betelgeuse1 wrote:I have a small question: when you talk about probability density in QM you call this abs(psi)^2. If you need the probability you multiply this with the volume element dV and if you whant the probability in a domain you integrate over that volume domain. If you whant to make your computations in spherical coordinates then your volume element is r^2*sin(theta)dr*d(theta)*d(phi) but the probability density still remains abs(psi)^2. If the function has spherical symmetry and it does not depend on theta and phi you can integrate out the sin(theta) and d(phi) and you get a 4*pi*r^2 out. Now, I think you call the probability density also abs(psi)^2 and 4*pi*r^2*dr is the volume element. My problem: There was a small confusion about this, somebody told me that the probability density was abs(psi)^2 *4*pi*r^2, and I still don't agree with that. I mean if you write in electromagnetism Integral(rho * dV)=Integral(rho * 4*pi*r^2 dr) then the charge density is still rho, or not? (I mean of course if rho depends only on r). I just whant to make sure the conventions are the same... would be silly to make mistakes like this in the GRE...
betelgeuse1 wrote:thanks, it's how I thought as well...
blackcat007 wrote:Probability density is abs(psi)^2
(abs(psi)^2)*d^r is the probability of finding the particle between r and r+dr
integrate this from a to b to get the probability between a and b
, for spherical symmetry probability between a and b is integral(abs(psi)^2 *4*pi*r^2)dr
abs(psi)^2 *4*pi*r^2 is just the term that we get after integrating twice for theta and phi, for spherical symmetric functions. I don't think there is any name to this term.
as for your second question, yes charge density is still rho, it is not rho * 4*pi*r^2
physics_auth wrote:The radial probability density (i.e. of finding the particle at a radial distance r independently of the angular direction in the three dimensional space) is always 4π*Ψ*Ψ r^2, where it is assumed that Ψ has spherical symmetry, i.e. Ψ=Ψ(r) only.
If you take 4π*Ψ*Ψ r^2 dr you have the RADIAL probability (not the density) of finding the particle within a concentric spherical shell of radial width equal to dr and with internal and external radii r and r+dr respectively.
If you integrate from r_a to r_b > r_a you find the probability that the particle is in a spherical shell (centered say at O) with internal radius r_a and external radius r_b > r_a. Radial probability density has meaning when there is spherical symmetry ... .
blackcat007 wrote: but in 3 dimension Integral{a->b}(Ψ*Ψ d^3r) is the probability of finding the particle between a and b where a = (x1, y1, z1) and b = (x2, y2, z2)
and by definition Ψ*Ψ is the probability density, is the word radial that is making the difference?
you probably mean d^3r = dV = dxdydz = {in spherical coordinates} = r^2 sinθ drdθdφ where
r = radial coordinate, θ = polar coordinate, φ = azimuthal coordinate -> that is how they are defined in maths.
the term "radial" refers to coordinate r in spherical coordinates. When you are asked for the radial probability density this means that they ask you about 4π * r^2 Ψ*(r)Ψ(r) = ρ(r) = radial probability density.
like the radial probability density is 4π*Ψ*Ψ r^2 for spherical symmetric case . in that case will this term lose its meaning if i used say cylindrical coordinates?
betelgeuse1 wrote:You can speak also about bi-elliptical coordinates the basic fact is that you integrate over all the variables you are not interested in. In spherical, if you are interested about the value of your quantity considering only its "radial" coordinate you "average" out the other variables. After integrating over them they become constant parameters and you got it. The fact is easy and quite used in stat-mech. I was somewhat confused about the term used in the problem "radial probability density"... probably because I did not read the term "radial"... AM I AN IDIOT!!!
physics_auth wrote:betelgeuse1 wrote:You can speak also about bi-elliptical coordinates the basic fact is that you integrate over all the variables you are not interested in. In spherical, if you are interested about the value of your quantity considering only its "radial" coordinate you "average" out the other variables. After integrating over them they become constant parameters and you got it. The fact is easy and quite used in stat-mech. I was somewhat confused about the term used in the problem "radial probability density"... probably because I did not read the term "radial"... AM I AN IDIOT!!!
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