The radial schrodinger equation


by MrMultiMedia
Tags: equation, radial, schrodinger
MrMultiMedia
MrMultiMedia is offline
#1
Mar28-12, 10:56 PM
P: 9
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
Phys.Org News Partner Physics news on Phys.org
Physicists consider implications of recent revelations about the universe's first light
Vacuum ultraviolet lamp of the future created in Japan
Grasp of SQUIDs dynamics facilitates eavesdropping
jtbell
jtbell is offline
#2
Mar28-12, 11:06 PM
Mentor
jtbell's Avatar
P: 11,251
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.
MrMultiMedia
MrMultiMedia is offline
#3
Mar28-12, 11:25 PM
P: 9
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?

jtbell
jtbell is offline
#4
Mar28-12, 11:31 PM
Mentor
jtbell's Avatar
P: 11,251

The radial schrodinger equation


Which textbook are you using?
MrMultiMedia
MrMultiMedia is offline
#5
Mar28-12, 11:33 PM
P: 9
University Physics with Modern Physics by Young and Freedman
12th edition
MrMultiMedia
MrMultiMedia is offline
#6
Mar29-12, 02:11 AM
P: 9
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
dextercioby
dextercioby is offline
#7
Mar29-12, 08:08 AM
Sci Advisor
HW Helper
P: 11,863
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.
MrMultiMedia
MrMultiMedia is offline
#8
Apr7-12, 10:37 PM
P: 9
Quote Quote by dextercioby View Post
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.
Bob S
Bob S is offline
#9
Apr8-12, 04:29 PM
P: 4,664
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/introducti...2up/search/135


Register to reply

Related Discussions
Show that the radial component of the wave function satisfies the radial equation Advanced Physics Homework 2
Show that R satisfies the radial Schrodinger equation Advanced Physics Homework 3
radial schrodinger and shooting method Quantum Physics 0
Numerical Solutions to the Radial Schrodinger's Equation? Advanced Physics Homework 4
numerially integrate the radial schrodinger equation Quantum Physics 5