; probability P(E) " e#E / kB T
&
4
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
4
Stefan's law : R = "T ; " = 5.67 #10 W /m K ; R = cU /4 , U =
' u(%)d%
0
Wien's displacement law : "m T = hc /4.96k B
h
h
Compton scattering :
"'- " =
(1# cos $ )
;
"c %
= 0.0243A
mec
mec
kq Q
1
"N # 4
Rutherford scattering: b = " 2 cot(# /2) ;
m" v
sin ($ /2)
kq q
kq
Electrostatics : F = 12 2 (force) ; U = q0 V (potential energy) ; V =
(potential)
r
r
1
1
1
1
Hydrogen spectrum
= R( 2 # 2 ) ! ;
R = 1.097 $10 7 m#1 =
! :
"
m
n
911.3A
Bohr atom : rn = r0 n 2 ; r0 =
!
2
$8
a0
Z2
h2
ke 2
; E n = "E 0 2 ; a0 =
=
0.529A
;
E
=
= 13.6eV ; L = mvr = nh
0
Z
n
mke 2
2a0
1
ke 2 Z
ke 2 Z
v2
E k = mv 2 ; E p = "
; E = Ek + E p ; F = 2 = m
; hf = hc/# = E n " E m
2
r
r
r
mM
Reduced mass : µ =
; X - ray spectra : f 1/ 2 = An (Z " b) ; K : b = 1, L : b = 7.4
m+M
h
E
2$
p2
de Broglie : " = ; f =
; # = 2$f ; k =
; E = h# ; p = hk ; E =
; hc = 1973 eV A
p
h
"
2m
group and phase velocity : v g =
d"
dk
;
vp =
"
k
;
Heisenberg
: #x#p ~ h
#t#E ~ h
;
"(x,t) =| "(x,t) | e i# (x,t ) ;
P(x,t) dx =| "(x,t) |2 dx = probability
E
-i t
h2 " 2#
"#
h
Schrodinger equation : +
V(x)#(x,t)
=
ih
;
#(x,t)
=
$
(x)e
2
2m "x
"t
%
h 2 # 2$
Time " independent Schrodinger equation : +
V(x)
$
(x)
=
E
$
(x)
;
& dx $ *$ = 1
2m #x 2
-%
Wave function
" square well : # n (x) =
2
n$x
$ 2h2n 2
sin(
) ; En =
;
L
L
2mL2
x op = x , pop =
h %
; < A >=
i %x
Eigenvalues and eigenfunctions : Aop " = a " (a is a constant) ; uncertainty :
Harmonic oscillator : "n (x) = Cn H n (x)e
#
m$ 2
x
2h
"
*
& dx# A
#
op
-"
#A = < A 2 > $ < A > 2
1
p2 1
1
; E n = (n + )h$ ; E =
+ m$ 2 x 2 = m$ 2 A 2 ; %n = ±1
2
2m 2
2
PHYSICS 4E
PROF. HIRSCH
QUIZ 4
(k1 " k2 ) 2
Step potential : R =
,
(k1 + k2 ) 2
T = 1" R
SPRING QUARTER 2010
MAY 21
;
k=
2m
(E " V )
h2
b
%
-2 # (x )dx
Tunneling :
" (x) ~ e -#x
;
T ~ e -2#$x ;
T~e
a
!
;
# (x) =
2m[V (x) - E]
h2
# 2 h 2 n12 n 22 n 32
( +
+ )
2m L12 L22 L23
Spherically symmetric potential: "n,l,m (r,#, $ ) = Rnl (r)Ylm (#, $ ) ; Ylm (#, $ ) = f lm (# )e im$
r r r
h #
Angular momentum : L = r " p ; Lz =
; L2Ylm = l(l + 1)h 2Ylm ; L z = mh
i #$
Z2
2
2
Radial probability density : P(r) = r | Rn,l (r) | ;
Energy : E n = !13.6eV 2
n
r
1
"e r
Spin 1/2 : s = , | S |= s(s + 1)h ; Sz = msh ; ms = ±1/2 ;
µs =
gS
2
2me
3D square well : "(x,y,z) = "1 (x)"2 (y)"3 (z) ; E =
!
!
!
!
!
!
!
!
r r r
Total angular momentum : J = L + S ; | J |= j( j + 1)h ; | l " s |# j # l + s ; " j # m j # j
r
r "e r
r r
Orbital + spin mag moment :
µ=
( L + gS ) ; Energy in mag. field : U = "µ # B
2m
Two particles : "(x1, x 2 ) = + /# "(x 2 , x1 ) ; 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
Justify all your answers to all problems. Write clearly.
Problem 1 (10 pts)
5 electrons are put in a two-dimensional infinite well of side lengths L1=2A, L2=3A.
Assume the electrons don't interact via the Coulomb interaction but they obey Pauli's
exclusion principle.
(a) Make an energy level diagram showing the quantum numbers of the occupied states
and of the lowest unoccupied state and the occupation of each state with the electron spin
orientation for the lowest possible energy state of the system (ground state).
(b) Find the ground state energy of this system in eV.
(c) Find the wavelength of the lowest energy photon that this system will absorbe, in A.
Assume there are no selection rules.
PHYSICS 4E
PROF. HIRSCH
!
!
QUIZ 4
SPRING QUARTER 2010
MAY 21
Problem 2 (10 pts)
An electron in a hydrogen-like ion is described by the wavefunction
"(r,# , $ ) = Cr 3e%r / 2a 0 sin 3 # e%i$
with C a constant and a0 the Bohr radius.
(a) Give the values of the quantum numbers n, l , m and of the ionic charge Z. Justify
your answers.
(b) Calculate the average value of 1/r for an electron described by this wavefunction.
Hint: you need to normalize the probability distribution.
!
$
s!
s " #r
Use % drr e = s+1 .
#
0
(c) Show that your result in (b) agrees with the prediction of the Bohr atom for this n and
Z. Alternatively, if you didn't find n and Z in (a), use the fact that your result in (b)
should agree with the Bohr atom prediction to deduce the values of n and Z.
Problem 3 (10 pts)
(a) Sodium has atomic number Z=11 and ionization energy 5.14 eV. The two lowest
energy photons that can be absorbed by a sodium atom in its ground state have
wavelengths 5890A and 5896A.
(a) Give the ground state electronic configuration of sodium.
(b) Assume a sodium atom is in an excited state where the outer electron occupies a 3p
orbital. Estimate the energy required to remove this electron from the sodium atom, in
eV.
(c) Assume a sodium atom is in an excited state where the outer electron occupies a 3d
orbital. Give the possible values for the magnitude of the total angular momentum for this
atom and state which state has higher energy, and explain clearly why. What can you say
about the energy required to remove this electron from the sodium atom? Explain.
Justify all your answers to all problems. Write clearly.