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# 237module2 - 2 What is the expectation value What is the...

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Quantum Mechanics I - Module 2 Getting Started 1. Three electrons are trapped in three diﬀerent inﬁnite potential wells of width (a) 50 pm, (b) 200 pm, and (c) 100 pm. Rank the electrons accord- ing to their zero-point energies, greatest ﬁrst. 2. An electron is trapped in an inﬁnite potential well in a state with n = 17. How many of the following does its matter wave have: (a) nodes (b) probability maxima 3. An electron is trapped in a ﬁnite potential well that is deep enough to allow the electron to exist in a state with n = 4. How many of the following does its matter wave have (within the well): (a) nodes (b) probability maxima 4. Consider the following potential: V ( x ) = 0 , x < 0 V 0 , x > 0 . where E > V 0 . You haven’t seen this potential before, but see if you can guess at what the boundary conditions should be for the wavefunction. Discussion Questions 1. What is an operator? What are the position, momentum, and energy operators in one dimension?

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Unformatted text preview: 2. What is the expectation value? What is the general expression for the expectation value of an operator? 3. A large part of quantum mechanics is devoted to the study of the solutions of the Schroedinger equation for diﬀerent potentials. What are the forms of the potential for the following: (a) a free particle (b) a particle in a box of length a (c) a harmonic oscillator 4. What are the conditions that the wavefunction must satisfy for the particle in a box problem? 1 Enrichment Problem Consider a system with potential energy given by: V ( x ) = 1 2 mω 2 x 2 1. What kind of potential is this? 2. What does it mean for a function to be an eigenfunction of a system? 3. Show that: φ = Ae-αx 2 is an eigenfunction of this system for a particular value of α . What is the value? 4. What is the value of the energy E for this eigenfunction? 2...
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237module2 - 2 What is the expectation value What is the...

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