Finding maximum of a wavefunction?

In summary, the conversation discusses a problem with finding the maximum value of r when using the product rule to differentiate a squared wavefunction. The solution is given, and the importance of the volume element dV in finding the probability of finding an electron is also mentioned. The individual asks for confirmation on their solution and acknowledges feeling foolish, but provides their updated work for future reference.
  • #1
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Homework Statement



http://[ATTACH=full]199743[/ATTACH] [ATTACH=full]199744[/ATTACH]

Question D.

[h2]Homework Equations[/h2]

[ATTACH=full]199745[/ATTACH]

[h2]The Attempt at a Solution[/h2]

The wavefunction is all real. So I can simply sqaure it... However when I do this and differentiate it using the product rule, I'm getting r = 2a(naught) as the maximum and not 4a(naught)... Any advice?

[ATTACH=full]199746[/ATTACH]
 

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  • #2
The probability of finding the electron is some volume element dV is |ψ|2 dV. The dV is important.

In spherical coordinates, dV is proportional to r2.
 
  • #3
Thanks - feel a little foolish now... Here is my updated workings incase it's useful for someone in the future:

2a4onjn.jpg


Please let me know if I've made any horrific mistakes.
 
  • #4
That looks good.
 

1. What is a wavefunction and how is it related to finding the maximum value?

A wavefunction is a mathematical description of a quantum system, which represents the probability amplitude of finding a particle in a specific state. By analyzing the wavefunction, one can determine the maximum value of a particle's position or momentum.

2. What is the significance of finding the maximum value of a wavefunction?

Finding the maximum value of a wavefunction can provide important information about the behavior and properties of a quantum system. It can also help in predicting the future evolution of the system and determining the most likely outcomes of measurements.

3. How is the maximum value of a wavefunction calculated?

The maximum value of a wavefunction is typically calculated using mathematical methods such as differentiation or integration. This involves solving the Schrodinger equation, which describes the evolution of quantum systems, to obtain the wavefunction and then determining its maximum value.

4. Can the maximum value of a wavefunction change over time?

Yes, the maximum value of a wavefunction can change over time as the quantum system evolves. This is because the wavefunction itself changes with time, and the maximum value is affected by the behavior of the system and any external influences.

5. How does the maximum value of a wavefunction relate to the uncertainty principle?

The maximum value of a wavefunction is related to the uncertainty principle, which states that the more precisely one knows the position of a particle, the less certain they can be about its momentum and vice versa. The maximum value of a wavefunction represents the most probable value of a particle's position or momentum, and the uncertainty principle shows the limitations of our knowledge about these variables.

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