Quantum Mechanics / Uncertainty Principle Question

[fazer2014]

Homework Statement

The square of a wave function gives the probability of finding a particle at a given point. What is the probability of finding an electron in a 1s orbital within a volume of 1pm^3, centred at:
a) the nucleus
b) 50pm away from the nucleus?

Homework Equations

Heisenberg Uncertainty Principle

The Attempt at a Solution

I sense that this is actually a straightforward question, but I just can't get my head around what it's asking. I feel like the first sentence is not actually relevant to solving the question, just a little ditty of information? I'm also thrown by the 'centred at' thing. If anyone can offer an explanation for how to think about this problem it would be greatly appreciated, thanks.
(Sorry, this is actually for a chemistry class, but I searched the forums and there are a few questions on this topic, though none I could find that answered this specific question).

 

[nrqed]

Homework Statement

The square of a wave function gives the probability of finding a particle at a given point. What is the probability of finding an electron in a 1s orbital within a volume of 1pm^3, centred at:
a) the nucleus
b) 50pm away from the nucleus?

Homework Equations

Heisenberg Uncertainty Principle

The Attempt at a Solution

I sense that this is actually a straightforward question, but I just can't get my head around what it's asking. I feel like the first sentence is not actually relevant to solving the question, just a little ditty of information? I'm also thrown by the 'centred at' thing. If anyone can offer an explanation for how to think about this problem it would be greatly appreciated, thanks.
(Sorry, this is actually for a chemistry class, but I searched the forums and there are a few questions on this topic, though none I could find that answered this specific question).

It is not really an uncertainty question.

First, note that the probability is not given by the square of the wave function! The probability of finding the particle in a small volume $dV$ is actually given by

$$\bigl| \psi (r, \theta, \phi) \bigr|^2 \, dV$$

So just square the wave function at the values of r given in the questions and multiply by the small volume.

 

[fazer2014]

I see... thanks, I understand in theory. But we weren't actually given a wave function. So is it just a thought experiment or something?

 

[Orodruin]

You were given the state (1s orbital). I suggest looking up the wave function for the ground state in a hydrogen-like potential.

 

[paranormal]

I can provide a response to this question. The first sentence is indeed a relevant piece of information as it relates to the probability of finding a particle at a given point, which is the main focus of the question. The Heisenberg Uncertainty Principle states that it is impossible to know both the exact position and momentum of a particle at the same time. This principle applies to all particles, including electrons.

To solve this problem, we can use the wave function for a 1s orbital, which is given by Ψ = (1/√πa^3) * e^(-r/a), where a is the Bohr radius and r is the distance from the nucleus. The square of this wave function gives the probability density of finding the electron at a given point.

a) Since the volume given is 1pm^3, we can use the probability density formula, P = |Ψ|^2 * dV, where dV is the volume element. For a spherical volume, dV = 4πr^2dr, where r is the radius. Since the volume is centered at the nucleus, r = 0. Plugging in the values, we get P = |Ψ|^2 * 4π(0)^2 * dr = 0. This means that the probability of finding the electron at the nucleus is 0, which makes sense since the electron is most likely to be found at a distance from the nucleus.

b) For a volume centered at 50pm away from the nucleus, we can use the same formula but with r = 50pm. Plugging in the values, we get P = |Ψ|^2 * 4π(50pm)^2 * dr = 4πa^3 * e^(-50/a)^2 * dr. This gives us the probability of finding the electron within a volume of 1pm^3 centered at a distance of 50pm from the nucleus.

In summary, the probability of finding an electron in a 1s orbital within a volume of 1pm^3 is dependent on the distance from the nucleus. The closer the distance, the lower the probability, and the farther the distance, the higher the probability. This is due to the nature of the wave function and the Heisenberg Uncertainty Principle, which states that the more precisely we know the position of a particle, the less we know about its momentum