Radial wave function and other graphs

[V Anirudh Sharma]

1.what is the difference between radial wave function(R),radial probability density(R^2) and radial probability function(4*π*r^2* R^2)?

 

[BvU]

Hi V,

The radial wave function can not be R (unnormalizable), so you want to be a bit clearer in formulating your question. Something with expectation values, perhaps ?

 

[V Anirudh Sharma]

thank you for paying attention towards my question.
the function R that i spoke about was the Ψ function.

 

[BvU]

I see. However, "the function R was the ##\Psi## function" still does not make sense to me. If you mean ##\Psi##, why call it R ?

Now, what exactly is it you want to ask ?

If the Schroedinger equation can be solved by separation of variables in position and time we have a time-independent SE and write ##\psi (\vec r, t) = \phi(\vec r) T(t)##.

If, furthermore, ##\phi(\vec r) ## can also be factored into a depends on ##|\vec r|## only, such as for the hydrogen atom, we can write ##\psi (\vec r, t) = R(|\vec r|) F(\phi)P(\theta) T(t)##.

Could that be the R you are referring to ? Then why not say so !​

For the hydrogen atom example, in the ground state the angular part is constant and

The radial probability density then only depends on ##\vec r|## and simplifies to ## \displaystyle {dP\over dr} = R(|\vec r|)^2 \; 4\pi r^2##
The most probable value for ##|\vec r|## is where ##\displaystyle {dP\over dr}{dP\over dr} = 0##

The expectation value for ##|\vec r|##, which is ##<|\vec r|> = \int \psi^* r psi d^3r ## then simplifies to ##<|\vec r|> = \int r \displaystyle {dP\over dr}{dP\over dr} dr ##​

Perhaps studying this example can help you soting out the subtle differences ?

 

[V Anirudh Sharma]

it was just that i had been reading a book on physical chemistry where there were 3 graphs related to schrodinger wave equation.
the first one was a graph of ' R vs r' of different orbitals( plot of radial wave function).
the second was of R^2 vs r(plot of radial probability density).
the third one was of " 4πr^2R^2 vs r " (plot of radial probability function).
the plots of the second one and third one mismatched drastically despite the fact that they both depict the radial probability.
all i know is that the first graph depicts the amplitude of the electron wave as a function of 'r'.
talking about the second and third graphs, especially what they talk about 1s orbital, the second one shows that as r tends to zero, R^2 tends to infinity whereas in the third one, 4πr^2R^2 tends to 0.
mathematically, i understood the plot. but what do both the graphs actually depict?