The ground state wavefunction of the Hydrogen atom (r) is a function of the radius r only, which is related to x, y, and z by r = √x² + y² + z². In terms of x, y, and z, this wavefunction is: where A is a constant and ao the 3D Schrödinger equation for the Coulomb potential 100 (x, y, z) = Ae¯√²+y²+z²/ao is the Bohr radius. Show that this wavefunction satisfies h² mke2 ħ² 2² ² 2² + + 2m მ2 dy² əz² = 2 + V (x, y, z)v(x, y, z) = Ev(x, y, z), V(x, y, z) ke² √x² + y² + z² for some constant E, and find this constant E. You can use Wolfram Alpha or a computer to do the derivatives, but show the rest of the steps.
The ground state wavefunction of the Hydrogen atom (r) is a function of the radius r only, which is related to x, y, and z by r = √x² + y² + z². In terms of x, y, and z, this wavefunction is: where A is a constant and ao the 3D Schrödinger equation for the Coulomb potential 100 (x, y, z) = Ae¯√²+y²+z²/ao is the Bohr radius. Show that this wavefunction satisfies h² mke2 ħ² 2² ² 2² + + 2m მ2 dy² əz² = 2 + V (x, y, z)v(x, y, z) = Ev(x, y, z), V(x, y, z) ke² √x² + y² + z² for some constant E, and find this constant E. You can use Wolfram Alpha or a computer to do the derivatives, but show the rest of the steps.
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Transcribed Image Text:The ground state wavefunction of the Hydrogen atom (r) is a function of the radius r
only, which is related to x, y, and z by r = √x² + y² + z². In terms of x, y, and z, this
wavefunction is:
where A is a constant and ao
the 3D Schrödinger equation
for the Coulomb potential
100 (x, y, z)
=
Ae¯√²+y²+2²/ao
is the Bohr radius. Show that this wavefunction satisfies
ħ²
mke2
ħ² 2² 2² 2²
+ +
2m
əx² dy² Əz²
=
2
+ V (x, y, z)v(x, y, z) = Ex(x, y, z),
V(x, y, z)
ke²
√x² + y² + z²
for some constant E, and find this constant E. You can use Wolfram Alpha or a computer
to do the derivatives, but show the rest of the steps.
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