Title Assignment-1_Foundation of Quantum Mechanics.pdf ✅

Assignment 1: Foundation of Quantum Mechanics 1 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 GATE Previous Years’ Questions 1. A proton is confined to a cubic box, whose sides have length 10–12 m. What is the minimum kinetic energy of the proton? The mass of proton is 1.67 × 10–27 kg and Planck’s constant is 6.63 × 10–34 Js. [GATE 2013] (a) 1.1 × 10–17 J (b) 3.3 × 10–17 J (c) 9.9 × 10–17 J (d) 6.6 × 10–17 J 2. The Compton wavelength of a proton is _____ fm. (up to two decimal places). [GATE 2017] 27 34 19 8 1 ( 1.67 10 kg, 6.626 10 Js, 1.602 10 C, 3 10 ms )     m h e c p         3. Consider a one-dimensional potential well of width 3 nm. Using the uncertainty principle    x p . /2  , an estimate of the minimum depth of the well such that it has at least one bound state for an electron is 31 34 19 ( 9.31 10 , 6.626 10 , 1.602 10 ) m kg h J s e C e          : [GATE 2017] (a) 1 μeV (b) 1 meV (c) 1 eV (d) 1 MeV 4. A matter wave is represented by the wave function: (4 3 5 10 ) ( , , , ) i x y z t x y z t Ae       where A is a constant. The unit vector representing the direction of propagation of this matter wave is (a) 4 3 1 ˆ ˆ ˆ 5 2 5 2 2 x y z   (b) 1 3 1 ˆ ˆ ˆ 5 2 5 2 2 x y z   [GATE 2021] (c) 3 4 1 ˆ ˆ ˆ 5 2 5 2 5 2 x y z   (d) 1 4 3 ˆ ˆ ˆ 5 2 5 2 5 2 x y z   ANSWER KEY 1. (c) 2. (1.30 to 1.34) 3. (b) 4. (a) JEST Previous Years’ Questions 1. The frequency dispersion relation of the surface waves of a fluid of density  and temperature T, is given by 3 2 Tk  gk    ; where  and k are the angular frequency and wavenumber, respectively, g is the acceleration due to gravity. The first term in r.h.s. describes the gravity waves and the second term describes the surface tension wave. What is the ratio of the first term to the second term, when the phase velocity is equal to the group velocity? [JEST 2022] ANSWER KEY 1. (1)