PHYSICS 4E
PROF. HIRSCH
QUIZ 4
Formulas:
Relativistic energy - momentum relation E =
SPRING QUARTER 2009
May 22
m2 c 4 + p 2 c2
c = 3 ! 108 m / s
;
Electron rest mass : me = 0.511 MeV / c2 ;Proton : mp = 938.26 MeV / c2 ;Neutron : m n = 939.55 MeV / c 2
_
8#
hc
1
;
E
( ") =
4
hc / "kB T
"
" e
$1
Energy in a mode/oscillator : E f = nhf ; probability P(E) " e#E / kB T
_
Planck's law :
u( ") = n( " ) E ( ")
;
n( ") =
&
!
!
!
!
!
!
Stefan's law : R = "T 4 ; " = 5.67 #10$8 W /m 2K 4 ; R = cU /4 , U =
0
hc
Wien's displacement law : "m T =
4.96k B
Photons :
E = pc ; E = hf ; p = h/!
;
f = c/!
1
Photoelectric effect : eV0 = ( mv 2 ) max = hf " # , # $ work function
2
h
Compton scattering :
! ' -! =
(1" cos # )
me c
kq Q
1
"N # 4
Rutherford scattering: b = " 2 cot(# /2) ;
m" v
sin ($ /2)
Constants : hc = 12,400 eV A ; hc = 1,973 eV A ; k B = 1/11,600 eV/K ; ke 2 = 14.4eVA
kq q
kq
Electrostatics : F = 12 2 (force) ; U = q0 V (potential energy) ; V =
(potential)
r
r
!
!
Hydrogen spectrum :
Bohr atom : E n = "
!
!
!
!
' u(%)d%
1
1
1
= R( 2 # 2 )
"
m
n
R = 1.097 $10 7 m#1 =
;
1
911.3A
ke 2 Z
Z 2E
ke 2 mk 2e 4
= " 2 0 ; E0 =
=
= 13.6eV ; E n = E kin + E pot , E kin = "E pot /2 = "E n
2rn
n
2a0
2h 2
hf = E i " E f ; rn = r0 n 2 ; r0 =
a0
Z
; a0 =
h2
= 0.529A ; L = mvr = nh angular momentum
mke 2
X - ray spectra : f 1/ 2 = An (Z " b) ; K : b = 1, L : b = 7.4
h
E
2$
de Broglie : " = ; f =
; # = 2$f ; k =
; E = h# ; p = hk ;
p
h
"
group and phase velocity : v g =
d"
dk
;
vp =
"
k
;
Heisenberg
: #x#p ~ h
E=
p2
2m
#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#
"#
Schrodinger equation : + V(x)#(x,t) = ih
;
#(x,t) = $ (x)e h
2m "x 2
"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
"
*
& dx# A
#
op
-"
PHYSICS 4E
PROF. HIRSCH
QUIZ 4
SPRING QUARTER 2009
May 22
Eigenvalues and eigenfunctions : Aop " = a " (a is a constant) ; uncertainty :
Harmonic oscillator : "n (x) = Cn H n (x)e
!
Step potential : R =
!
(k1 " k2 ) 2
,
(k1 + k2 ) 2
m$ 2
#
x
2h
#A = < A 2 > $ < A > 2
2
1
p
1
1
; E n = (n + )h$ ; E =
+ m$ 2 x 2 = m$ 2 A 2 ; %n = ±1
2
2m 2
2
T = 1" R
;
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
Radial probability density : P(r) = r 2 | Rn,l (r) |2 ;
Energy : E n = !13.6eV 2
n
1 Z 3 / 2 $Zr / a 0
Ground state of hydrogen and hydrogen - like ions : "1,0,0 = 1/ 2 ( ) e
#
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
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
Problem 1 (10 pts)
A two-dimensional infinite square well with side lengths L1=L2=4A has 4 electrons.
Assume the electrons don't interact with each other.
(a) Find the ground state energy (in eV), taking into account that electrons are fermions
with spin ½.
(b) Give the degeneracy of the ground state (number of different states with the same
energy).
(c) Taking into account now that electrons interact, do you expect the total spin of the
ground state to be zero or nonzero? Justify.
(d) Answer the questions (b) and (c) for the case where L1=4A, L2=4.2A.
Use h 2 /2me = 3.81 eV A 2
!
PHYSICS 4E
PROF. HIRSCH
QUIZ 4
SPRING QUARTER 2009
May 22
Problem 2 (10 pts)
The n=3, l=2 radial wavefunction for an electron in a hydrogen-like ion is
R(r) = Cr 2e"2r / a 0
where a0 is the Bohr radius and C is a constant.
(a) Find the most probable r for this electron, in terms of a0.
(b) Find the average r for this electron, in terms of a0.
(c) Give the value of the radius of this orbit within Bohr theory and compare with the
results of (a) and (b). Discuss similarities and differences.
$
s!
Use % drr se" #r = s+1
#
0
!
!
Problem 3 (10 pts)
Boron has atomic number Z=5, and electronic configuration 1s22s22p1.
(a) Give a qualitative explanation for why the ionization energy of boron, IB=8.3 eV, is
lower than both the ionization energy of Be (Z=4, IBe=9.3 eV) and that of C (Z=6, IC=11.3
eV).
(b) Calculate by how much (in eV) the energy of a Boron atom will change in an external
magnetic field Bext=15T, ignoring spin-orbit coupling. Will the energy increase or
decrease if Bext is (i) along the z direction, (ii) along the -z direction, or (iii) along the x
direction?
(c) Explain what are the consequences (if any) of spin-orbit coupling for the electronic
states of the Boron atom in the absence of an external magnetic field.
(d) Give an example of an atom (other than hydrogen) for which spin-orbit coupling does
not affect the ground state.
Justify all your answers to all problems