2. A free particle (a particle that has zero potential energy) has mass 8 eV/c² and total energy 10 eV and is traveling to the right. At x = 0, the potential jumps from zero to Vo = 5 eV and remains at this value for all positive x. (a) In classical mechanics, what happens to the particle when it reaches x = 0? (b) What is the wavenumber of the quantum particle in the region x > 0? (c) Find the reflection coefficient R and the transmission coefficient T for the quantum particle.

10. A particle is represented (at time t = 0) by the wave function ¥(x,0) = {4(a² ¯ 0, JA(a²-x²), if- a ≤x≤+a otherwise (a) Determine the normalization constant A. (b) What is the expectation value of x (at time t = 0)? d (c) What is the expectation value of p (at time t = 0)? (Note that you cannot get it from p = m² .Why dt not?) (d) Find the expectation value of x². (e) Find the expectation value of p².

3. Plane waves and wave packets. In class, we solved the Schrodinger equation for a "free particle" (e.g. when U(x,t) = 0). The correct[solution is (x, t) = Ae(px-Et)/ħ This represents a "plane wave" that exists for all x. However, there is a strange problem with this: if you try to normalize the wave function (determine A by integrating * for all x), you will find an inconsistency (A has to be set equal to 0?). This is because the plane wave stretches to infinity. In order to actually represent a free particle, this solution needs to be handled carefully. Explain in words (and/or diagrams) how we can construct a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane waves for a limited range of momentum/energy. Hint 2: have a look at the brief discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)

B7

a) Find the de Broglie wavelength of electrons moving with an average speed of 1.55 x 105 m/s. b) Suppose these electrons are described by a wave packet of width 2.55 nm. What is the uncertainty of the momentum of the electrons?