Physical meaning of probability density

In summary, the probability density and radial probability of finding an electron in a hydrogen atom can be calculated using different approaches, but they must be equal. The maximum probability density occurs at the nucleus (r = 0), but the probability of actually finding the electron at this point is zero due to the volume of the shell going to zero. This has practical implications, such as in electron capture, and is a purely geometrical artifact.
  • #1
Absentee
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Hi guys. I'm trying to get the idea of probability density for 1s hydrogen atom.

I just don't understand that probability density reaches maximum at nucleus (r → 0) if the most probable radius where electron can be found is at Bohr radius according to radial probability (Which also states probablity of finding electron is 0 at r → 0.

Could you please give me a more physical, visual kind of interpretation? Is there a physical meaning anyways? I've seen a lot of 'that's how the math works' but it doesn't quite work for me. Thanks!
 
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  • #2
Absentee said:
I just don't understand that probability density reaches maximum at nucleus (r → 0) if the most probable radius where electron can be found is at Bohr radius according to radial probability (Which also states probablity of finding electron is 0 at r → 0.

The probability density P(r) = |ψ|2 is the probability per unit volume.

The radial probability (which I think most books call R(r)) is the probability per unit radius (distance from center).

If you have a spherical shell of radius r and thickness dr, the probability that the electron can be found in the shell, i.e. in the range dr can be calculated two ways:

1. using the probability density and the volume of the shell: Pshell = P(r)4πr2dr

(note this is the volume of the shell itself, not the volume inside the shell!)

2. using the radial probability and the thickness of the shell: Pshell = R(r)dr

The two probabilities have to be equal, so

R(r) = P(r)4πr2

At the center (r = 0), R(0) must be zero even though P(0) is not zero (provided P(0) is not infinite, of course).

Think of this as due to the volume of the shell (itself) going to zero as r goes to zero, for a fixed dr.
 
  • #3
Thanks for the answer, but I just can't wrap my head behind PRACTICAL meaning of this. If I said that most probable location where I would find the electron would be at Bohr radius and there is practically no chance of finding it at radius 0 (as the radial probability is 0 at that location) how can I even think about other probability that states the probabilty is infinite at radius 0? What is PRACTICAL meaning of this maximum at r = 0? Is there an analogy?
 
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  • #4
Consider a small volume V, the size of an atomic nucleus, around the r = 0 point. The probability of the electron being in that volume is nonzero. This is what allows electron capture to happen:

http://en.wikipedia.org/wiki/Electron_capture

Using the volume probability density P, this probability is P(0)*V. Approximately, of course. The approximation becomes better as V becomes smaller and smaller.

The fact that R(0) = 0 is a purely geometrical artifact. Any volume probability distribution P that isn't infinite at r = 0 will give you R(0) = 0.

As an example, consider a uniform volume probability distribution, P = (some constant) everywhere in some region around r = 0. Using the equation in my previous post, R(r) = (that constant) * 4πr2, which is zero at r = 0.
 

Related to Physical meaning of probability density

1. What is the physical meaning of probability density?

Probability density refers to the likelihood of a continuous random variable taking on a particular value within a given range. It is a measure of the relative frequency of occurrence of a certain value in a continuous distribution.

2. How is probability density different from probability?

While probability refers to the likelihood of a specific outcome occurring, probability density describes the likelihood of a range of outcomes. Probability is a single value, while probability density is a function that can vary across a range of values.

3. What is the importance of understanding probability density in science?

Probability density is a fundamental concept in statistics and science. It allows us to make predictions and draw conclusions based on the likelihood of certain events occurring. It is used in a wide range of fields such as physics, biology, and economics.

4. How is probability density used in research and experiments?

In research and experiments, probability density is used to analyze and interpret data. It can help determine the likelihood of certain outcomes and can be used to make predictions and draw conclusions from the data collected.

5. How does probability density relate to the concept of uncertainty?

Probability density is closely related to uncertainty, as it represents the range of possible outcomes and their corresponding likelihood. The higher the probability density, the more certain we can be about a particular outcome, while a lower probability density implies a larger degree of uncertainty.

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