68110927-Midterm-1-Solutions

68110927-Midterm-1-Solutions - Section A November 11, 2010...

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Section A November 11, 2010 EP 341 Quantum Physics Exam-1; Time Interval: 17:00-18:30 Name: ID No: Lecturer: Prof. Dr. Ramazan Koç 01 02 03 04 05 06 07 08 Total Questions 1-3-4-5 are 15 points and 2-6-7-8 are 10 points. Answer the following questions 1. Write down time dependent and time independent Schrödinger equation, Heisenberg Uncertainty principle and de Broglie relation. Briefly explain how to use the equations. Answer 1 (Time dependent Schrödinger equation. It is used to calculate energy levels and time dependent wave function of the particles) (Time independent Schrödinger equation. It is used to calculate energy levels and wave function of the particles) (Heisenberg Uncertainty relation, Means that position and momentum of a particle can not be measured simultaneously.) (Wavelength of a particle (matter wave) is inversely proportional to the momentum of the particle) 2. Show that is one of the solutions of the time dependent Schrödinger equation with zero potential. Answer 2. We substitute given wave function in to time dependent Schrödinger equation with V(x)=0: Differentiation of the functions yields: Then the terms lhs=rhs . Barrier Barrier Well Figure 1 Hints: ; ; ; ; ;
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Section A November 11, 2010 3. Evaluate the following commutators ( is momentum and is position) a) ; b) ; c) What is the physical meaning of cummutators in quantum mechanics? Answer 3 . Using definition of commutator of two operator; and operator representation of . We can write: a) then the commutator b) then the commutator c) If two operators can commute then they can be measured simultaneously, there is no uncertainty between operators and they have the same eigenfunctions. If two operator cannot commute then they cannot be measured simultaneously, there is an uncertainty between
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68110927-Midterm-1-Solutions - Section A November 11, 2010...

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