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Phys chem II CHM4411 Exam 2 Solutions

# Phys chem II CHM4411 Exam 2 Solutions -...

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Unformatted text preview: CHM4411=02 [Phys-Chem. ii], Spring 2012 Instructor: H. Mattoussi (6 pages total) Exam 2 (Monday Feb, 7, 2012) You name: --------- ' i-é;£‘-*~:§—--- 1 i' Exercise 1 (5 points) The wavefunction inside a long (infinite) potential barrier of finite height V0 is an exponentially decaying function of the form: (1)00 = A exp(—ax) ; with A and on being constants (see drawing). (a) is this wavefunction normalized? if not, find the normalization constant? (b) Calculate the probability of finding that particle inside the barrier. ‘ (c) Calculate the average penetration depth of the particle inside the barrier (hint: think of the expectation value). a V(x) . V , ’ g , s i) \Qx Vivsl’ [a géttgﬁg/E g aigfiﬂim 3% 3 ‘ £0 m2 ‘2" he; 2%“ '1‘ #5513? .1: .1? a: @zw I g! A R} v ,3, ﬂ «2 é“&' r’" :2 Kg m wish g W: 53%; \$3? ; ’ "" 9?; :4 J; 3 a. o .53} 1 r g: i M N i 33 M free/f4 .4» 163.7479 Mf Time in a“? a4. eds. test???“ W w ‘ j , soft. A?» WL ‘5‘ . g, 020% :9 gal/Nit!) g; T04 g :5? - 1,“. «. ‘ (l3) ém \$395.54 a} was“ igiifgi’fPéZt‘iﬂfl mg; (Let. i he“ I , ‘ r a 2 j "5’1 "as ‘* 94 1754:, {at g; 5' § 3? 6 1 n } J’ 7’: f g r . r“? 3 « «a t M , a f,» i 1,, a.» W 5 . i '9 Exercise 2 (5 points) 7 Consider a one dimension potential square well of width L (V = O for O s x S L and V = 0° for x > L and x<O). 1) Write the Schrodinger equation fOr this problem. 2) Find a common solution for the wavefunction. 3) Find the corresponding energy (ies). 4 Calculate the expectation values for the operators 13 and 132 for a particle in the state having n =2. (Extra info: sinzly) = [l-cos(2y)]/2). , _3// in; f .9 I L Q may (2} Y M a .,, 4; fix}: IJ Aggy/x Ea ,, a M £53“ mum“. M \M a.“ .29 a 5W» m bur. r. Hunt f .L “Earn JIM @ m; Wm. x k ,L M :::::: A 2 ; x w z} W . 4 m J 3,. f g g q , g ﬁ E a . L v a (a) _ ‘‘‘‘‘‘‘‘‘‘ g 9 , E, , A a. ._ Mr“ \I W Z A!“ W «ex 5 J W 7w a 5 S mi \ W L W? m PTA C 9: m , V, o , 52 La 2 Z. L, o w ? . L Iii 49L / \tNV «W n a O C J a m . S w w inﬁﬁﬁnﬁxaaeriisﬁyéwa . . , L 5 \a/ 6; AU? an, 1/,\ A Exercise 3 (5 points) To the movement of a particle in one dimension x (x varies between ~00 and +00) we associate three . -- . _GX2 wavefunctions: (a) (I) = e ‘1“; (b) ([1 = sm(kx); (c) (,1) = e . . A 75d 1) Determine which one of these functions is an eigen-function of the momentum operator, p = ~77. I X A 132 2) Determine which eye of these functions is an eigen~function of the energy operator, H = m 3) Calculate the average linear momentum, of the particle for the wavefunctions (b) and (c) above. (Hint: think of the expectation value). “NEE f daft/ea “(ft-t, ‘ it ,0” cl Aiﬁ 52a 7L )0 ,‘ £43m tlﬁﬂaes : c.”— ' M a .v ‘ “far” {134? 3 f 3% gmikk’gi f ééﬂﬁggﬁe :. 053%}; Exercise 4 (5 points) 1) Provide the definition of the density of probability for a particle. 2) When two distinct wavefunctions are considered degenerate? 3) What property do two distinct wavefunctionsCorresponding to two different eigenvalues have to satisfy? 4) Explain in a few sentences the meaning of particle tunneling through a potential barrier in quantum mechanics. 5) Provide two simple examples that involve electron tunneling through a potential barrier. j r a l I I’ a a _ \ ,a. 1“ {Ci E? 7W vi) i it; a?” We W“ 635,»! t? if; a??? a was) get. 5’ e taxi/a4: Mm get re d. its lititiﬁt‘gw ' Wﬁija at 4 New @Vfﬁgjmiﬂ' ...
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Phys chem II CHM4411 Exam 2 Solutions -...

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