Spectrum of hydrogenlike atoms#

Equation of hydroen atom energy can be expressed in units (\(m^{-1}\); usually \(cm^{-1}\) is used):

\[{\tilde{E}_n = \frac{E_n}{hc} = \frac{E_n}{2\pi\hbar c} = -\overbrace{\frac{m_ee^4}{4\pi c(4\pi\epsilon_0)^2\hbar^3}}^{\equiv R} \times\frac{Z^2}{n^2}\textnormal{ }}\]

where \(R\) is the Rydberg constant and we have assumed that the nucleus has an infinite mass. To be exact, the Rydberg constant depends on the nuclear mass, but this difference is very small. For example, \(R_H = 1.096 775 856 \times 10^7\) \(m^{-1} = 1.096 775 856 \times 10^5\) \(cm^{-1}\), \(R_D = 1.097 074 275 \times 10^5\) \(cm^{-1}\), and \(R_\infty = 1.097 373 153 4 \times 10^5\) \(cm^{-1}\). The latter value is for a nucleus with an infinite mass (i.e., \(\mu = m_e\)).

Eq of H atom energy can be used to calculate the differences in the energy levels:

\[\Delta\tilde{v}_{n_1,n_2} = \tilde{E}_{n_2} - \tilde{E}_{n_1} = -\frac{R_HZ^2}{n_2^2} + \frac{R_HZ^2}{n_1^2} = R_HZ^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\]

In the previous figure, the Lyman series is obtained with \(n_1 = 1\), Balmer series with \(n_1 = 2\), and Paschen series with \(n_1 = 3\). The ionization energy (i.e., when the electron is detached from the atom; see previous figure) is given by:

\[{E_i = R_HZ^2\left(\frac{1}{1^2} - \frac{1}{\infty}\right)}\]

For a ground state hydrogen atom (i.e., \(n = 1\)), the above equation gives a value of 109678 cm\(^{-1}\) = 13.6057 eV. Note that the larger the nuclear charge \(Z\) is, the larger the binding energy is.

Recall that the wavefunctions for hydrogenlike atoms are \(R_{nl}(r)Y_l^m(\theta,\phi)\) with \(l < n\). For the first shell we have only one wavefunction: \(R_{10}(r)Y_0^0(\theta,\phi)\). This state is usually labeled as \(1s\), where 1 indicates the shell number (\(n\)) and \(s\) corresponds to orbital angular momentum \(l\) being zero. For \(n = 2\), we have several possibilities: \(l = 0\) or \(l = 1\). The former is labeled as \(2s\). The latter is \(2p\) state and consists of three degenerate states: (for example, \(2p_x\), \(2p_y\), \(2p_z\) or \(2p_{+1}\), \(2p_0\), \(2p_{-1}\)). In the latter notation the values for \(m\) have been indicated as subscripts. Previously, we have seen that:

\[{m = -l, -l+1, ..., 0, ..., l-1, l}\]

For historical reasons, the following letters are used to express the value of \(l\):

\[{\phantom{\textnormal{symbo}}l = 0, 1, 2, 3, ...}{\textnormal{symbol} = s, p, d, f, ...}\]

To summarize the quantum numbers in hydrogenlike atoms:

\[{n = 1, 2, 3, ...}\]
\[{l = 0, 1, 2, ..., n-1}\]
\[{m = 0, \pm 1, \pm 2,...,\pm l}\]

For a given value of \(n\), the level is \(n^2\) times degenerate. There is one more quantum number that has not been discussed yet: Spin quantum number For one-electron systems this can have values \(\pm\frac{1}{2}\) (will be discussed in more detail later). In absence of magnetic fields the spin levels are degenerate and therefore the total degeneracy of the levels is \(2n^2\).

The total wavefunction for a hydrogenlike atom is (\(m\) is usually denoted by \(m_l\)):

}

\[{\psi_{n,l,m_l}(r,\theta,\phi) = N_{nl}R_{nl}(r)Y_l^{m_l}(\theta,\phi)}\]
\[{N_{nl} = \sqrt{\left(\frac{2Z}{na_0}\right)^3\frac{(n - l - 1)!}{2n\left[(n + l)!\right]}}}\]
\[R_{nl}(r) = \rho^le^{-\rho/2}{L_{n-l-1}^{2l+1}(\rho)}\]

Table of Wavefunctions in cartesian coordinates#

\(n\)

\(l\)

\(m\)

Wavefunction expressed in \(\sigma= \frac{zr}{a_0}\)

1

0

0

\(\psi_{1s} = \frac{1}{\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}e^{-\sigma}\)

2

0

0

\(\psi_{2s} = \frac{1}{4\sqrt{2\pi}}\left(\frac{Z}{a_0}\right)^{3/2}(2 - \sigma)e^{-\sigma/2}\)

2

1

0

\(\psi_{2p_z} = \frac{1}{4\sqrt{2\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\sigma e^{-\sigma/2} \textnormal{cos}(\theta)\)

2

1

\(\pm 1\)

\(\psi_{2p_x} = \frac{1}{4\sqrt{2\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\sigma e^{-\sigma/2}\textnormal{sin}(\theta)\textnormal{cos}(\phi)\)

\(\psi_{2p_y} = \frac{1}{4\sqrt{2\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\sigma e^{-\sigma/2}\textnormal{sin}(\theta)\textnormal{sin}(\phi)\)

3

0

0

\(\psi_{3s} = \frac{1}{81\sqrt{3\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\left(27 - 18\sigma + 2\sigma^2\right)e^{-\sigma/3}\)

3

1

0

\(\psi_{3p_z} = \frac{\sqrt{2}}{81\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\left(6 - \sigma\right)\sigma e^{-\sigma/3} extnormal{cos}(\theta)\)

3

1

\(\pm 1\)

\(\psi_{3p_x} = \frac{\sqrt{2}}{81\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\left(6 - \sigma\right)\sigma e^{-\sigma/3}\textnormal{sin}(\theta)\textnormal{cos}(\phi)\)

\(\psi_{3p_y} = \frac{\sqrt{2}}{81\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\left(6 - \sigma\right)\sigma \)

3

2

0

\(\psi_{3d_{z^2}} = \frac{1}{81\sqrt{6\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\sigma^2e^{-\sigma/3}\left(3\textnormal{cos}^2(\theta) - 1\right)\)

3

2

\(\pm 1\)

\(\psi_{3d_{xz}} = \frac{\sqrt{2}}{81\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\sigma^2 e^{-\sigma/3}\textnormal{sin}(\theta)\textnormal{cos}(\theta)\textnormal{cos}(\phi)\)

\(\psi_{3d_{yz}} = \frac{\sqrt{2}}{81\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\sigma^2 \)

3

2

\(\pm 2\)

\(\psi_{3d_{x^2-y^2}} = \frac{1}{81\sqrt{3\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\sigma^2 e^{-\sigma/3}\textnormal{sin}^2(\theta)\textnormal{cos}(2\phi)\)

\(\psi_{3d_{xy}} = \frac{1}{81\sqrt{3\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\sigma^2\)

Exaples of Laguerre polynomials#

\(L_0^k(x)\)

\(1\)

\(L_1^k(x)\)

\(k-x+1\)

\(L_2^k(x)\)

\(\frac{1}{2} \left(k^2+3 k+x^2-2 (k+2) x+2\right)\)

\(L_3^k(x)\)

\(\frac{1}{6} \left(k^3+6 k^2+11 k-x^3+3 (k+3) x^2-3 (k+2)(k+3) x+6\right)\)

\(L_4^k(x)\)

\(\frac{1}{24} (x^4-4 (k+4) x^3+6 (k+3) (k+4) x^2-4 k(k (k+9)+26) x\)

\(-96 x+k (k+5) (k (k+5)+10)+24)\)