Analyze the radial distribution of the H-atom at the 2s state. How are the maxima

situated with respect to the minimum?
Is the “golden ratio” involved?

The 2s orbital has a probability distribution function that is a function of r only.

There is a maximum at r = 0 and a zero-probability node at r = 2 ao, where ao is the Bohr radius. There is a secondary spherical-shell relative maximum at r = 4 ao. In don't see where the "golden ratio" (1.618) is involved.

The probability distribution function of that state is

u^2(r) = [1/(32 pi)]*(1/ao)^3 *[(2 -(r/ao)]^2 *exp(r/ao)

See if you can find a golden ratio in the min and max locations

The probability distribution function of that state is

u^2(r) = [1/(32 pi)]*(1/ao)^3 *[(2 -(r/ao)]^2 *exp(-r/ao)

To analyze the radial distribution of the hydrogen atom at the 2s state, we need to understand its radial probability density function (RPDF). The RPDF gives the probability of finding the electron at a particular distance from the nucleus.

For the hydrogen atom, the 2s state refers to the second energy level (n = 2) and the s orbital. The wave function for the 2s orbital can be expressed as:

Ψ(2s) = (1/(2√2a₀³/2)) * (2 - r/a₀) * e^(-r/(2a₀))

where a₀ is the Bohr radius (approximately 0.529 Å) and r is the radial distance from the nucleus.

To analyze the radial distribution, we square the wave function to obtain the RPDF:

RPDF = |Ψ(2s)|² = [(1/(2√2a₀³/2)) * (2 - r/a₀) * e^(-r/(2a₀))]²

Simplifying this equation gives us:

RPDF = (1/(32πa₀³)) * (2 - r/a₀)² * e^(-r/a₀)

Now, let's analyze the behavior of the RPDF and observe the positioning of the maxima and the minimum.

To locate the maxima and minimum, we can differentiate the RPDF equation with respect to r and set it equal to zero. Unfortunately, this leads to a complex equation that cannot be easily solved analytically.

However, we can analyze the behavior qualitatively. The RPDF is a combination of exponential decay (e^(-r/a₀)) and a quadratic factor (2 - r/a₀)². As r increases, the exponential term rapidly decreases, while the quadratic term reaches its maximum at r=a₀/2.

Therefore, the maximum of the RPDF is situated approximately at r=a₀/2, which is the average distance of the electron from the nucleus in the 2s state. The minimum occurs at r=0, where the electron has the highest probability of being found at the nucleus itself.

Regarding the involvement of the "golden ratio," it does not play a direct role in the radial distribution of the hydrogen atom in the 2s state. The golden ratio (approximately 1.618) is a mathematical constant that relates two quantities in a specific ratio, and it is not directly related to the behavior or positioning of the maxima and minimum in the hydrogen atom's 2s orbital.