Quantum mechanics - probability of finding an electron

In summary, the wave function of the electron in the lowest (ground) state of the hydrogen atom is given by: \psi(r) = (\frac{1}{\pi a_0^3})^{1/2} exp(-\frac{r}{a_0}) a_0 = 0.529 \times 10^{-10} m. The probability of finding the electron inside a sphere of volume 1.0 pm3, centered at the nucleus (1pm = 10-12m), is 1.137 * 10-16. The probability of finding the electron in a volume of 1.0 pm3 at a distance of 52.9 pm from the nucleus, in a fixed but arbitrary direction
  • #1
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Homework Statement


The wave function of an electron in the lowest (that is, ground) state of the hydrogen atom is
[tex]\psi(r) = (\frac{1}{\pi a_0^3})^{1/2} exp(-\frac{r}{a_0})[/tex]
[tex]a_0 = 0.529 \times 10^{-10} m[/tex]
(a) What is the probability of finding the electron inside a sphere of volume 1.0 pm3, centered at the nucleus (1pm = 10-12m)?
(b) What is the probability of finding the electron in a volume of 1.0 pm3 at a distance of 52.9 pm from the nucleus, in a fixed but arbitrary direction?
(c) What is the probability of finding the electron in a spherical shell of 1.0 pm thickness, at a distance of 52.9 pm from the nucleus?

Homework Equations


[tex]|\psi(r)|^2[/tex]

The Attempt at a Solution


(a) [tex] volume = 1.0 \times 10^{-36} m^3[/tex]
using r = 0, the probability is 1.137 * 10-16.
(b), (c) What equations should I use here?
[tex]R^2|\psi(r)|^2[/tex] ?
[tex]4\pi r^2 R^2|\psi(r)|^2[/tex] ?
but I don't have R...
 
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  • #2
a) The probability of finding the electron with a real wave function [tex]\psi[/tex] in a small volume element [tex] d\tau[/tex] is [tex]|\psi|^2 d\tau[/tex]. To obtain the probability of finding the electron inside the sphere, you integrate. Can you take it from here?
 
  • #3
I don't know how to integrate it, but my teacher said although the correct way is to integrate, we won't need to integrate...
 
  • #4
I would like some help on the same problem too... I'm not sure if I'm doing it correctly. The probability of finding the electron is given by ([tex]\Psi[/tex])[tex]^{2}[/tex]dV... Though I know how to integrate I don't think its necesssary (we're not supposed to use integration). I am solving it by setting r=0 in the wave function, then squaring it, and multiplying it by dV, which I am taking to be 1.0 pm^3. I'm not sure if this is the correct way of doing it. Any help appreciated.
 

1. What is the probability of finding an electron at a specific location in an atom?

The probability of finding an electron at a specific location in an atom is described by the wave function, which is a mathematical function that represents the electron's position and momentum. The square of the wave function, known as the probability density, gives the probability of finding the electron within a certain region.

2. How does the probability of finding an electron change as it moves through an atom?

The probability of finding an electron changes as it moves through an atom due to the wave-like nature of electrons. The electron's position and momentum are described by its wave function, which spreads out and changes as the electron moves. This means that the probability of finding the electron in a specific location can vary depending on its position within the atom.

3. Can the probability of finding an electron be measured experimentally?

No, the probability of finding an electron cannot be measured directly. However, it can be indirectly measured through experiments such as electron diffraction, which can give information about the electron's wave function and probability density. The results of these experiments can be used to make predictions about the probability of finding an electron at a given location in an atom.

4. How is the probability of finding an electron related to its energy level?

The probability of finding an electron is related to its energy level through the quantum numbers. The energy level of an electron is determined by its principal quantum number, and the shape of its wave function is determined by its angular momentum quantum number. These quantum numbers also affect the probability of finding the electron at a specific location in an atom.

5. Can the probability of finding an electron ever be 100%?

No, the probability of finding an electron can never be 100% in quantum mechanics. This is due to the Heisenberg uncertainty principle, which states that it is impossible to know both the exact position and momentum of a particle at the same time. Therefore, there will always be some uncertainty in the electron's position, and the probability of finding it in a specific location will never be 100%.

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