Expectation values of the electron.

In summary, the expectation value <r> of the electron-nucleus separation distance 'r' is given by the integral <r> = ʃ r |ψ|² dV. The value for ψ in the 1, 0, 0 state of hydrogen is (1/πa³)^1/2 exp(-r/a). To solve for <r>, the integral is rewritten as 4πr² dr and then integrated by parts. The resulting integral should be in the form of u dv = exp(-2r/a) r^3 dr, which can be solved by assigning u and v and integrating several times.
  • #1
hhhmortal
176
0

Homework Statement



The expectation value <r> of the electron-nucleus separation distance 'r' is:

<r> = ʃ r |ψ|² dV.

(a) Determine <r> for the 1, 0, 0 state of hydrogen.





The Attempt at a Solution



Right, I've obtained the value for ψ = (1/πa³)^1/2 exp(-r/a)

I then replace dv by 4πr² dr.

I then put all of that in the equation above and try to integrate but I can't seem to go any further from here. Any help?

Thanks very much.
 
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  • #2
Are you having trouble with the integral? Remember how to integrate by parts?
 
  • #3
I think it might be the integral. The problem is, I have an exponential term in the integral which I can't solve by parts? if so how?
 
  • #4
How are you trying to integrate, and what's the problem? Show us what you are doing.
 
  • #5
Your integral should be of the form:

u dv = exp(-2r/a) r^3 dr

All you have to do is assign u and v. Hint: you will have to integrate by parts several times.
 

Related to Expectation values of the electron.

1. What are expectation values of the electron?

Expectation values of the electron refer to the average value of a physical observable, such as position or energy, that is predicted by a quantum mechanical system. It is a mathematical concept used to describe the behavior of electrons in an atom or molecule.

2. How are expectation values calculated?

Expectation values are calculated using the mathematical principle of integration, where the value of a physical observable is multiplied by the probability of obtaining that value and then summed over all possible values. This calculation is based on the wave function of the system, which describes the probability distribution of the electron's position or energy.

3. What is the significance of expectation values in quantum mechanics?

In quantum mechanics, expectation values are important because they provide a way to make predictions about the behavior of subatomic particles, such as electrons. They allow us to calculate the most likely values of physical observables, which can then be compared to experimental data to test the accuracy of the theory.

4. How do expectation values relate to uncertainty principle?

The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. Expectation values play a role in this principle by providing a way to calculate the average position and momentum of a particle, but not the exact values. This uncertainty is inherent in the quantum nature of particles and is a fundamental aspect of quantum mechanics.

5. Can expectation values change over time?

Yes, expectation values can change over time as the wave function of a system evolves. This evolution is described by the Schrödinger equation, which dictates how the quantum state of a system changes over time. As the wave function changes, the probabilities of obtaining different values for physical observables also change, causing the expectation values to change accordingly.

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