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The Schrödinger equation for the hydrogen-like atom

This is a revision section.
For full details of setting up and solving the hydrogen Schrödinger equation,
see my Quantum notes.
Fig.: Geometry of the hydrogen-like atom.

A hydrogen-like atom
is an atom consisting of a nucleus and just one electron; the nucleus can be bigger than just a single proton, though. H atoms, He+ ions, Li2+ ions etc. are all hydrogen-like atoms. This means the system consists of only two particles (left). Therefore the positions of both masses relative to the centre of gravity remain constant, and we can simplify the situation to that of the reduced mass, $\mu=\frac{mM}{m+M}$
, in a point located at the centre of gravity. This is not possible in a system of more than two particles (right) because the distances keep changing.

Using the reduced mass, the Hamiltonian
consists of a kinetic energy term, $-\frac{\hbar^2}{2\mu}\hat{\nabla}^2$
, and the potential energy part derives from the Coulomb potential, $-\frac{Ze^2}{4\pi\epsilon_0r}$, where $Z$ acknowledges the fact that the nucleus of a hydrogen-like atom may contain more than one positive charge.

With the del operator in spherical co-ordinates
, the Schrödinger equation, $\hat{H}\psi=E\psi$ is, then

$$-\frac{\hbar^2}{2\mu}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial\psi}{\partial r}\right)+\frac{1}{r^2\sin\vartheta}\frac{\partial}{\partial\vartheta}\left(\sin\vartheta\frac{\partial\psi}{\partial\vartheta}\right)+\frac{1}{r^2\sin^2\vartheta}\frac{\partial^2\psi}{\partial\varphi^2}\right]-\frac{Ze^2}{4\pi\epsilon_0r}\psi=E\psi$$

This can be solved by double separation of variables, and the resulting wave functions are

$$\psi(r,\vartheta,\varphi)=NR_{n,l}(r)P_l^m(\cos\vartheta){\rm e}^{{\rm i}m\varphi}$$

where $N$ is a normalisation constant ensuring that $\int\psi^{\ast}\psi{\rm d}V$ when integrating over all space, $R_{n,l}$ is the radial wave function
, $P_l^m$ is a Legendre polynomial
describing the dependence on the polar angle $\vartheta$, and ${\rm e}^{{\rm i}m\varphi}$ is the azimuth part
of the wave function. The latter two are often listed together as spherical harmonics, $Y_{l,m}$
. The precise functional dependence of $R_{n,l}(r)$ and $Y_{l,m}(\varphi,\vartheta)$ on the quantum numbers $n,l,m$
and the coordinates $r,\varphi,\vartheta$ is known and can be looked up.

n= 1 2 3 ...
l= 0 0 1 0 1 2 0...(n-1)
m= 0 0 -1 0 +1 0 -1 0 +1 -2 -1 0 +1 +2 -l...+l

The quantum numbers distinguish each state uniquely: $n$ determines the radial spread of the wave function and its energy, $l$ describes the shape of the wave function, and $m$ its orientation in space relative to some external axis such as a magnetic field or a chemical bond. The quantum numbers are not independent; the choice of $n$ limits the choice of $l$, which in turn limits the choice of $m$. A fourth quantum number, $s$, does not follow directly from solving the Schrödinger equation but is to do with spin
. The possible combinations of quantum numbers are given in the table.

Note that the energy of a state (i.e. of a wave function) depends only on $n$ but not on the other quantum numbers. This degeneracy is only strictly true for the hydrogen-like atom; any approximate solutions for higher atoms cause a dependence of the energy eigenvalue of a state on all quantum numbers.

Hydrogen spectra

Fig.: Transitions in the hydrogen spectrum.

The energy of a state with quantum number $n$ is $$E_n=-\frac{\mu Z^2e^4}{8\epsilon_0^2h^2n^2}\qquad.$$ For transitions
between two states, the energy difference between the two states must either be supplied (excitation) or emitted. Since the states have discrete energy levels, only fitting energy quanta can be involved in the transition. The energy difference between two adjacent states is $$\Delta E_{n_1,n_2}=-\frac{\mu Z^2e^4}{8\epsilon_0^2h^2}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$; thus spectral lines occur only at the corresponding energies (frequencies, wavelengths, wavenumbers,...).

This had been observed long before quantum mechanics - the first observations concern the Sun's spectrum. The constants in front of the bracket are collectively known as the Rydberg constant, $R_H$; its value was determined experimentally before quantum physics provided the link to the fundamental constants.

The different series of spectral lines discovered by Lyman, Balmer, and Paschen, correspond to the excitations from the $n$=1,2,3 states.

Wave functions, measurements, and electron density

The wave function itself, a complex function with positive and negative values, doesn't tell us much about the structure of the atom or any connectivity it may have with other atoms. The complex square of the wave function represents the probability density
of finding the electron at a given point in space when one looks (i.e. does an experiment). It does not say anything about where the electron actually is at any moment, the solutions of the Schrödinger equation only refer to which states are observable. The act of measuring throws the system into one of the eigenstates
that are solutions of the Schrödinger equation.

A recurrent idea of statistics is that temporal averages are equivalent to ensemble averages, i.e. look for a while at one object moving randomly and you will see the same set of orientations that you would see when taking a snapshot of many identical objects moving randomly. In the same way, we can interpret the probability density as a representation of the electron density in an atom, molecule, or solid.

Spherical and planar nodes

Fig.: The radial eigenfunctions of the hydrogen atom.

The radial solutions of the Schrödinger equation of the hydrogen atom, $R(r)$, are plotted on the right. Each time the quantum number $n$ increases, an additional node is created. At $n$=1, the radial function is all positive. Its maximum is at $r$=0, i.e. the point in space with the highest probability density of finding the electron is actually inside the nucleus! That is why the term probability density is used: As we move outward along the radius, the volume of a shell of equal thickness is getting larger and larger, thereby spreading out the probability over a larger volume.

Each time the quantum number $l$ is increased, one of the spherical nodes disappears again. It is replaced by a planar node that goes through the nucleus. Therefore, only $l$=0 electrons have a finite probability density at the nucleus.

Fig.: Polar cross sections of the eigenfunctions of the hydrogen atom.

The diagram on the right shows cross sections of the full wave function $\psi(r,\vartheta,\varphi)$ in the polar ($r$-$\vartheta$) plane. This representation highlights the transformation of spherical nodes into planar nodes as $l$ increases.

It is also apparent that the wave function is spreading out into space as $n$ increases, i.e. that electrons with a small $n$ are, on balance, nearer the nucleus. Given that the energy eigenvalue increases with $n$, that matches the semi-classical expectation that electrons have a lower energy if they are deep down in the Coulomb potential.

One more observation: As $l$ increases, the additional planar nodes cause the wave function to become less and less symmetric. This is compensated by the increasing number of equivalent states having the same $n$ and $l$ but different $m$. (In the diagram, only $m$=0 states are shown.) All (2$l$+1) states with the same value of $n$ and $l$ together form a perfectly spherical distribution of probability density.

Note that since the probability density is the square of the wave function, it doesn't make any difference if the wave function is positive or negative.

Spectroscopic notation

l= 0 1 2 3 4...
letter: s p d f g (then continue alphabetically)

Instead of describing a state by listing the values of its quantum numbers, a common practice is to refer to them by main quantum number, $n$, followed by a letter representing the value of $l$ as shown in the table.

Thus, a 2p state is one with $n$=2 and $l$=1 (the $m$=-1,0,+1 cases are sometimes distinguished as 2px, 2py, and 2pz); a 3d state is one with $n$=3 and $l$=2.

We now have accurate wave functions and their energies for hydrogen-like atoms. Solving the Schrödinger equation analytically is impossible for more complex systems. Instead, we can use the known system as a base and add complexity gradually, adjusting the wave functions and energies step by step. This perturbation approach also allows us to calculate the effect of interactions with the electron system, such as the interaction of radiation with matter on which spectroscopy is based.