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Exam1_key

# Exam1_key - EXAM 1 INFORMATION/xé ‘7 m h De Broglie...

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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 J-s 1 J = 1 kg-m2/32 h = h/21t =1.05x10'34 J-s 1A = 104° m c = 3.00x1o8 mls = 3.00x1o‘° cm/s k-NA = 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/mol-K 1 eV = 1.60x10‘19 J R = 8.31 Pa-m3/mol-K 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 non-degenerate 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 non-interacting particles in a semi-inﬁnite 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 (dﬂk-Og‘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 inf-mg 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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