Hi, I'm doing a homework problem in my modern physics class and I'm stuck at a point. The question is "Show that the radial probability density of the 1s level in hydrogen has its maximum value at r = a0, where a0 is the Bohr radius" I know that the radial schrodinger equation will give me the part of the answer that I need. I know that ψ(r,θ,phi) is found by separation of variables and that once I find ψ I can find the probability at any r by using P(r)dr = abs(ψ)^2dV = (abs(ψ)^2)*4∏(r^2)dr I know what my r is. My problem is solving the radial schrodinger equation. I have no idea what to do. The book gives boundary conditions: lim(R(r)) r-->∞ = 0 and the angular components must be periodic (f(θ) = f(θ+2∏n)) Thanks in advance for any advice, -MMM
Are you really required to solve the radial Schrödinger equation for this exercise, instead of looking up the appropriate wave function from a table that your textbook probably has? Solving the radial equation is messy (it involves associated Laguerre polynomials), and you generally see the gory details only at the advanced undergraduate or even graduate level, not in an introductory modern physics textbook.
There is no table in the textbook. I think need to find P(r) and find the maximum. There's no simpler way to solve the radial schrodinger equation?
Ok, I found a simple definition in terms of a0 for the radial wave function in the textbook. It was in one of the examples, but not explicitly shown in the main text, so it took some extra searching. I used: dP = (4(r^2)*e^(-2r/a0))/a0^3dr and solved for dP/dr. After that I just had to set dP/dr to find the maximum probability (probability functions contain no minima), and voila! It equaled a0 Thanks for the help guys, -MMM
Radial probability functions do generally contain minima. It's thus compulsory to compute the second derivative for the probability density at the value you found to check whether you have a minimum, maximum or saddle point.
Oh you're right. They do. But not for the 1s level. So in the case of that problem I didn't have to worry about second derivatives.
Pauling and Wilson's book Intro to Quantum Mechanics contains all the hydrogen radial wave functions through n = 6. See pages 135-136. The book is free on the web for reading, or downloading, or for Kindles. See http://archive.org/stream/introductiontoqu031712mbp#page/n143/mode/2up/search/135