3-Dimension Expectation Values (QM)

In summary, the expectation value of <r> <r^2> for the ground state of hydrogen can be found by integrating over all three dimensions, taking into account the normalization of the wave function and the variation in spatial volume at any given r. Alternatively, the function u(r) = rR(r) can be used for a better representation of the radial wave function.
  • #1
moo5003
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Hello, I have a problem that wants me to find the expectation value of <r> <r^2> for the ground state of hydrogen (part a.). My friend and I already completed the exercise but I'm concerned about how we found the expectation value. Since the ground state of hydrogen is only dependent on r do we only integrate over r? I notice that if we integrate over psi and phi we will add an extra 2pi^2 multiplied with what we had previously. Any help would be appreciated:

Recap- Do you integrate over all three dimensions if the wave function is only dependent on one?
 
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  • #2
moo5003 said:
Hello, I have a problem that wants me to find the expectation value of <r> <r^2> for the ground state of hydrogen (part a.). My friend and I already completed the exercise but I'm concerned about how we found the expectation value. Since the ground state of hydrogen is only dependent on r do we only integrate over r? I notice that if we integrate over psi and phi we will add an extra 2pi^2 multiplied with what we had previously. Any help would be appreciated:

Recap- Do you integrate over all three dimensions if the wave function is only dependent on one?
You need to make sure the wave functions are normalized and that you are accounting for the variation of spatial volume in the vecinity of any given r. If you are using the full wave function with its normalization constant then you need the angular integrals to get the normalization correct. There are also factors of r in the volume element dV that are important. People often look at the function u(r) = rR(r) as a better representation of the radial wave function because the amount of 3-D space associated with any given dr is proportional to r². This all comes together naturally if you use the full normalized wave function integrated over all space.
 

Related to 3-Dimension Expectation Values (QM)

What is the concept of 3-Dimension Expectation Values in Quantum Mechanics?

In quantum mechanics, 3-Dimension Expectation Values refer to the average values of physical quantities in three-dimensional space. These quantities can include position, momentum, energy, and other observable properties of a quantum system.

How are 3-Dimension Expectation Values calculated?

To calculate the 3-Dimension Expectation Values, we use the mathematical framework of quantum mechanics, specifically the Schrödinger equation, to determine the probability distribution of a quantum system. The expectation value is then calculated by taking the integral of the probability distribution multiplied by the corresponding observable property.

What is the significance of 3-Dimension Expectation Values?

3-Dimension Expectation Values provide us with a way to understand and predict the behavior of quantum systems. By calculating the expectation values, we can make predictions about the average values of physical quantities in a given three-dimensional space, which can help us to better understand the underlying principles of quantum mechanics.

How do 3-Dimension Expectation Values relate to Heisenberg's uncertainty principle?

Heisenberg's uncertainty principle states that it is impossible to know both the position and momentum of a quantum particle with absolute certainty. 3-Dimension Expectation Values help us to understand this principle by showing that the more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa.

Can 3-Dimension Expectation Values be measured experimentally?

Yes, 3-Dimension Expectation Values can be measured experimentally through a process called quantum state tomography. This involves performing multiple measurements on a quantum system and using statistical analysis to determine the expectation values of the desired properties.

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