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Unformatted text preview: EXAM 1: INFORMATION [/xé ‘7 m h
De Broglie Wavelength: ’1 = ; Uncertainty Principle: 0' o 2:: x I7 An Hermitian operator satisﬁes the relation: IV}; #11617 = I'M/:1 1,11 jdr 712 2 One Dimensional Schrodinger Equation: H = _E (fix? + V(x)w = E w nzh2
One Dimensional PIB Energies: En = 8 2 n = 1, 2,3,... ma 7: d A d2
Momentum Operators: ‘ = ._ 2 = 712 __
p i dx 1’ dx2 Constants and Conversions:
h = 6.63x10'34 Js 1 J = 1 kgm2/32
h = h/21t =1.05x10'34 Js 1A = 104° m
c = 3.00x1o8 mls = 3.00x1o‘° cm/s kNA = R
NA = (3.02x1o23 mor1 1 amu = 1.66x10‘27 kg
k = 1.38x10‘23 J/K 1 atm. = 1.013x1o5 Pa
R = 8.31 J/molK 1 eV = 1.60x10‘19 J R = 8.31 Pam3/molK
me = 9.10x10‘31 kg (electron mass)
mF, = 1.67x10'27 kg (proton mass) CHEM 5210 — Exam 1 February 18, 2011
Name l/cé ’/ (10) 1. The wavefunction for a particle conﬁned to 0 s x s a was found to be: _ 'W
w(x) = A sin(E) A :
a
Where A is the normalization constant. Find A. N.B. sin2(ax)dx = éx‘zl'smaw‘)
V a
A 7 7 x 7. A
’1: ASIA Lcjﬁ : A L1_L‘;§M211 “ 1 4 W 2' o 0 : IA  fr‘(Mﬂﬁ , ’30 T gab
Z 4 ﬁ /,»’f '9’ g,
k, 4.",— ‘ll \‘)
\J’
b
"t
\f‘
D
II
f—\
l’\3
\_/ (06) 2. Indicate whether each operator is linear (Yes or No  no explanation needed).
(a) d/dx (take the ﬁrst derivative) ya ,
(b) sine (take the sine) No (C) xd2/dx2 (take the second derivative and then multiply by x ) "/0 (14) 3. NB. (15) 4. Using the wavefunction: w(x)= Ae‘a“ ’2 z "(J/z. ,_
Evaluate the expectation value <x2 > ( 1 D — WA L 1 A e, [x 2e'ﬂxzdx— — —J:
4:3 : A1J111_de7( : _/_’\____ TV
1.1 A So far we have covered 5 of the 6 postulates in quantum theory. What are they? (10) 5. Short Answer Questions (1 or 2 sentences, if correct, is sufﬁcient) (a) Why is it important that an operator in quantum mechanics be Hermitian? Km} (Wis, (b) What is the condition on the potential energy, V, that permits us to use the time
independent form of the Schrodinger equation? (C) Is a linear combination of two nondegenerate wavefunctions (i.e. eigenfunctions
of the Hamiltonian) also a valid wavefunction (Yes or No  no explanation needed)?
V50 (10) 6. Consider a particle in a three dimensional box of dimensions, axaxa/JE Draw a diagram with the ﬁrst ﬁve energy levels of a particle in this box (in units of
h2/8ma2), including the sets of quantum numbers and the degeneracies of each level.
L4. 1 ,2 1 lé ’— 9:1
én, (\th ; A“ 4' :3 + :2— ﬂ .._— __,_ c511
8M 0‘1 92 0:71 ‘0 .
M — (3'1
 L1 (A7. . 2 ’l j H ’— 3:2
' / 1 *A)+ It2 »
3M “*1 > 4’ 3:1
{\1/“3 OZ é/kz/ A} l; A1,") '01; € )LZ/gm‘z
(Di l l 4r (9 2 2 2 H
G)
@2 l i 7 3 ‘ ‘ '2
@i l l to \‘ 3 :0
a ' l l
C91 2 l \O 4 ‘ (10) 7. Consider two noninteracting particles in a semiinﬁnite box (right)
V1
Suggest a suitable Hamiltonian equation. Brieﬂy discuss methods for solving the problem.
No solution is necessary. \R’L // ’L 31 X—>
’” l2 mil:
2» 311 02; gr. Mk mum, L‘qu (dﬂkOg‘f... Q, V's [d2 ] (10) 8. Evaluate the commutator: —2,x
a1 L
{2331’ka : 9’6”” .— 1&2“)
dzz /’{
l C);
v 2 Z
2‘1" * “a! 7C» r 29,,
w” a Mw_é4/5%:’/M‘)f m infmg WA &1
, i’
QM : M + x21: land,
32 1 ()2 J1
ﬂ (15) 9. Consider a particle in a box deﬁned by the potential: V(x)=0 L3xs+L
V(X)~—)oo x<L,x>+L 00
An approximate normalized wavefunction for the ground
state of the particle is:
15 L
=BL2— 2 B: —L_<. s+L
w ( x ) 1/16]; x
911:0 x<L and x>+L
Calculate the average (i.e. expectation) value of the kinetic energy as a function of
h, m and L.
1 ’i
.. t ‘l’
k g : / &/l’
.2 "" a 1
7. + L
’z ’1 2
<L<€7z «3% (9.17 .AZ $0.111) &} a/l .d’
: ~31 1 dJ
: T‘E/ 1_1)()1
z / L
5) _ D ”232 ' I Q” ..
— , , 2% J)
011 @x / 1 1; 2 3 L
/ :E )2 )L ,'j
M 3 —L
1/31? .,’Z/3Lr5 3
2 ’L ,3“ ’12 ’3
{3L Mm): Ui"
3 m é
: 4 AR )D/ L3
cfims ...
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