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(PHYS234)[2006](s)final~fhshiu^_14031.pdf
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PHYS234 Elementary Quantum Mechanics I
Spring 2006
Final Examination
12:30 C3:30pm, 27/5/2006 Rm LG1
Important:
a) This exam paper has FIVE questions. Answer ALL questions.
b) This is a close book examination. You are not allowed to bring any book or
paper to your seat. c) Please put your student ID on the upper-right-hand corner of your desk. d) Please leave your other belongings, including your cell phone and
telecommunication devices, by the teachers desk. You are not allowed to carry your cell phone any time during the examination.
e) You can find a list of formulae below. Take a quick look through them before you start answering questions. Note that not every one of them is needed to answer this exam paper. If you need any formula not listed here, you can raise your hand any time during the examination. I will give it to you if appropriate.
---- Buena Suerte / Viel Glueck / Bon Chance / Good Luck /\ -----
Formulae: NOTE: not every formula listed here is needed in answering this exam paper
1D time-independent Schr.dinger equation
. =2 d 2 .() ()x =E x
.+Vx () (1)
. 2 .
2mdx
..
Time-dependence of stationary state
.iEt / =
(,) xt =e ( ) (2)
x
3D time-independent Schr.dinger equation with a central potential:
2
. =.2+ ()=E
Vr (3)2m
Laplacian operator in Cartesian and spherical coordinates:
222 .2
2 ... 1 .. 2 .. L
.= + + = r .. (4)
2 222 . 22
.x .y .zr .r ..r . r =
1
.A., B. .
Uncertainty principle
(5)
AB 2
..
Orbital angular momentum operator in spherical coordinates: 22 . 1 .. .. 1 .2 .
L. =.=..sin .+ 22 . (6)
.sin . . sin ..
..
Commutation relations of the angular momentum operators: .. iL., [ . . =iL. .,. iL=.
[,LL ] = = LL ,] = , [ LL ] =
xyzyzxzx y
(7)[..
2, ] =0 where = ,,
LL xyz
.2
Eigenstates and eigenvalues of L and L. z: .2 m 2 m . mm
LY + LY mY . (8)
l =ll (1) = Yl , zl = =l
Orbital wave function of hydrogen atom :
nlm
() =R () m(,
rrY ) (9)
nlm nl l
First few Rnl and Ylm : 2 .ra/ 0
1
R =
=
e Y0
10 a34
1 . 1 r..ra/2 03
cos (10)
R20 =
.1..eY1 =
3
2a . 2 a. 4
1 r . /2
3 i
R21 =
era Y11 =.
sine
24a3 a 8
Volume integral in 3D:
2
dr3 = r2 sindrd d (11)
000
Orthogonality and normalization
2 *
2 mm
= , 0 (Y )Y sin = (12)
rRR dr dd
0 nl nl nn ll 0 ll ll mm
Energy levels of a hydrogen atom:
. 22 . 2
m . e . 1 = 1 13.6eV
E . =. 2 (13)
n =.. 2 .. 2 22 =.
2=4 n 2ma n n
.. 0 ..
..
Bohr radius
2
40 =.10
a 2 =0.529 10 m (14)
me
Orthogonality of hydrogen atom wavefunction:
'' '
nlm
nlm = (15)
nn' ll ' mm
'
Coupling of two spin s= 1/2 particles,
Sms
: 00
=
1(.) singlet
2
11
= .
(16)
.
11
.triplet
.= 1 .
10
=
(+)
2 .
Mathematical formula