# Expectation values r and x for electron in H2 ground state

• gfd43tg
In summary, the conversation discusses finding the expectation values for different variables in a given equation. The first part involves calculating the expectation values for ##r## and ##r^2##, which come out to be ##\frac{3}{2}a## and ##3a^2## respectively. In the second part, the conversation discusses finding the expectation value of ##x^2## and the confusion around finding the expectation value of ##x##. It is concluded that the expected value of ##x^2## cannot be zero due to the positive nature of the measurement.
gfd43tg
Gold Member

## Homework Equations

$$\psi_{100} = \frac {1}{\sqrt{\pi a^{3}}} e^{-r/a}$$

## The Attempt at a Solution

a)
$$\langle r \rangle = \frac {1}{\pi a^{3}} \int_0^{2 \pi} d \phi \int_{0}^\pi d \theta \int_0^{\infty} r^{3} e^{-2r/a} dr$$
This comes out to be ##\frac {3}{2}a##

$$\langle r^{2} \rangle = \frac {1}{\pi a^{3}} \int_0^{2 \pi} d \phi \int_{0}^\pi d \theta \int_0^{\infty} r^{4} e^{-2r/a} dr$$
Which comes out as ##3a^{2}##

b)
I know ##r^{2} = 3x^{2}##, so the answer for the expectation value of ##x^{2}## is one third the expectation value of ##r^{2}##, therefore ##\langle x^{2} \rangle = a^{2}##

However, I am confused how to find ##\langle x \rangle##. Do I just say ##x = r sin \theta##, therefore ##dx = sin \theta dr + r cos \theta d \theta##?

Last edited:
<x> is easy to find by symmetry, but you can calculate the integral if you want.

Yes, I am confused how to exploit the symmetry

Do you expect the electron to be on average more on the left or the right side? Does that question even make sense as you don't know where left and right are?

Okay, I suppose that makes sense, but then why would the square x coordinate not be zero then? By the same reasoning

Maylis said:
Okay, I suppose that makes sense, but then why would the square x coordinate not be zero then? By the same reasoning

Because a measurement of ##x^2## is positive so the expected value can't be 0.

## 1. What is the meaning of "expectation values r and x for electron in H2 ground state"?

The expectation values r and x refer to the average position and momentum of an electron in the ground state of the H2 molecule. These values are calculated using quantum mechanics and provide information about the electron's behavior in the molecule.

## 2. How are the expectation values r and x calculated for the electron in H2 ground state?

The expectation values r and x are calculated using the wave function of the electron in the ground state of the H2 molecule. The wave function is an equation that describes the probability of finding the electron at a certain position and with a certain momentum. By integrating the wave function over all possible positions and momenta, the expectation values can be determined.

## 3. What do the expectation values r and x tell us about the electron in the H2 ground state?

The expectation values r and x provide important information about the electron's behavior in the H2 molecule. The value of r tells us the average distance of the electron from the nucleus, while the value of x tells us the average momentum of the electron. These values can give insight into the stability and properties of the H2 molecule.

## 4. How do the expectation values r and x change with different quantum states of the H2 molecule?

The expectation values r and x can change with different quantum states of the H2 molecule. This is because the wave function, and therefore the probability of finding the electron at a certain position and with a certain momentum, can vary for different energy levels of the molecule. However, the overall trend of the expectation values is similar for all quantum states of H2.

## 5. Can the expectation values r and x be measured experimentally?

Yes, the expectation values r and x can be measured experimentally using techniques such as electron spectroscopy. These measurements can provide confirmation of the theoretical calculations and help to further our understanding of the behavior of electrons in molecules.

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