phy3113s22009
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phy3113s22009

Course Number: P 3113, Fall 2009

College/University: Lake City CC

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1 PHY3113: Quantum mechanics I Problem set 2 1. (10 points) Find the probability current for the following wave functions: a) (x, t) = Aeipx/ + Be-ipx/ e-ip 2 Winter 2009 due January 30th, 2009 t/2m , where A and B are complex numbers, b) (x, t) = (A0 (x, t) + B1 (x, t)), where k (x, t) is the k'th harmonic oscillator wavefunction, and where A and B are complex numbers so that |A|2 + |B|2 = 1. 2. (40 points)...

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Quantum 1 PHY3113: mechanics I Problem set 2 1. (10 points) Find the probability current for the following wave functions: a) (x, t) = Aeipx/ + Be-ipx/ e-ip 2 Winter 2009 due January 30th, 2009 t/2m , where A and B are complex numbers, b) (x, t) = (A0 (x, t) + B1 (x, t)), where k (x, t) is the k'th harmonic oscillator wavefunction, and where A and B are complex numbers so that |A|2 + |B|2 = 1. 2. (40 points) A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: (x, 0) = A (1 (x) + 2 (x)) . a) (5 points) Normalize (x, 0). b) (5 points) Find (x, t) and |(x, t)|2 , and express the result in terms of sine and cosine functions of time, using = 2 /2ma2 . c) (5 points) Compute x . What is the frequency of oscillation of x ? What is the amplitude of oscillation? d) (5 points) Compute p e) (3 points) Find the expectation value of H. How does it compare with E1 and E2 ? f) (2 points) Find E. Although the overall phase constant of the wave function is of no physical significance, the relative phase of the coefficients in Eqn. (1) does matter. For example, suppose we change the relative phase of 1 (x) and 2 (x) to (x, 0) = A 1 (x) + ei 2 (x) , where is a constant. g) (10 points) Find (x, t), |(x, t)|2 , x and p , and compare your results with the results obtained using Eqn.(1), where was set to 0. 3. (35 points) The "radial" part of Schrdinger's equation in three dimensions is given by: o 1 d R(r) dr r2 dR dr - 2mr2 2 (1) (2) (V (r) - E) = ( + 1), (3) where R(r) is the radial wave function and r 0 is the radius. a) (5 points) Show that the change of variable (r) = rR(r) will transform the radial equation into an equivalent onedimensional problem subject to the condition r 0. What is the equivalent onedimensional potential for this problem? The nuclear potential can be modeled as a well having a finite depth of 57MeV and radius r0 1.25 A1/3 with A the total number of nucleons. b) (15 points) How many states with angular momentum these states? = 0 can you find for 56 26 F e? What are the energies of c) (10 points) Write down explicitly the normalized wave function for the lowest energy state of 56 F e. Find r for 26 this state if the radial probability density is given by P (r) = r2 |R(r)|2 . d) (5 points) What is the probability of finding a nucleon "outside" the nucleus if it is described by an wavefunction having the lowest possible energy in the above well? (4) = 0 2 4. (35 points) The Morse potential. To investigate the vibrational spectrum of a diatomic molecule, Morse has introduced the potential V (x) = D e-2x - 2e-x , x= r - r0 . r0 (5) a) (5 points) Sketch this potential, and show that it has a minimum at r = r0 . This is the equilibrium point. b) (5 points) By expanding V in series about r0 , write an approximate expression for V up to and including terms in (r - r0 )2 : 1 V (x) V0 + 2 m 2 (r - r0 )2 . (6) c) (10 points) Using the reduced mass formula: 1/m = 1/mCl + 1/mH and the parameters given below, evaluate for an electron in potential specified by the parameters describing the HCl molecule if: D = 37244 cm-1 , 2 2 /2mr0 = 10.5930 cm-1 , = 2.380. To convert from cm-1 to eV, use E(eV ) = E(cm-1 ) 1.2398 10-4 ). What is the equilibrium separation distance r0 between the two atoms in the molecule? d) (5 points) By comparing with the harmonic oscillator, obtain the (approximate) values for the lowest two energy levels of HCl. e) (10 points) By comparing with the wave functions of the harmonic oscillator, write the (approximate) radial wave function R(r) for the state of lowest energy. 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CONCORDIA UNIVERSITYDEPARTMENT OF MUSICSelf Appraisal Report Summary of Strengths &amp; Weaknesses and RecommendationsMay 2003(Extracts)Brief discussion of the overall strengths and weaknesses of each of the programs under reviewSeveral of the
East Los Angeles College - P - 420
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Practical Applications of Game Theory in Sociological ResearchRich Lafferty (9416113) for Prof. Smith 166-480B, Honours Project August 9, 2001Practical Applications of Game TheoryIntroductionPractical Applications of Game Theory in Sociologica
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East Los Angeles College - PHYS - 30800
Particle Interactions section 2, page 14. Energy loss processes for fast particlesIn this section we describe the various physical processes which occur when fast particles of different types interact with bulk matter. The majority of the discussi
East Los Angeles College - PHYS - 30800
Particle interactions: page 495. Measuring particlesIn this section we discuss in rather general terms the types of consideration which go into the design of particle detection systems. Then in the final section of the course we describe some comm
Wilfrid Laurier - CPSC - 581
I!&amp;iaConference Companion ~ CHI94~ Boston, Massachttsdts April24-2%, USA~ 1994Toolglass and Magic Lenses: The See-ThroughEric A. Bier, Maureen C. Stone, Ken Pier, Ken Fishkin, Thomas William Buxton* and Tony DeRose* Baudelf, MattInterfaceConw
Wilfrid Laurier - CPSC - 581
NotificationsSaul GreenbergStrategiesPolling interfaces A fundamental issue with user interfaces is how to help users stay aware of information without being overly intrusive or distracting.-from Sideshow: Providing Peripheral Awareness of Impor
Wilfrid Laurier - CPSC - 581
Physical User InterfacesWhat they are and how to build themSaul GreenbergUniversity of Calgary CPSC 581New disciplines and genres of computing ubiquitous computing pervasive computing context-aware computing mixed / augmented real
Wilfrid Laurier - CPSC - 581
Visual Presentation of Magnetic Resonance ImagesJ. E. van der Heyden, M. S. T. Carpendale, K. Inkpen, M. S. Atkins School of Computing Science Simon Fraser UniversityABSTRACTMedical image analysis is shifting from current film-oriented light scre
Wilfrid Laurier - CPSC - 581
Published in Human Factors in Computing Systems CHI `86 Conference Proceedings, 16-23. 1986.Generalized Fisheye ViewsGeorge W. Furnas Bell Communications Research 435 South St. Morristown, New Jersey 201-829-4289 Abstract In many contexts, humans
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Wilfrid Laurier - CPSC - 581
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Wilfrid Laurier - CPSC - 581
CRAPcontrast,repetition,alignment,proximityMajorsources:DesigningVisualInterfaces,Mullet&amp;Sano,PrenticeHall/RobinWilliamsNonDesignersDesignBook,PeachpitPressSlidedeckbySaulGreenberg.Permissionisgrantedtousethisfornoncommercialpurposesaslongasgene