V Anirudh Sharma
- 3
- 0
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)?
Radial wave function and other graphs
- Thread starter V Anirudh Sharma
- Start date
Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
4 replies · 3K views
- 16,219
- 4,934
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 ?
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
- 3
- 0
thank you for paying attention towards my question.
the function R that i spoke about was the Ψ function.
the function R that i spoke about was the Ψ function.
- 16,219
- 4,934
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)##.
Perhaps studying this example can help you soting out the subtle differences ?
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 ##
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
- 3
- 0
it was just that i had been reading a book on physical chemistry where there were 3 graphs related to Schrödinger 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?
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?