• MrMultiMedia
In summary: The%20book%20is%20free%20on%20the%20web%20for%20reading%2C%20or%20downloading%2C%20for%20Kindles.%21%22University Physics with Modern Physics by Young and Freedman12th editionI looked up the radial wave function for hydrogen in Pauling and Wilson's book Intro to Quantum Mechanics. It's on pages 135-136.
MrMultiMedia
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))

-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?

Which textbook are you using?

University Physics with Modern Physics by Young and Freedman
12th edition

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.

dextercioby said:
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.

1. What is the radial Schrodinger equation?

The radial Schrodinger equation is a mathematical equation that describes the behavior of a quantum system in terms of its radial distance from a central point. It is a fundamental equation in quantum mechanics and is used to calculate the probability of finding a particle at a specific distance from the central point.

2. How is the radial Schrodinger equation derived?

The radial Schrodinger equation is derived from the Schrodinger equation, which is a general equation that describes the behavior of a quantum system. The radial Schrodinger equation is obtained by separating out the radial component of the Schrodinger equation and solving it separately.

3. What is the role of the radial Schrodinger equation in quantum mechanics?

The radial Schrodinger equation is a fundamental equation in quantum mechanics and is used to describe the behavior of particles at the atomic level. It is used to calculate the probability of finding a particle at a specific distance from a central point, which is essential in understanding the behavior of atoms and molecules.

4. What are the key assumptions made in the radial Schrodinger equation?

The radial Schrodinger equation assumes that the potential energy of the system only depends on the radial distance from the central point, and that the wave function describing the system can be separated into a radial component and an angular component. It also assumes that the system is in a stationary state, meaning that its properties do not change over time.

5. Can the radial Schrodinger equation be applied to all quantum systems?

No, the radial Schrodinger equation can only be applied to spherically symmetric systems, meaning that the potential energy of the system is the same in all directions from the central point. It cannot be applied to more complex systems, such as molecules with multiple atoms, as these systems do not exhibit spherical symmetry.

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