Problem Set #9 (Keeping it Real in 3
Transcription
Problem Set #9 (Keeping it Real in 3
Physics 322: Modern Physics Spring 2015 Problem Set #9 (Keeping it Real in 3-D, getting to Hydrogen) Due WEDNESDAY, April 8 in Lecture ASSUMED READING: Before starting this homework, you should have read Chapter 7, Sections 1 to 6 of Harris’ Modern Physics. 1. [Harris 7.1 modified] What is a quantum number, and how does it arise? In answering this question, clarify what it physically represents. 2. [Harris 7.3 tweaked] Consider a 2D infinite well whose sides are of unequal length with the “horizontal” (x-axis) length being longer than the “vertical” (y-axis) length. a. Sketch the probability density – as density of shading – for the ground state (in other words, darker shades in a sketch mean higher probability density). b. There are two likely choices for the next lowest energy. Sketch the probability density and explain how you know this must be the next lowest energy. (Focus on the qualitative idea, avoiding unnecessary reference to calculations). 3. [Harris 7.21 tweaked] An electron is trapped in a (cubical) quantum dot, in which it is confined to a very small region in all three dimensions. If the lowest-energy transition is to produce a photon of 450 nm wavelength, what should be the width of the well (assumed cubic)? NOTE: You can treat the quantum dot as a 3D infinite well. Also, explain how you know the transition you picked is the lowest-energy transition. 4. [Harris 7.22] (10 points) Consider a cubic 3D infinite well. a. How many different wave functions have the same energy as the one for which (nx, ny, nz) = (5, 1, 1)? HINT: Table 7.1 might help speed things up. b. Into how many different energy levels would this level split if the length of one side (say the z-axis) were increased by 5%? c. Make a scale diagram, similar to Figure 7.3, illustrating the energy splitting of the previous degenerate wave functions. d. Is there any degeneracy left? If so, how might it be “destroyed”? – Page 1! of 3! – Physics 322: Modern Physics Spring 2015 5. [Harris 7.37 extended] An electron is in the " ℓ = 3 state of the hydrogen atom. a. What possible angles might the angular momentum vector make with the z- axis? b. What possible angles might the angular momentum vector make with the x- axis? y-axis? NOTE: To be clear, I am not suggesting you know the angle between the angular momentum vector and the z-axis while also knowing the other ones. I am suggesting if you were to measure along the x- or y- axis instead of the z- axis, what would you measure? 6. [Harris 7.39 extended] In Section 7.5, " eimℓφ is presented as our preferred solution to the azimuthal equation, but there is a more general one that need not violate the smoothness condition, and that in fact covers not only complex exponentials, but also, with suitable redefinitions of multiplicative constants, sine and cosine " Φ mℓ (φ ) = Ae+imℓφ +B e −imℓφ a. Show that the complex square of this function is not, in general, independent of" φ . b. What conditions must be met by A and/or B or the probability density to be rotationally symmetric – that is, independent of " φ ? c. The original version of this problem in Harris parenthetically notes “This highlights another reason, besides their being of well-defined Lz, why we like our preferred solutions.” Why is rotational symmetry a desired property for a solution to the wave equation for a central force? 7. [Harris 7.45 tweaked] An electron is in an n=4 state of the hydrogen atom. a. What is its energy (Verify equation 7-14 comes from equation 7-12 and you can use 7-14)? b. What properties besides energy are quantized, and what values might be found if these properties were to be measured? 8. [Harris 7.48] Show that the normalization constant " 15 / 32π given in Table 7.3 for the angular parts of the " ℓ = 2, mℓ = ±2 wave function 15 sin 2 θ e±2iφ is correct. Hint: You are working in 32π spherical coordinates, so you need the normalization condition given in equation (7-37). Justify your use of this equation, don’t just use it because " Θ 2, ±2 (θ ) Φ ±2 (φ ) = – Page 2! of 3! – Physics 322: Modern Physics Spring 2015 I told you to. To “justify” your use, you must answer “why would this equation be at all appropriate” and “what does an equation need to do to be a normalization condition?” Hint #2: sin5θ can be written as (1cos2θ)2sinθ, which if you expand, can make for an easier to tackle integral. – Page 3! of 3! –