Physical Chemistry First Examination with Useful Formuals Sheet | Fall 2017 MIT, Exams of Physical Chemistry

Download Physical Chemistry First Examination with Useful Formuals Sheet | Fall 2017 MIT and more Exams Physical Chemistry in PDF only on Docsity! MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.61 Physical Chemistry Fall, 2017 Professor Robert W. Field FIFTY MINUTE EXAMINATION I Thursday, October 5 Question Possible Score My Score I 25 II 10 III 10 IV 20 V 35 Total 100 Name: 5.61 Exam I Fall, 2017 Page 1 of 16 pages I. Tunneling and Pictures (25 POINTS) V(x) = ∞ |x| > a/2 Regions I and V V(x) = 0 a/4 ≤ |x| ≤ a/2 Regions II and IV V (x)=V0 = h2 8ma2 ⎡ ⎣⎢ ⎤ ⎦⎥ 9 |x| < a/4 Region III The energy of the lowest level, En n = 1 is near E1 (0) = h2 8ma2 ⎡ ⎣⎢ ⎤ ⎦⎥ and the second level, En n = 2, is near E2 (0) = h2 8ma2 ⎡ ⎣⎢ ⎤ ⎦⎥ 4 . A. (8 points) Sketch ψ1(x) and ψ2(x) on the figure above. In addition, specify below the qualitatively most important features that your sketch of ψ1(x) and ψ2(x) must display inside Region III and at the borders of Region III. 5.61 Exam I Fall, 2017 Page 1 Region I Region II Region III Region IV Region V 6161 V (x) h2 8ma2 (9) h2 8ma2 (4) h2 8ma2 (1) 6 - a/2 a/4 0 a/4 a/2 0 5.61 Exam I Fall, 2017 Page 4 of 16 pages II. Measurement Theory (10 POINTS) Consider the Particle in an Infinite Box "superposition state" wavefunction, ψ1,2 = (1/3)1/2 ψ1 + (2/3)1/2ψ2 where E1 is the eigen-energy of 𝜓1 and E2 is the eigen-energy of 𝜓2. A. (5 points) Suppose you do one experiment to measure the energy of 𝜓1,2 Circle the possible result(s) of your measurement: (i) E1 (ii) E2 (iii) (1/3)E1 + (2/3) E2 (iv) something else. B. (5 points) Suppose you do 100 identical measurements to measure the energies of identical systems in state 𝜓1,2 What will you observe? 5.61 Exam I Fall, 2017 Page 5 of 16 pages (Blank page for Calculations) 5.61 Exam I Fall, 2017 Page 6 of 16 pages III. Semiclassical Quantization (10 POINTS) Consider the two potential energy functions: V1 |x| ≤ a/2, V1(x) = –|V0| |x| > a/2, V1(x) =0 V2 |x| ≤ a/4, V2(x) = –2|V0| |x| > a/4, V2(x) = 0 A. (5 points) The semi-classical quantization equation below 2 h ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ pEx− (E ) x+ (E )∫ (x)dx = n pE = 2m E −V (x)( )[ ] 1/2 describes the number of levels below E. Use this to compute the number of levels with energy less than 0 for V1 and V2. B. (5 points) V1 and V2 have the same product of width times depth, V1 is (a)|V0| and V2 is (a/2)(2|V0|), but V1 and V2 have different numbers of bound levels. Which has the larger fractional effect, increasing the depth of the potential by X% or increasing the width of the potential by X%?? 5.61 Exam I Fall, 2017 Page 9 of 16 pages D. (10 points) Derive the commutation rule N̂ , â[ ] starting from the definition of N̂ . 5.61 Exam I Fall, 2017 Page 10 of 16 pages (Blank page for Calculations) 5.61 Exam I Fall, 2017 Page 11 of 16 pages V. 〈x〉 , 〈p〉 , σx, σp and Time Evolution (35 POINTS) of a Superposition State x̂ = ! 2µω ⎡ ⎣⎢ ⎤ ⎦⎥ 1/2 a†" + â( ) p̂ = !µω 2 ⎡ ⎣⎢ ⎤ ⎦⎥ 1/2 i a†" − â( ) A. (5 points) Show that x2! = " 2µω ⎡ ⎣⎢ ⎤ ⎦⎥ â2 +a†! 2 +2N̂ +1( ) . B. (5 points) Derive a similar expression for p 2! . (Be sure to combine a†!â and âa†! terms into an integer times N̂ plus another integer. 5.61 Exam I Fall, 2017 Page 14 of 16 pages (Blank page for Calculations) 5.61 Exam I Fall, 2017 Page 15 of 16 pages Some Possibly Useful Constants and Formulas h = 6.63 × 10–34 J · s ! = 1.054 × 10–34 J · s ε0 12 2 1 38854 10= × − − −. Cs kg m c = 3.00 × 108 m/s c = λν λ = h/p me = 9.11 × 10–31 kg mH = 1.67 × 10–27 kg 1 eV = 1.602 x 10-19 J e = 1.602 x 10-19 C E = hν a0 = 5.29 x 10-11 m e± iθ = cosθ ± isinθ ν = 1 λ = RH 1 n1 2 − 1 n2 2 $ % & & ' ( ) ) where RH = me4 8ε0 2h3c =109,678 cm-1 Free particle: E =  2k2 2m ψ x( ) = Acos kx( )+ Bsin kx( ) Particle in a box: En = h2 8ma2 n2 = E1 n 2 ψ 0 ≤ x ≤ a( ) = 2a # $ % & ' ( 1 2 sin nπ x a # $ % & ' ( n = 1, 2, … Harmonic oscillator: En = n+ 1 2 ! " # $ % &ω [units of ω are radians/s] ψ0 x( ) = α π $ % & ' ( ) 1 4 e−αx 2 2 , ψ1 x( ) = 1 2 α π $ % & ' ( ) 1 4 2α1 2x( )e−αx2 2 ψ2 x( ) = 18 α π $ % & ' ( ) 1 4 4αx2 −2( )e−αx2 2 !̂x ≡ mω " x̂ !̂p ≡ 1 "mω p̂ [units of ω are radians/s] a ≡ 1 2 !̂x+ i !̂p( ) Ĥ !ω = aa† − 1 2 = a†a+ 1 2 N̂ = a †a a† = 1 2 !̂x − i !̂p( ) 2πc !ω =ω [units of !ω are cm –1] 5.61 Exam I Fall, 2017 Page 16 of 16 pages Semi-Classical λ = h/p pclassical(x) = [2m(E – V(x))]1/2 period: τ = 1/ν = 2π/ω For a thin barrier of width ε where ε is very small, located at x0, and height V(x0): Hnn (1) = ψ n (0)* x0−ε/2 x0+ε/2∫ V (x)ψ n(0)dx = εV (x0 ) ψ n(0)(x0 ) 2