- #1
Bosonichadron
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Hi everyone, This question is from my problem set this week in my Phys 371 class. Any help, hints or ideas would be very much appreciated!
"Use the Heisenberg Uncertainty Principle to estimate the ground state energy in the hydrogen atom. Since the wave function that solves this problem is not a Gaussian, it will work best if you use [tex]\sigma_{r}[/tex][tex]\sigma_{p}[/tex]=[tex]\hbar[/tex]."
Where [tex]\sigma_{r}[/tex] is the standard deviation of the radius centered at the nucleus and [tex]\sigma_{p}[/tex] is the standard deviation of the momentum of the electron.
What I tried so far is to get the momentum in terms of the kinetic energy p=sqrt(2m(E-V)) [where V is the potential energy and E-V is the kinetic] and then put V in terms of r, since it would just be the coulomb potential energy...the trouble is that the algebra is devastatingly complicated and rather tedious when I try to solve for E--so it seems like there should be an easier way. Also, seems dubious to have E in terms of r without knowing what r is.
"Use the Heisenberg Uncertainty Principle to estimate the ground state energy in the hydrogen atom. Since the wave function that solves this problem is not a Gaussian, it will work best if you use [tex]\sigma_{r}[/tex][tex]\sigma_{p}[/tex]=[tex]\hbar[/tex]."
Where [tex]\sigma_{r}[/tex] is the standard deviation of the radius centered at the nucleus and [tex]\sigma_{p}[/tex] is the standard deviation of the momentum of the electron.
What I tried so far is to get the momentum in terms of the kinetic energy p=sqrt(2m(E-V)) [where V is the potential energy and E-V is the kinetic] and then put V in terms of r, since it would just be the coulomb potential energy...the trouble is that the algebra is devastatingly complicated and rather tedious when I try to solve for E--so it seems like there should be an easier way. Also, seems dubious to have E in terms of r without knowing what r is.