most probable value of r in ground state of hydrogen

[yxgao]

What is the most probable value of r for the grounds tate of hydrogen, and why? Is it just the Bohr radius?

[Tom Mattson]

Yes, it is the Bohr radius (which I'll call 'a'). The ground state wavefunction is:

ψ1s(r)=(1/π1/2a3/2)e-r/a.

The probability density is |ψ1s(r)|2, which is:

ρ1s(r)=|ψ1s(r)|2=(1/πa3)e-2r/a.

But that function is not going to give you the most probable radius. You have to take into account the fact that ρ1s is in spherical coordinates, whose volume element is:

dV=r2sin(φ)dr dθ dφ.

So, when you integrate ρ1s over all space, it gets multiplied by r2. Furthermore, since ψ1s is spherically symmetric, you can integrate over θ and φ to get what is called the radial probability density P1s(r):

P1s(r)=(4/a3)r2e-2r/a.

If you optimize this function, you will find that it has a relative maximum at r=a, the Bohr radius.

[yxgao]

Thank you so much!! That was very helpful!

[Orion1]

Quantum Hydrogen

Hydrogen wave function:
\psi_{1s} = \frac{1}{\sqrt{\pi r_0^3}} e^{-r/r_0}

Bohr radius:
r_0 = \frac{\hbar}{\alpha M_e c}

Probability density:
|\psi_{1s}|^2 = \left( \frac{1}{\pi r_0^3} \right) e^{-2r/r_0}

P(r)dr = |\psi|^2 dV

dV = 4 \pi r^2 dr

P(r)dr = |\psi|^2 4 \pi r^2 dr

Beta cloud probability:
P(r) = 4 \pi r^2 |\psi|^2

Hydrogen probability density function:
P_{1s}(r) = \left( \frac{4r^2}{r_0^3} \right) e^{-2r/r_0}

The most probable value of r corresponds to the peak of the plot of P(r) versus r. The slope of the curve at this point is zero. To evaluate the most probable value of r is by setting dP/dr = 0 and solving for r:

\frac{dP}{dr} = 0

\frac{dP}{dr} = \frac{d}{dr} \left[ \left( \frac{4r^2}{r_0^3} \right) e^{-2r/r_0} \right] = 0

Derivative operation and simplification:
e^{-2r/r_0} \frac{d}{dr} (r^2) + r^2 \frac{d}{dr} (e^{-2r/r_0}) = 0

2re^{-2r/r_0} + r^2(-2/r_0)e^{-2r/r_0} = 0

2re^{-2r/r_0} - (2r^2/r_0)e^{-2r/r_0} = 0

2r[1 - (r/r_0)]e^{-2r/r_0} = 0

Expression satisfied if:
1 - \frac{r}{r_0} = 0

Therefore:
\boxed{r = r_0}