# Asymptotic properties of Hydrogen atom wave function

• andrewkirk
In summary, Shankar is discussing constraints on the Hamiltonian operator, which must be placed on U in order for it to be Hermitian. He derives an equivalent condition that, for any two functions obeying these constraints, we must have: U_1^*\frac{dU_2}{dr}-U_2\frac{dU_1^*}{dr}_0=0. This condition is satisfied if ##lim_{r\rightarrow 0}U=c## where '##c## is constant.
andrewkirk
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I am working through an explanation of the wave function of the Hydrogen atom.

I have got as far as deriving the version of Schrodinger's equation for the one-dimensional problem in which only the radial coordinate can vary:

##[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial^2 r}+\frac{\hbar^2 l (l+1)}{2mr^2}+V(r)]U_{El}(r)=EU_{El}(r)##

It is assumed that ##V(r)=-\frac{e^2}{r}##.

The presentation I am working through says:

'Find the asymptotic behavior of ##U_{El}(r)## as ##r\rightarrow 0##...
Solution: At ##r\rightarrow 0##, the first and second terms in the Hamiltonian will dominate, so [the above equation] becomes:

##\frac{\partial^2}{\partial^2 r}U_{El}(r)=\frac{l(l+1)}{r^2}EU_{El}(r)##'

I don't see why this follows. Certainly the second term in the square brackets will dominate the third term, which is only divided by ##r##, not ##r^2##, and the right-hand side of the equation, but why would it not also dominate the first term, which is not divided by ##r## at all?

Is it possible to derive the second equation above in a more convincing way?

Thank you.

Suppose it did not exist, you'd have a static thing (U would be constant or something).

The derivative will give you how things change...

1 person
This is a standard technique in differential equations. To derive an approximate solution valid in a particular region (r → 0 in this case), assume a general form for the solution. Examine the magnitudes of each term in the equation, and keep only the leading ones.

So we assume u ~ rn and compare magnitudes of the four terms:

u'' ~ rn-2
u/r2 ~ rn-2
u/r ~ rn-1
Eu ~ rn

As r → 0, the first two terms have equal magnitude and dominate the others, so those are the ones we keep.

One could ask instead for an approximate solution for r → ∞. In that case, assume u ~ e-ar. Plugging it in, you'll find that the first and fourth terms dominate the others.

Last edited:
1 person
Thank you very much. That has completely cleared it up for me.

If I may, I would like to ask a supplementary question on the same topic area, although this time from a different text - Shankar.

Shankar (p341-2 of 2nd edition of 'Principles of Quantum Mechanics') discusses what constraints must be placed on ##U_{El}## in order for the Hamiltonian operator (part in square brackets in 1st equation - which he abbreviates to ##D_l(r)##) to be Hermitian with respect to U, ie for ##<D_l(r)U_{El}|=<U_{El}|D_l(r)## to hold.

He derives an equivalent condition that, for any two functions ##U_1## and ##U_2## obeying these constraints, we must have:

##[U_1^*\frac{dU_2}{dr}-U_2\frac{dU_1^*}{dr}]_0=0## [12.6.9]

He then states that 'this condition is satisfied if ##lim_{r\rightarrow 0}U=c## where '##c## is constant'.

I don't understand what this statement means - constant with respect to what? Obviously it is constant with respect to r, because it is a limit at a particular value of r, so that can't be his meaning. He might mean constant with respect to time but, given that we are talking about eigenvectors of the time-independent Schrodinger equation, the time dimension doesn't seem relevant. The only other interpretation I can think of is that he means:

##\exists c## such that ##\forall U: lim_{r\rightarrow 0}U(r)=c##.

But if that's what he means, I can't see how he proves that the above condition 12.6.9 follows from it.

Can anybody suggest what Shankar might be trying to say here, and how it can be proven?

Thank you very much.

## 1. What are the asymptotic properties of the Hydrogen atom wave function?

The asymptotic properties of the Hydrogen atom wave function refer to its behavior at large distances from the nucleus. This includes its decay behavior, symmetry, and overall shape.

## 2. How does the Hydrogen atom wave function behave at the origin?

At the origin, the Hydrogen atom wave function has a non-zero value, indicating that there is a non-zero probability of finding the electron at the nucleus. However, the wave function also has a singularity at the origin, meaning that it is not well-defined at that point.

## 3. What is the relationship between the energy levels of the Hydrogen atom and its wave function?

The energy levels of the Hydrogen atom are directly related to the wave function through the Schrödinger equation. The wave function describes the probability of finding the electron in a particular energy state.

## 4. How does the Hydrogen atom wave function change with increasing energy levels?

As the energy level increases, the Hydrogen atom wave function becomes more complex and oscillatory, with more nodes (points where the wave function crosses zero). The wave function also becomes more extended and less localized around the nucleus.

## 5. What is the physical significance of the asymptotic properties of the Hydrogen atom wave function?

The asymptotic properties of the wave function have important physical implications. For example, they determine the behavior of the electron as it approaches the nucleus, which is crucial for understanding chemical bonding and atomic processes. They also provide insight into the quantization of energy levels in the Hydrogen atom.

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