Calculating the expectation of a quantity using wavefunctions

In summary, the conversation is about finding the expectation value of kinetic energy in the fine structure of the hydrogen atom. The expectation value is calculated using the unperturbed wavefunction ψ0, but the exact form of ψ0 is unknown. The speaker is asking if there is a general formula for the wavefunction and the response is that the expectation value is typically calculated using the ground state wavefunction, which can be found in any book on quantum mechanics.
  • #1
spaghetti3451
1,344
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I am reading the fine structure article from Wikipedia at http://en.wikipedia.org/wiki/Fine_structure.

Under the heading 'Kinetic energy relativistic correction', we have the following:

For the hydrogen atom, V = e2/r. This implies that the expectation of V = -e2/a0n2.

Now, I know that you use the unperturbed wavefunction ψ0 to find the expectation of V, but I am not sure of the exact form of ψ0. Any help on this will be greatly appreciated.
 
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  • #2
I need the form as I want to calculate the expectation myself. Is there a general formula for the wavefunction?
 
  • #3
Usually the expectation value is computed with the ground state wavefunction. You can find the n=1, l=0, m=0 wavefunction in any book on QM.
 

FAQ: Calculating the expectation of a quantity using wavefunctions

What is the expectation value of a quantity in quantum mechanics?

The expectation value of a quantity in quantum mechanics is a mathematical calculation that represents the average value that is expected to be measured for a given physical quantity in a quantum system.

How is the expectation value calculated using wavefunctions?

The expectation value is calculated by taking the integral of the product of the wavefunction and the operator associated with the quantity of interest. This integral is then normalized by dividing by the integral of the square of the wavefunction.

What is the significance of calculating the expectation value?

Calculating the expectation value allows us to make predictions about the behavior of a quantum system. It provides insight into the most probable outcome of a measurement and can help us understand the overall behavior and characteristics of the system.

Can the expectation value ever be negative?

No, the expectation value is always a positive number or zero. This is because the wavefunction and the operator are both real-valued functions, and the integral of their product will always result in a non-negative value.

Are there any limitations to using wavefunctions to calculate the expectation value?

One limitation is that the wavefunction must be normalized in order for the expectation value to be a meaningful calculation. Additionally, the wavefunction must be well-behaved and continuous for the calculation to be accurate.

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