Wave function of hydrogen in 2s state

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SUMMARY

The discussion centers on calculating the wave function of a hydrogen atom in the 2s state, specifically evaluating \(\psi_{2s}(a_0)\) at the Bohr radius \(r = a_0\). Participants confirmed that the radial wavefunction for \(n=2\) and \(l=0\) aligns with the solution manual, but discrepancies arose in the final numerical results. The calculated value was reported as \(1.57 \times 10^{14} \, m^{3/2}\), while the author questioned the derivation of the constant \(e^{-1/2}/(4(2\pi)^{1/2}) = 0.380\), which was consistent across multiple editions of the referenced textbook.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly wave functions.
  • Familiarity with hydrogen atom quantum states and their corresponding wavefunctions.
  • Knowledge of spherical harmonics and their application in quantum mechanics.
  • Basic proficiency in mathematical manipulation of exponential and polynomial functions.
NEXT STEPS
  • Review the derivation of the radial wavefunction for hydrogen in quantum mechanics.
  • Study the properties and applications of spherical harmonics in quantum systems.
  • Explore the differences between various editions of quantum mechanics textbooks for consistency in solutions.
  • Practice calculating wave functions for different quantum states of hydrogen and other atoms.
USEFUL FOR

Students of quantum mechanics, physicists working with atomic models, and educators teaching wave functions and atomic structure concepts.

tatiana_eggs
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Homework Statement


Suppose that a hydrogen atom is in the 2s state. Taking r = a0, calculate value for \psi2s(a0)

Homework Equations


I did spherical harmonics for l=0 ml=0 times the radial wavefunction for n=2 l=0. Got the same thing as the solution manual attached but when I started calculating I did not get the same final answer. I would just like someone else to punch in the numbers and see if they get the final answer in the book or perhaps something different, like I did.

The Attempt at a Solution


My final answer turned out to be 1.57*10^14 m^(3/2)

My main question is how the author got e^(-1/2) over 4(2pi)^(1/2) = 0.380, see underlined terms in pic (latex is not working for me today)
 

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That's weird, I agree with your answer.
 
Phew, that's a relief. I thought I was going crazy. What's funny is I have both the second and third edition of this book and they have that same answer.

Thanks for your help.
 

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