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That is,
$$\chi = \chi_{+}\alpha + \chi_{-}\beta \tag{2.200}$$
where $\chi_+$ and $\chi_-$ are complex coefficients such that $|\chi_+|^2$ is the probability of finding the electron in the 'spin up' state $\alpha$ , while $|\chi_-|^2$ is the probability of finding it in the 'spin down' state $\beta$ . The... | {
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Finally, one can prove (Problem 2.14) the identity
$$(\boldsymbol{\sigma} \cdot \mathbf{A})(\boldsymbol{\sigma} \cdot \mathbf{B}) = \mathbf{A} \cdot \mathbf{B} + i\boldsymbol{\sigma} \cdot (\mathbf{A} \times \mathbf{B})$$
[2.218]
where **A** and **B** are any two vectors, or two vector operators whose components co... | {
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We shall consider in this section the important case of *central* potentials, that is potentials V(r) which depend only upon the magnitude $r = |\mathbf{r}|$ of the vector $\mathbf{r}$ . Since V(r) is spherically symmetric, it is natural to use the spherical polar coordinates defined
<sup>[5]</sup> In general we s... | {
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Denoting the corresponding eigenfunctions by $\psi_+$ and $\psi_-$ , we have
$$\mathcal{P}\psi_{+}(\mathbf{r}) = \psi_{+}(\mathbf{r}), \qquad \mathcal{P}\psi_{-}(\mathbf{r}) = -\psi_{-}(\mathbf{r})$$
[2.245]
or
$$\psi_{+}(-\mathbf{r}) = \psi_{+}(\mathbf{r}), \qquad \psi_{-}(-\mathbf{r}) = -\psi_{-}(\mathbf{r})... | {
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A straightforward calculation (Problem 2.17) then
yields
$$e^{i\mathbf{k}\cdot\mathbf{r}} = \sum_{l=0}^{\infty} (2l+1)i^l j_l(kr) P_l(\cos\theta)$$
[2.260]
Using the addition theorem of the spherical harmonics (see the equation [A4.23] of Appendix 4) we can also write the above formula in the form of equation [2.... | {
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The Schrödinger equation [2.273] may
also be obtained by introducing the relative momentum
$$\mathbf{p} = \frac{m_2 \mathbf{p}_1 - m_1 \mathbf{p}_2}{m_1 + m_2}$$
[2.276]
together with the total momentum
$$P = p_1 + p_2 ag{2.277}$$
Since
$$\frac{\mathbf{p}_1^2}{2m_1} + \frac{\mathbf{p}_2^2}{2m_2} = \frac{\ma... | {
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If we let the operator P act on a wave function $\psi(q_1, \ldots, q_N)$ , we have
$$P\psi(q_1,\ldots q_N) = \psi(q_{P1},\ldots q_{PN})$$
[2.293]
It is worth stressing that except for the case N=2 the N! permutations P do not commute among themselves. This is due to the fact that the interchange operators $P_{ij}... | {
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The coefficient of $\lambda$ then gives
$$H_0\psi_k^{(1)} + H'\psi_k = E_k\psi_k^{(1)} + E_k^{(1)}\psi_k$$
[2.305]
while that of $\lambda^2$ yields
$$H_0\psi_k^{(2)} + H'\psi_k^{(1)} = E_k\psi_k^{(2)} + E_k^{(1)}\psi_k^{(1)} + E_k^{(2)}\psi_k$$
[2.306]
and so on.
In order to obtain the first energy correc... | {
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That is
$$\Psi_{kr} = \chi_{kr} + \lambda \psi_{kr}^{(1)} + \lambda^2 \psi_{kr}^{(2)} + \cdots$$
[2.321]
We shall also write the perturbed energy $\mathscr{E}_{kr}$ as
$$\mathscr{E}_{kr} = E_k + \lambda E_{kr}^{(1)} + \lambda^2 E_{kr}^{(2)} + \cdots$$
[2.322]
with $E_k \equiv E_{kr}$ $(r = 1, 2, ..., \alph... | {
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From [2.330] and the fact that $H_0\psi_k = E_k\psi_k$ , we then have
$$i\hbar \sum_{k} \dot{c}_{k}(t)\psi_{k}e^{-iE_{k}t/\hbar} = \sum_{k} c_{k}(t)\lambda H'(t)\psi_{k}e^{-iE_{k}t/\hbar}$$
[2.335]
where the dot indicates a derivative with respect to the time. Taking the scalar product with a particular function ... | {
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If the transition is such that the unperturbed energy is strictly conserved $(\omega_{ba}=0)$ , then we see from [2.346] that
$$P_{ba}(t) = \frac{|H'_{ba}|^2}{\hbar^2} t^2$$
[2.351]
so that the transition probability increases as $t^2$ .
2. If on the contrary one has $\omega_{ba} \neq 0$ , then one sees from [... | {
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To see how this comes about, we replace the arbitrary variation $\delta\phi$ by i $\delta\phi$ in [2.365] so that we have
$$-i\int \delta\phi^{\star}(H-E)\phi \,d\tau + i\int \phi^{\star}(H-E)\,\delta\phi\,d\tau = 0 \qquad [2.366]$$
Upon combining [2.366] with [2.365] we then obtain the two equations
$$\int \... | {
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If $\phi_1$ is a trial function orthogonal to $\phi_0$ (that is, if $\langle \phi_0 | \phi_1 \rangle = 0$ ) it may be shown (Problem 2.23) that
$$E_1 - \varepsilon_0(E_1 - E_0) \le E[\phi_1]$$
[2.380]
where $\varepsilon_0$ is the positive quantity
$$\varepsilon_0 = 1 - |\langle \psi_0 | \phi_0 \rangle|^2 \... | {
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Let $\phi_k^{(1)}$ be an arbitrary trial function and $F_1[\phi_k^{(1)}]$ the functional
$$F_{1}[\phi_{k}^{(1)}] = \langle \phi_{k}^{(1)} | H_{0} - E_{k} | \phi_{k}^{(1)} \rangle + 2 \langle \phi_{k}^{(1)} | H' - E_{k}^{(1)} | \psi_{k} \rangle \qquad [2.389]$$
We now express that this functional is *stationary*... | {
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Suppose that the normalised wave function of the system at t=0 is given by
$$\Psi(x, t = 0) = \frac{1}{\sqrt{2}} e^{i\alpha_1} \psi_1(x) + \frac{1}{\sqrt{3}} e^{i\alpha_2} \psi_2(x) + \frac{1}{\sqrt{6}} e^{i\alpha_3} \psi_3(x)$$
where the $\alpha_i$ are constants.
- (a) Write down the wave function $\Psi(x, t)... | {
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An electric field $\mathscr{E}(t)$ , directed along the X axis, is switched on at time t=0 so that the system is perturbed by an interaction
$$H'(t) = -qx\mathscr{E}(t)$$
If $\mathscr{E}(t)$ has the form
$$\mathscr{E}(t) = \mathscr{E}_0 \exp(-t/\tau)$$
where $\mathscr{E}_0$ and $\tau$ are constants, and ... | {
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In this chapter we begin our quantum mechanical study of atomic structure by considering the simplest atom, namely the hydrogen atom, which consists of a proton and an electron. Apart from small corrections, which we shall discuss in Chapter 5, the hydrogen atom may be considered as a non-relativistic system of two par... | {
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In terms of the new quantities $\rho$ and $\lambda$ , the equation [3.8] now reads
$$\left[\frac{d^2}{d\rho^2} - \frac{l(l+1)}{\rho^2} + \frac{\lambda}{\rho} - \frac{1}{4}\right] u_{E,l}(\rho) = 0$$
[3.13]
As in the case of the harmonic oscillator (see Section 2.4) we first analyse the asymptotic behaviour of $... | {
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Using
atomic units (a.u.) defined in Appendix 11, we also have
$$E_n = -\frac{Z^2}{2n^2} \left(\frac{\mu}{m}\right) \tag{3.31}$$
where we have written explicitly the electron mass (which is equal to unity in a.u.) for future convenience.
The energy values $E_n$ , which we have obtained here by solving the Schr... | {
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To this end we first define the Laguerre polynomials $L_q(\rho)$ by the relation
$$L_q(\rho) = e^{\rho} \frac{d^q}{d\rho^q} \left( \rho^q e^{-\rho} \right)$$
[3.35]
and we note that these Laguerre polynomials may also be obtained from the generating function
$$U(\rho, s) = \frac{\exp[-\rho s/(1-s)]}{1-s}$$
$$... | {
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The result is (see [A3.26])
$$\int_0^\infty e^{-\rho} \rho^{2l} [L_{n+l}^{2l+1}(\rho)]^2 \rho^2 d\rho = \frac{2n[(n+l)!]^3}{(n-l-1)!}$$
[3.52]
so that the normalised radial functions for the bound states of one-electron atoms may be written as [2]
$$R_{nl}(r) = -\left\{ \left( \frac{2Z}{na_{\mu}} \right)^{3} \fra... | {
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In some applications it is convenient to consider a different set of hydrogenic wave functions, in which the real form of the spherical harmonics is used for the angular part. As we saw in Section 2.5, the spherical harmonics in real form exhibit a directional dependence and behave like simple functions of Cartesian ... | {
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Let us return to the hydrogenic wave functions [3.48]. First of all, we note that
$$|\psi_{nlm}(r, \theta, \phi)|^2 d\mathbf{r} = \psi_{nlm}^{\star}(r, \theta, \phi)\psi_{nlm}(r, \theta, \phi)r^2 dr \sin\theta d\theta d\phi$$
[3.56]
represents the probability of finding the electron in the volume element dr (given ... | {
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$(r, \theta, \phi) \to (r, \pi - \theta, \phi + \pi)$ in spherical polar coordinates) leaves the radial part $R_{nl}(r)$ of the hydrogenic wave function unaffected, while the angular part $Y_{lm}(\theta, \phi)$ has the parity of l, as shown by [2.252]. As a result, under the parity operation $\Re \mathbf{r} = -\... | {
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The result [3.77] is a particular case of the *virial theorem*, which we shall now prove.
#### The virial theorem
Let us denote by H the Hamiltonian of a physical system and by $\Psi$ its state vector, solution of the time-dependent Schrödinger equation [2.46]. We have proved in Section 2.3 that the time rate of ... | {
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Let us recall some of the key results we have obtained for hydrogenic systems. The energy eigenvalues are given by [3.29] and the frequencies of the transitions by [3.34]. In particular, the ionisation potential $I_P = |E_{n=1}|$ is just
$$I_{\rm P} = \frac{e^2}{(4\pi\varepsilon_0)a_\mu} \frac{Z^2}{2}$$
[3.85]
an... | {
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The masses of the particles considered are $M_p = M_{\bar{p}} \approx 1836$ , $M_{\mu-} = M_{\mu+} \approx 207$ , $M_{\pi-} \approx 273$ , $M_{K-} \approx 966$ , $M_{\Sigma^-} \approx 2343$ , the unit of mass being the electron mass m.
| System | Reduced mass<br>μ | 'Radius'<br>a ... | {
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On the other hand, the electron in a high Rydberg state is very weakly bound, its binding energy being smaller than the binding energy of the ground state by a factor $n^2$ . For example the energy required to ionise a hydrogen atom with n=100 is only $1.36 \times 10^{-3}$ eV. We also remark that the energy separati... | {
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In this chapter, we shall first discuss the interaction of hydrogenic atoms with electromagnetic radiation and show how spectral lines arise, and at a later stage we shall study the photoelectric effect. In considering the interaction of an atom with radiation, there are three processes to analyse. First, just as a cla... | {
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The classical electromagnetic field in vacuo is described by electric and magnetic field vectors $\mathcal{E}$ and $\mathcal{R}$ , which satisfy Maxwell's equations [1]. We shall express these and other electromagnetic quantities in rationalised MKS units, which form part of the standard SI system. The electric fiel... | {
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radiation, which is given by
$$I(\omega) = 2\varepsilon_0 \omega^2 c A_0^2(\omega)$$
$$= \left[ \frac{N(\omega)\hbar \omega}{V} \right] c$$
$$= \rho(\omega)c$$
[4.11]
A general pulse of radiation can be described by taking $\phi = 0$ and representing $\mathbf{A}(\mathbf{r}, t)$ as a superposition of the p... | {
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The coefficients $c_k(t)$ satisfy the coupled equations [2.336] with $\lambda = 1$ ,
$$\dot{c}_b(t) = (i\hbar)^{-1} \sum_k H'_{bk}(t) c_k(t) e^{i\omega_{bk}t}$$
[4.25]
where
$$H'_{bk}(t) = \langle \psi_b | H'(t) | \psi_k \rangle$$
$$= \int \psi_b^{\star}(\mathbf{r}) H' \psi_k(\mathbf{r}) \, d\mathbf{r}$$
[4.... | {
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Hence we have
$$|c_b^{(1)}(t)|^2 = 2 \left[ \frac{eA_0(\omega_{ba})}{m} \right]^2 |M_{ba}(\omega_{ba})|^2 \int_{-\infty}^{+\infty} F(t, \, \tilde{\omega}) \, d\omega$$
[4.35]
and using the result [2.348], we obtain
$$|c_b^{(1)}(t)|^2 = 2\pi \left[\frac{eA_0(\omega_{ba})}{m}\right]^2 |M_{ba}(\omega_{ba})|^2 t$$
[4... | {
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The corresponding part of the vector potential describing the *creation* of a photon, adding a single photon to a N photon state, is
$$\mathbf{A}_{2} = \hat{\varepsilon} \left[ \frac{(N(\omega) + 1)\hbar}{2V\varepsilon_{0}\omega} \right]^{1/2} e^{-i(\mathbf{k}\cdot\mathbf{r} - \omega t + \delta_{\omega})}$$
[4.47] ... | {
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Therefore
$$\langle \psi_b | \dot{\mathbf{r}} | \psi_a \rangle = (i\hbar)^{-1} \langle \psi_b | \mathbf{r} H_0 - H_0 \mathbf{r} | \psi_a \rangle$$
$$= (i\hbar)^{-1} (E_a - E_b) \langle \psi_b | \mathbf{r} | \psi_a \rangle$$
[4.58]
or, in a more compact notation,
$$\mathbf{p}_{ba} = im\omega_{ba}\mathbf{r}_{ba} \t... | {
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Since $\rho = I/c$ (see [4.11]) and the transition rate for absorption (per atom) is $W_{ba}$ , we have from (4.69)
$$B_{ba} = \frac{W_{ba}}{\rho} = \frac{4\pi^2}{3\hbar^2} \left( \frac{e^2}{4\pi\epsilon_0} \right) |\mathbf{r}_{ba}|^2$$
[4.73]
where in the last step we have used the dipole approximation [4.69] f... | {
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In the last section, we found the probability of a radiative transition between two levels a and b, in the electric dipole approximation. For stimulated emission or absorption of radiation with a particular polarisation vector $\hat{\boldsymbol{\varepsilon}}$ , the basic expression is given by [4.63] and for spontaneo... | {
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$\Delta l = \pm 1$ [4.91]
This is the orbital angular momentum selection rule for electric dipole transitions. This rule can also be deduced in a more elementary fashion by using the recurrence relations satisfied by the associated Legendre functions $P_I^m(\cos \theta)$ . Indeed, using the expressions [2.181] for... | {
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In terms of complex exponentials, AL and AR can be written as
$$\mathbf{A}^{L} = A_{0}(\omega) [\hat{\mathbf{e}}^{L} \mathbf{e}^{i(kz-\omega t + \delta \omega)} + \text{c.c.}]$$
$$\mathbf{A}^{R} = A_{0}(\omega) [\hat{\mathbf{e}}^{R} \mathbf{e}^{i(kz-\omega t + \delta \omega)} + \text{c.c.}]$$
[4.99]
where $\ha... | {
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From the behaviour of the vector potential A under reflections, it can be inferred that the photon carries negative parity, which is consistent with the selection rule [4.87] showing that an electric dipole transition causes a change in parity of the atom.
#### Magnetic dipole and electric quadrupole transitions
Wh... | {
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"Header 2": "4.5 SELECTION RULES AND THE SPECTRUM OF ONE-ELECTRON ATOMS",
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From [4.59], we have
$$x_{ka} = \langle k|x|a\rangle = -\frac{i}{m\omega_{ka}}\langle k|p_x|a\rangle$$
[4.114a]
$$x_{ak} = \langle a|x|k\rangle = \frac{i}{m\omega_{ba}}\langle a|p_x|k\rangle$$
[4.114b]
and hence
$$f_{ka}^{x} = \frac{2i}{3\hbar} \langle a|p_{x}|k\rangle\langle k|x|a\rangle$$
[4.115a]
$$= -\fra... | {
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Therefore there is a finite probability that photons will be emitted with energies in an interval about $(E_b - E_a)$ of width $(\hbar/\tau_a + \hbar/\tau_b)$ , where $\tau_a$ and $\tau_b$ are the lifetimes of the states a and b, respectively.
Let us consider for example the spontaneous decay of an excited sta... | {
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$$\lim_{\eta \to +\infty} \int_{-n}^{+\eta} \frac{1}{x - \alpha + i\beta} \, \mathrm{d}x = -i\pi$$
[4.134a]
$$\int_{-\infty}^{+\infty} \frac{e^{-ixt}}{x - \alpha + i\beta} dx = -2\pi i e^{-i(\alpha - i\beta)t}$$
[4.134b]
with $x = \omega$ , $\alpha = \omega_{ba}$ and $\beta = 1/2\tau$ , we find that
$$\dot{... | {
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We also note from [4.145] that the Doppler broadening is proportional to the frequency $\nu_0$ and inversely proportional to the square root of the atomic mass M.
Although pressure broadening does not alter the line shape, which remains Lorentzian, the Gaussian shape produced by the Doppler effect is quite differen... | {
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when $\hbar\omega \gg |E_{1s}|$ , we have from [4.146]
$$\hbar\omega \simeq \frac{\hbar^2 k_{\rm f}^2}{2m} \tag{4.160}$$
so that
$$\frac{k}{k_{\rm f}} \simeq \frac{\hbar k_{\rm f}}{2mc} = \frac{v_{\rm f}}{2c} \tag{4.161}$$
where $v_f$ is the velocity of the ejected electron. In the non-relativistic regime, f... | {
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1) and where
$$\Psi = \Phi \exp \left[ -\frac{ie}{\hbar} \mathbf{A}(t) \cdot \mathbf{r} \right] \text{ and } \mathbf{\mathscr{E}} = -\frac{\partial \mathbf{A}}{\partial t}.$$
- 4.3 Starting from the transformed Schrödinger equation of Problem 4.2 and taking $\mathscr{E} = 2\omega A_0 \sin(\omega t)$ , use first-or... | {
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Our discussion of the energy levels and wave functions of one-electron atoms in Chapter 3 was based on the simple, non-relativistic Hamiltonian
$$H = \frac{p^2}{2\mu} - \frac{Ze^2}{(4\pi\epsilon_0)r}$$
[5.1]
where the first term represents the (non-relativistic) kinetic energy of the atom in the centre of mass syst... | {
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Moreover, it commutes with the components of the orbital angular momentum (see Problem 2.12) so that the
<sup>[3]</sup> When no confusion is possible, we shall continue to write m instead of $m_l$ for the magnetic quantum number associated with the operator $L_z$ .
perturbation $H_1$ is already 'diagonal' in l... | {
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Since $\mathbf{L} \cdot \mathbf{S}$ commutes with $\mathbf{L}^2$ , $\mathbf{S}^2$ , $\mathbf{J}^2$ and $\mathcal{J}_z$ it is apparent that the new zero-order wave functions $\psi_{nljm_i}$ form a satisfactory basis set in which the operator $\mathbf{L} \cdot \mathbf{S}$ (and
[4] Specifically, if we use th... | {
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We note that the dimensionless constant $\alpha = 1/137$ controls the scale of the splitting, and it is for this reason that it has been called the fine structure constant.
The fine structure splitting of the energy levels corresponding to n = 1, 2, 3 is illustrated in Fig. 5.1. We have used in that figure the spec... | {
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$$n = 2 \frac{\sum_{p_{1/2}, p_{3/2}} \frac{0.21 \text{ cm}^{-1}}{\sum_{p_{1/2}, p_{3/2}} \frac{0.12 \text{ cm}^{-1}}{\sum_{p_{1/2}, p_{3/2}} \frac{0.09 \text{ cm}^{-1}}{\sum_{p_{1/2}, p_{3/2}} \frac{0.09 \text{ cm}^{-1}}{\sum_{p_{1/2}, p_{1/2}} \frac{0.09 \text{ cm}^{-1}}{\sum_{p_{1/2}, p_{1/2}} \frac{0.09 \text{ cm... | {
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The result is
$$\delta E(j_1 = l + 1/2, j_2 = l - 1/2) = |E_n| \frac{(Z\alpha)^2}{nl(l+1)}$$
$$= \frac{\alpha^2 Z^4}{2n^3 l(l+1)} \text{ a.u.}$$
[5.32]
For example, in the case of atomic hydrogen the splitting of the levels j=3/2 and j=1/2 for n=2 and n=3 is, respectively, 0.365 cm<sup>-1</sup> $(4.52 \times 10^... | {
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Nevertheless, and in spite of the fact that precise optical measurements of fine structure are verv difficult to perform, small deviations from the theoretical predictions of [5.28] were observed as early as 1934. In particular, detailed experimental studies of the $H_{\alpha}$ line of atomic hydrogen indicated that ... | {
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The spin magnetic moment $\mathcal{M}_s$ gives rise to an additional interaction energy
$$H_2' = -\mathbf{M}_s \cdot \mathbf{98} = g_s \mu_{\rm B} \mathbf{98} \cdot \mathbf{S}/\hbar$$
[5.48]
The complete Schrödinger equation for a one-electron atom in a constant magnetic field, including the spin-orbit interactio... | {
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From the discussion given in Section 4.5 we see that in this case $x_{ba} = y_{ba} = 0$ and we are only concerned with $z_{ba}$ . The transition rate for emission in the solid angle $d\Omega$ of a photon with
5.8 The unit vectors $\hat{\mathbf{e}}_1$ , $\hat{\mathbf{e}}_2$ and ... | {
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"token_count": 1969,
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When the interaction caused by the external magnetic field is small compared with the spin-orbit term, the unperturbed Hamiltonian can be taken to be
$$H_0 = -\frac{\hbar^2}{2m} \nabla^2 - \frac{Ze^2}{(4\pi\varepsilon_0)r} + \xi(r)\mathbf{L} \cdot \mathbf{S}$$
[5.65]
The unperturbed wave functions are eigenfuncti... | {
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Using [5.71] and the commutation relations for the components of J, namely
$$[\mathcal{J}_x, \mathcal{J}_y] = i\hbar \mathcal{J}_z, \qquad [\mathcal{J}_y, \mathcal{J}_z] = i\hbar \mathcal{J}_x, \qquad [\mathcal{J}_z, \mathcal{J}_x] = i\hbar \mathcal{J}_y \qquad [5.72]$$
it may be shown after some manipulation that ... | {
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"token_count": 1978,
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#### Linear Stark effect
Since the ground state (100) is non-degenerate, we see from [2.308] and [5.83] that the first-order correction to its energy is given by
$$E_{100}^{(1)} = e \, \mathcal{E} \, \langle \psi_{100} | z | \psi_{100} \rangle$$
$$= e \, \mathcal{E} \, \int |\psi_{100}(r)|^2 z \, d\mathbf{r}$$
... | {
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Thus the level n=2 splits in a symmetrical way into three sublevels, one of which (corresponding to $m=\pm 1$ ) is twofold degenerate.
At this point it is worth recalling again that a classical system having an electric dipole moment **D** will experience in an electric field $\mathscr E$ an energy shift $-\mathb... | {
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where $E_{n-2} = -mc^2\alpha^2/8$ and $\Delta E = |H'_{12}| = 3e \%$ $a_0$ is the absolute value of the (first-order) energy shift. The coefficients $c_1$ and $c_2$ are easily found from the initial condition
$$\Psi(\mathbf{r}, t = 0) = \psi_{200}(r)$$
[5.99]
Using [5.92] and [5.99], we find that $c_1 ... | {
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Upon differentiation of the expression [5.102] with respect to the electric field strength, we obtain for the magnitude of the dipole moment the result
$$D = -\frac{\partial E_{100}^{(2)}}{\partial \mathcal{E}} = \bar{\alpha}\mathcal{E}$$
[5.111]
where
$$\bar{\alpha} \doteq 2e^2 \sum_{\substack{n \neq 1 \\ l,m}... | {
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Indeed, as we have seen in Chapter 4, the electric dipole transition from the state $2s_{1/2}$ to the ground state $1s_{1/2}$ is forbidden by the selection rule $\Delta l = \pm 1$ . The most probable decay mechanism of the $2s_{1/2}$ state is two-photon emission, with a lifetime of 1/7 s. Thus, in the absence of... | {
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Perot in 1897 are called *hyperfine effects*, because they produce shifts of the electronic energy levels which are usually much smaller than those corresponding to the fine structure studied in Section 5.1.
It is convenient to classify the hyperfine effects into those which give rise to splittings of the electronic ... | {
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The notation is such that #X represents a nucleus with a total of a nucleons, b of which are protons
| Nucleus | Spin<br>I | Landé factor<br>g <sub>I</sub> | Magnetic moment M <sub>N</sub><br>(in nuclear ma <b>gnetons)</b> |
|---------------------------------|-----------|----------------------... | {
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The magnetic field associated with the vector potential [5.125] is
$$\mathfrak{R} = \nabla \times \mathbf{A}$$
$$= -\frac{\mu_0}{4\pi} \left[ \mathbf{M}_{\mathbf{N}} \nabla^2 \left( \frac{1}{r} \right) - \nabla (\mathbf{M}_{\mathbf{N}} \cdot \nabla) \frac{1}{r} \right]$$
[5.128]
The spin magnetic moment of the el... | {
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We begin by considering the case $l \neq 0$ , and write [5.134] more simply as
$$H'_{\rm MD} = \frac{\mu_0}{4\pi} \frac{2}{\hbar^2} g_I \mu_B \mu_N \frac{1}{r^3} \mathbf{G} \cdot \mathbf{I}$$
[5.138]
where
$$\mathbf{G} = \mathbf{L} - \mathbf{S} + 3 \frac{(\mathbf{S} \cdot \mathbf{r})\mathbf{r}}{r^2}$$
[5.139]
... | {
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Thus
$$C_0 = \frac{\mu_0}{4\pi} \frac{16}{3} g_I \mu_B \mu_N \frac{Z^3}{a_\mu^3 n^3}$$
[5.155]
Comparing [5.146] and [5.153], and recalling that j = s for s-states, we see that for both cases $l \neq 0$ and l = 0 we have
$$\Delta E = \frac{C}{2} \left[ F(F+1) - I(I+1) - j(j+1) \right]$$
[5.156a]
with
$$C = ... | {
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It is a symmetric, second-order tensor whose components $Q_{ij}$ are defined in the following way. Let $\mathbf{R}_p$ be the coordinate of a proton with respect to the centre of mass of the nucleus, and let $X_{p1} = X_p$ , $X_{p2} = Y_p$ , $X_{p3} = Z_p$ be its Cartesian components. Then
$$Q_{ij} = \sum_{p} ... | {
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Thus all other effects (such as the relativistic corrections) are neglected and we have
$$H' = \begin{bmatrix} \frac{Ze^2}{(4\pi\varepsilon_0)2R} \left(\frac{r^2}{R^2} + \frac{2R}{r} - 3\right) & r \leq R \\ 0 & r \geq R \end{bmatrix}$$
[5.173]
The first-order energy shift due to this perturbation is
$$\Delta E =... | {
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In this chapter we begin our study of many-electron atoms by considering the simplest ones, namely atoms (or ions) consisting of a nucleus of charge Ze and two electrons. These include the negative hydrogen ion $H^-(Z=1)$ , the helium atom (Z=2), the singly ionised lithium atom $Li^+(Z=3)$ , and so on. These systems ... | {
"Header 1": "6 Two-electron atoms",
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Let us consider an atom (or ion) consisting of a nucleus of charge Ze and mass M and two electrons of mass m. As in the case of one-electron atoms, we shall begin our treatment by neglecting all but the Coulomb interactions between the particles, and by writing down the Schrödinger equation for the spatial part of the ... | {
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Until now we have not taken into account the spin of the two electrons. In the case of one-electron atoms we have seen in Chapter 5 that the electron spin only affects the fine and hyperfine structure of the spectrum. On the contrary, for two-electron atoms, we shall see that spin effects directly influence the spectru... | {
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Using the results of Table 6.1, we see that the symmetric spin function $\chi_+$ is an eigenstate of both operators $S^2$ and $S_z$ , with quantum numbers given by S=1 and $M_S=0$ , respectively. The antisymmetric spin function $\chi_-$ is also an eigenstate of both $S^2$ and $S_z$ , with corresponding quant... | {
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The configuration of each level is of the form 1s nl. The doubly excited states (for example 2s nl) are at positive energies on this scale, within the $He^+(1s) + e^-$ continuum.
and so on. In addition, a superscript to the left gives the value of the quantity 2S + 1, or *multiplicity*, which is equal to 1 for sing... | {
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The proper (zero-order) spatial wave functions of our simple independent-particle model must therefore be the symmetric (+) and antisymmetric (-) linear combinations
$$\psi_{\pm}^{(0)}(\mathbf{r}_{1}, \mathbf{r}_{2}) = \frac{1}{\sqrt{2}} \left[ \psi_{n_{1}l_{1}m_{1}}(\mathbf{r}_{1})\psi_{n_{2}l_{2}m_{2}}(\mathbf{r}_{... | {
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As pointed out above, they also exhibit the exchange degeneracy, according to which the para (+) and ortho (-) levels are degenerate in the 'zero-order' approximation [6.38]. The electron-electron repulsion term $1/r_{12}$ , which is ignored in the very simple approach leading to [6.39], will clearly raise these energ... | {
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In terms of the individual electron orbitals $u_{nlm}(\mathbf{r})$ , our new zero-order spatial wave functions, which are the properly symmetrised eigenstates of [6.41] are given by
$$\tilde{\psi}_0^{(0)}(r_1, r_2) = u_{100}(r_1)u_{100}(r_2)$$
[6.49]
for the ground state and by
$$\widetilde{\psi}_{\pm}^{(0)}(\ma... | {
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correspond to L = 0, 1, 2, . . . as we have seen in Section 6.3. Thus in this notation the ground state is denoted by $1^1S$ and the following energy levels (by order of increasing energy) are $2^3S$ , $2^1S$ , $2^3P$ , $2^1P$ , and so on. In the following two sections we shall study successively the ground state... | {
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the polar angles $(\theta_1, \phi_1)$ and $(\theta_2, \phi_2)$
$$E_{0}^{(1)} = \frac{Z^{6}}{\pi^{2}} \sum_{l=0}^{\infty} \sum_{m=-l}^{+l} \frac{(4\pi)^{2}}{2l+1} \int_{0}^{\infty} dr_{1} r_{1}^{2} \int_{0}^{\infty} dr_{2} r_{2}^{2} e^{-2Z(r_{1}+r_{2})} \frac{(r_{<})^{l}}{(r_{>})^{l+1}}$$
$$\times \int d\Omega_... | {
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Their mutual interaction energy is therefore roughly given by $e^2/(4\pi\epsilon_0)a = Ze^2/(4\pi\epsilon_0)a_0$ , which is indeed proportional to Z.
<sup>[3]</sup> The 'exact' results quoted in Table 6.3 are accurate values of the ground state energy $E_0$ of the Hamiltonian [6.2], obtained by using the Rayleigh-... | {
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"Header 2": "6.2 SPIN WAVE FUNCTIONS AND THE ROLE OF THE PAULI EXCLUSION PRINCIPLE",
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This corresponds to a reduction by more than a factor of two with respect to the relative error made in the first-order perturbation calculation. For the delicate case of H<sup>-</sup> the variational result is also a marked improvement over the first-order perturbation value, although the ground state energy of -0.473... | {
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For example, in helium this radial limit is about -2.879 a.u., and differs from the correct value of -2.904 a.u. by 0.025 a.u. (=0.68 eV). This difference is due to radial and angular correlations distributed among the higher relative partial waves in the expansion [6.83].
An important drawback of the CI approach is ... | {
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"Header 2": "6.2 SPIN WAVE FUNCTIONS AND THE ROLE OF THE PAULI EXCLUSION PRINCIPLE",
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From Appendix 8 and equation [6.1], we see that the Schrödinger equation in which this effect is ignored (i.e. in which one sets $M=\infty$ ) is then modified in two ways. Firstly, as in the case of one-electron atoms, the reduced mass $\mu=mM/(m+M)$ of the electron with respect to the nucleus replaces the mass m of... | {
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**Table 6.5** Values of the non-relativistic ionisation potential $I_{\rm p}^{*}$ , the reduced mass correction $\Delta E_1$ , the mass polarisation correction $\Delta E_2$ , the relativistic
| | $I_{\mathbf{p}}^{\infty}$ | $\Delta E_1$ | $\Delta E_2$ | $\Delta E_3$ | $\Delta E_4$ | $I_{\rm ph}^{\rm ... | {
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Adding the corrections [6.103] to the zero-order energies [6.39], we thus obtain the 'first-order' energy values
$$E_{nl,\pm} \simeq E_{1,n}^{(0)} + E_{nl,\pm}^{(1)} = -\frac{Z^2}{2} \left( 1 + \frac{1}{n^2} \right) + \mathcal{J}_{nl} \pm K_{nl}$$
3.104
The integrals $\mathcal{J}_{nl}$ and $K_{nl}$ can be eva... | {
"Header 1": "6.1 THE SCHRÖDINGER EQUATION FOR TWO-ELECTRON ATOMS. PARA AND ORTHO STATES",
"Header 2": "6.2 SPIN WAVE FUNCTIONS AND THE ROLE OF THE PAULI EXCLUSION PRINCIPLE",
"token_count": 1981,
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On the basis of the independent-particle model discussed in Section 6.4 and of our first-order perturbation theory calculation, it is reasonable to adopt for the spatial part of the wave function a simple trial function which is the antisymmetrised product of an 'inner' (1s) orbital $u_{1s}$ corresponding to the 'eff... | {
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"Header 2": "6.2 SPIN WAVE FUNCTIONS AND THE ROLE OF THE PAULI EXCLUSION PRINCIPLE",
"token_count": 1980,
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In the language of the independent particle model introduced in Section 6.4, the 'genuinely discrete' excited states which we have studied in the preceding section all correspond to singly excited states, for which one electron occupies the ground state (1s) orbital and one occupies an excited orbital. However, we noti... | {
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"token_count": 1971,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Compare your results with the 'exact' values $\langle r_1^2 + r_2^2 \rangle = 2.39$ , $\langle \delta(\mathbf{r}_1) \rangle = 1.81$ and $\langle \delta(\mathbf{r}_{12}) \rangle = 0.106$ (in a.u.).
- Evaluate explicitly the Coulomb integral $\mathcal{J}_{nl}$ and the exchange integral $K_{nl}$ , given respective... | {
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"token_count": 350,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
We have seen in the preceding chapter that Schrödinger's equation cannot be solved exactly for two-electron atoms or ions, so that approximation methods must be used. We also saw in Chapter 6 that very accurate results can be obtained for the energy levels and wave functions of helium-like atoms by performing variation... | {
"Header 1": "Many-electron atoms",
"token_count": 1972,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Moreover, since the overall effect of the (N-1) other electrons is to screen the central Coulomb attraction between an electron and the nucleus, it is clear that the inter-electron repulsion term $\sum_{i < j} 1/r_{ij}$ contains a large spherically symmetric component, which we shall write as $\sum_i S(r_i)$ . A goo... | {
"Header 1": "Many-electron atoms",
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"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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+ l$ [7.18]
As we already noted in connection with [6.48], the central field orbitals $u_{nim}(\mathbf{r})$ should not be confused with the hydrogenic wave functions $\psi_{nlm}(\mathbf{r})$ of Chapter 3, since the radial functions $R_{nl}(r)$ , solutions of [7.17] differ from the hydrogenic radial functions [3... | {
"Header 1": "Many-electron atoms",
"token_count": 1877,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
According to [7.22], the two-electron wave function $\Psi_c(q_1, q_2)$ describing the helium ground state in the central field approximation is
$$\Psi_{c}(q_{1}, q_{2}) = \frac{1}{\sqrt{2}} \begin{vmatrix} u_{100}(r_{1})\alpha(1) & u_{100}(r_{1})\beta(1) \\ u_{100}(r_{2})\alpha(2) & u_{100}(r_{2})\beta(2) \end{vmat... | {
"Header 1": "Many-electron atoms",
"token_count": 834,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
We have shown that within the framework of the central field approximation the energy levels $E_c$ of an atom (ion) having N electrons are given by summing the individual electron energies $E_{nl}$ , while the N-electron wave functions $\Psi_c(q_1, q_2, \ldots, q_N)$ are obtained by forming Slater determinants (or... | {
"Header 1": "Many-electron atoms",
"Header 3": "Electron states in a central field. Configurations, shells and subshells",
"token_count": 2018,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We are now equipped with all the necessary information to discuss the electronic structure and the 'building up' (aufbau) of atoms. For the sake of simplicity we shall only consider the ground state of neutral atoms. The Z
electrons of an atom of atomic number (nuclear charge) Z then occupy the lowest individual ener... | {
"Header 1": "Many-electron atoms",
"Header 3": "7.2 THE PERIODIC SYSTEM OF THE ELEMENTS",
"token_count": 960,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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| Electronic<br>configuration <sup>†</sup> | Term <sup>†</sup> | Ionisation<br>potential (eV) |
|----------|----------|---------------------|---------------------------------------------------------... | {
"Header 1": "Many-electron atoms",
"Header 3": "7.2 THE PERIODIC SYSTEM OF THE ELEMENTS",
"token_count": 1216,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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| 6.77 |
| 25 | Mn | manganese | [Ar]4s <sup>2</sup> 3d <sup>5</sup> | <sup>6</sup> S <sub>5/2</sub> | 7.44 ... | {
"Header 1": "Many-electron atoms",
"Header 3": "7.2 THE PERIODIC SYSTEM OF THE ELEMENTS",
"token_count": 4410,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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configuration $1s^22s^22p^6$ ). Since the 2p individual energy level is somewhat higher than the 2s level, the ionisation potential of boron (8.30 eV) is smaller than that of beryllium. For neon the ionisation potential reaches the value of 21.56 eV, which is larger than any other one, except helium. We note that si... | {
"Header 1": "Many-electron atoms",
"Header 3": "7.2 THE PERIODIC SYSTEM OF THE ELEMENTS",
"token_count": 1749,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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The chemical properties of an atom are related to the possible interactions of this atom with other ones, and in particular to its possibility of being bound with other atoms to form a molecule. At the low energies involved in chemical reactions, the interactions between atoms are mostly determined by the least tightly... | {
"Header 1": "Chemical properties and the Mendeleev classification of the elements",
"token_count": 585,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We see that each of the periods begins with an alkali element and ends with a noble gas atom, except for the seventh period,
Table 7.3 The periodic table of the elements
| Period | l IA | IIA | ШВ | IVB | VB | VIB | VIIB ... | {
"Header 1": "Chemical properties and the Mendeleev classification of the elements",
"token_count": 2025,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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If it were not obeyed, all the electrons of an atom would be in the 1s subshell (which has the lowest energy), and all atoms would be more or less alike, with spherically symmetric charge distributions having very small radii.
#### 7.3 THE THOMAS-FERMI MODEL OF THE ATOM
We now turn to a basic problem in the central... | {
"Header 1": "Chemical properties and the Mendeleev classification of the elements",
"token_count": 2017,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
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