Quantum Mechanics and the Hydrogen Atom

In summary, to calculate the expectation value of the potential energy for an electron in a 1s orbital for a hydrogen atom, we use the potential energy operator V=-e2/4∏ε0r and the wave function ψ= (1/4∏)1/2. By taking the integral of ψ*Vψr2sin∅drd∅dphi from 0 to r, 0 to pi, and 0 to 2pi, we can factor out terms not dependent on r,θ, or∅ and evaluate the remaining integrals. This results in the final answer of <v> = -e2/4∏εoao, which is half
  • #1
a.11en
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Calculate the expectation value of the potential energy for an electron in a 1s orbital for a hydrogen atom





Ive determined the potential energy operator to be V=-e2/4∏ε0r
and a wave function of

ψ= (1/4∏)1/2

therefore i get
<V> = ∫∫∫ψ*Vψr2sin∅drd∅dphi
integrals from 0 to r, 0 to pi, 0 to 2pi


not sure where to go from here.
 
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  • #2
Nevermind, i got it.

but if anyone is interested ill explain.

For a 1s orbital of a hydrogen the wavefunction is ψ=root(1/∏ao3 e-r/ao)

this gives
∫∫∫ψVψr2sinθdrdθd∅
integrals are zero to 2pi, zero to pi, and zero to infinity.

then factor out any terms that are not a function of r,θ, or∅.
This gives several terms outside of the integral: ∫∫∫e-2r/aorsinθdrdθd∅

then you can separate the integrals and evaluate. they were pretty easy to do.

the final answer was <v> = -e2/4∏εoao
 
  • #3
You can use the virial theorem. The expectation value you're looking for is then half the ground state energy.
 

1. What is quantum mechanics?

Quantum mechanics is a branch of physics that deals with the behavior and interactions of subatomic particles, such as electrons and protons. It explains how these particles behave and interact with each other on a microscopic level.

2. What is the hydrogen atom and why is it important in quantum mechanics?

The hydrogen atom is the simplest atom, consisting of just one proton and one electron. It is important in quantum mechanics because it was the first system to be studied and understood using quantum theory, leading to the development of this branch of physics.

3. How does quantum mechanics explain the behavior of the hydrogen atom?

Quantum mechanics explains the behavior of the hydrogen atom through the use of mathematical equations, such as the Schrödinger equation, which describe the probability of finding the electron in a particular location around the nucleus. This probabilistic behavior is different from classical physics, where the position and momentum of a particle can be precisely measured.

4. What is the significance of the energy levels in the hydrogen atom?

The energy levels in the hydrogen atom refer to the different allowed energy states that the electron can occupy around the nucleus. These energy levels are quantized, meaning they can only take on specific values, and are responsible for the unique spectral lines observed in the emission or absorption of light by hydrogen atoms.

5. How does the hydrogen atom contribute to our understanding of the universe?

The hydrogen atom, being the simplest and most well-understood system in quantum mechanics, serves as a basis for understanding more complex systems in the universe. Its behavior and interactions provide insights into the fundamental laws of nature and have led to many technological advancements, such as the development of transistors and lasers.

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