# Hydrogen Atom

### From FSUPhysicsWiki

We now discuss the solution of the Schrödinger equation for the hydrogen atom. We are especially interested in this system because it is of direct physical interest and because it is possible to solve it exactly.

The effective one-dimensional problem for this system is

where represents the hydrogen atom, represents the helium ion, and so on. Here, is the effective mass of the atom.

We will focus on the bound states here. To solve the above equation, let us begin by writing down its solutions in the limits of small and large

In the limit of small i.e., the equation becomes

The only solution to this equation that is not divergent as is

In the opposite limit, we obtain

The only solution to this equation that does not diverge as is where

Let us now define The above asymptotic limit for large can now be written as

We now assume that the full expression for has the form,

To simplify the equation, let us introduce the dimensionless radial coordinate, so that may now be written as

Substituting this into the effective Schrödinger equation and simplifying, it becomes

where

Let us now try a series solution for

Our equation now becomes

or, upon simplification,

Setting the coefficients of all powers of to zero, we obtain

or

In the limit of large the recursion relation becomes

or

These are the coefficients of the series for We therefore see that, unless we terminate the series at a finite order (i.e., we make a polynomial), the wave function will diverge as If we wish to terminate the series at order, we set This yields the condition,

If we now solve this for the energy, we obtain the energy eigenvalues of the system,

The recursion relation now becomes

The function, can be expressed in terms of the confluent hypergeometric function,

which is a solution of Kummer's equation,

If we assume a power series solution as before, then the recursion relations are

Comparing this form to the recursion relations for our solution for we see that

and

Therefore, we may write as

where

The full normalized wave function is given by

where we have introduced the more commonly-used principal quantum number

The first few normalized wave functions for the hydrogen atom are as follows

The energy may be written as

where the Rydberg for the hydrogen atom and The degeneracy of each energy level is

To the side is a chart that depicts the energy levels for the hydrogen atom graphically for in units of . The parenthesis indicates the degeneracy due to possibile values of the magnetic quantum number *m* from − *l* to + *l*.

## Problems

**(1)** (N. Zettili, *Quantum Mechanics: Concepts and Applications*, Exercise 6.3)

An electron in a hydrogen atom is in the energy eigenstate,

**(a)** Find the normalization constant,

**(b)** What is the probability per unit volume of finding the electron at and

**(c)** What is the probability per unit radial interval of finding the electron at *r* = *a*?

**(d)** What are the expectation values of and

**(2)** (Griffiths, *Quantum Mechanics*, Problems 4.13 and 4.14)

**(a)** Find and for an electron in the ground state of hydrogen. Express your answers in terms of the Bohr radius

**(b)** What is the most probable value of in the ground state of hydrogen?