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For Rutherford scattering (3.21) implies $\cot(\theta_{\min}/2) = mv^2 b_{\max}/\kappa_c$ , and so
$$\sigma(b < b_{\text{max}}) = \pi b_{\text{max}}^2. \tag{3.24}$$
Third, although not realistic in practice (since beams are not infinitely large), notice that in principle the integrated cross section diverges if it... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "3 Calculational tools II",
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Then, for a b, the impulse approximation predicts
$$F_{\perp,\pm} = \pm \frac{\kappa_c}{x^2 + (b \mp a/2)^2} \frac{b}{\sqrt{x^2 + b^2}} \simeq \pm \frac{\kappa_c b}{(x^2 + b^2)^{3/2}} \left[ 1 \pm \frac{ab}{x^2 + b^2} - \frac{a^2(x^2 - 3b^2)}{4(x^2 + b^2)^2} + \cdots \right],$$
(3.29)
and so these sum to give
$$F... | {
"Header 1": "2.2.2 Decay rates",
"Header 2": "3 Calculational tools II",
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We start by converting the two-body scattering problem into a one-body problem, by isolating the centre of mass. To this end define as before R = (mAr<sup>A</sup> + mBrB)/(m<sup>A</sup> + mB) and r = r<sup>A</sup> − rB, and change variables from Ψ(rA, rB, t) to Ψ(R, r, t) in the Schr¨odinger equation, to get
$$i\frac... | {
"Header 1": "3.2.1 The equivalent one-body problem",
"token_count": 1886,
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In the absence of a potential the Schr¨odinger equation in spherical coordinates is
$$-\nabla^2 \psi = -\left[\frac{\partial^2 \psi}{\partial r^2} + \frac{2}{r} \frac{\partial \psi}{\partial r} + \frac{1}{r^2} \Delta \psi\right] = 2mE \psi, \qquad (3.41)$$
where ∆ is the following differential operator that depends... | {
"Header 1": "3.2.1 The equivalent one-body problem",
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In principle this is done by explicitly solving [\(3.38\)](#page-76-0) and fixing the integration constants by requiring agreement with the asymptotic form [\(3.44\)](#page-78-0) at large r. In practice this must often be done numerically, though it is possible to solve explicitly in closed form for some special cases ... | {
"Header 1": "3.2.1 The equivalent one-body problem",
"token_count": 2014,
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For this to be possible it must be that all of the in-coming waves in [\(3.60\)](#page-82-2) are equal to those in [\(3.61\)](#page-82-3) (for each `) once we expand the sine and cosine in terms of e <sup>±</sup>ikr. Once this is done we collect the terms in front of the out-going wave in the difference between [\(3.60... | {
"Header 1": "3.2.1 The equivalent one-body problem",
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As our first example consider again scattering from a hard sphere: U = 0 for r > R and U → ∞ for r < R. In this case the radial solution exterior to the sphere is given by [\(3.56\)](#page-81-2) and we must impose the boundary condition that ψ(r = R, θ, φ) = 0 for all values of θ and φ. This implies that R`(r = R) = 0 ... | {
"Header 1": "3.2.5 Hard-sphere scattering",
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Consider next a finite square well, with U = 0 for r > R and $U = -U_0$ for r < R. Besides being solvable, as we shall see this is a poor man's model of nuclear forces: attractive but with finite range. When solving the Schrödinger equation for this potential the heavy lifting comes when we solve the radial equation,... | {
"Header 1": "3.2.5 Hard-sphere scattering",
"Header 2": "3.2.7 An attractive square well",
"token_count": 1998,
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(3.86)
Because ` = 0 the scattering is isotropic, and the total cross section is
$$\sigma \simeq \sigma_0 = \frac{4\pi}{k^2} \sin^2 \delta_0 \simeq \frac{4\pi \delta_0^2}{k^2} \simeq 4\pi R^2 \left[ \frac{\tan(k_{\rm in}R)}{k_{\rm in}R} - 1 \right]^2,$$
(3.87)
where kin = p 2m(E + U0) ' √ 2mU0. At low energies th... | {
"Header 1": "3.2.5 Hard-sphere scattering",
"Header 2": "3.2.7 An attractive square well",
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The above discussion seems to make it inevitable that (in the absence of resonances) lowenergy nuclear cross sections become energy-independent, but Fig. [21](#page-90-0) also shows that this is not true for neutron absorption cross sections – which describe reactions where the incoming neutron is absorbed by the nucle... | {
"Header 1": "3.2.10 Low-energy absorption cross sections",
"token_count": 2049,
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That is, formally we write U = U and take ψ = ψ<sup>0</sup> + ψ<sup>1</sup> + <sup>2</sup>ψ<sup>2</sup> +· · · and substitute this into [\(3.104\)](#page-93-0). Demanding the solution to hold for all allows us to separately set to zero the coefficient of each power, leading to the sequence of equations
$$\nabla^2 \ps... | {
"Header 1": "3.2.10 Low-energy absorption cross sections",
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This amounts to solving the tower of equations, (3.105), for the corrections $\psi_1$ , $\psi_2$ and so on, which can be done using the substitutions $\mu \to k = \sqrt{2mE}$ and $J(\mathbf{x}) = 2mU(\mathbf{x})\psi_0(\mathbf{x})$ in (3.106), whose solutions, (3.111), we have just constructed. This leads to the ... | {
"Header 1": "3.2.10 Low-energy absorption cross sections",
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We now apply the above series solution to the scattering problem, and in so doing generate a perturbative Born expansion for the scattering state. To this end we start with the zerothorder (free) solution describing the incoming wave: ψ0(x) = e ikz = e <sup>i</sup>ki·<sup>x</sup> where k<sup>i</sup> is the initial mome... | {
"Header 1": "3.3.3 The Born approximation",
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This says that the energy cost (imposed by the uncertainty principle) to be localized in the area of size a should be larger than the energy available in the potential there. (This seems reasonable given our experience with the square well, which shows that in this regime we do not expect bound states to exist — as mig... | {
"Header 1": "3.3.3 The Born approximation",
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Another useful application of the Born approximation is to the scattering from a continuous charge distribution, ρ(x), rather than a point charge. In this case electrostatics tells us that the interaction potential with an incident point particle with charge Q becomes
$$U(\mathbf{r}) = \int d^3 \mathbf{x} \, \frac{\r... | {
"Header 1": "3.3.6 Scattering from charge distributions",
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Before diving into the particulars about the substructure of protons, neutrons and nuclei it is first worth understanding what it means for a particle not to have substructure.
Colloquially, what is meant is intuitive: a particle is elementary when there is no evidence for it being built from constituents (as we shal... | {
"Header 1": "4 Nucleon substructure",
"Header 2": "What is an 'elementary' particle?",
"token_count": 2048,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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In this convention g is negative for the electron rather than positive, as it is here.
<sup>25</sup>The corrections arise because the magnetic moment is defined by the energy shift of different spin states when a magnetic field is applied. But a magnetic field is itself really a quantum operator, Bˆ , which in QED do... | {
"Header 1": "4 Nucleon substructure",
"Header 2": "What is an 'elementary' particle?",
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Ground-state mesons built from u and d quarks | |
|--------------------------------------------------------|--|
|--------------------------------------------------------|--|
| Particle | spin | charge | isospin | mass | decay width | quark content |
|----------|------|--------|---------|--------... | {
"Header 1": "4 Nucleon substructure",
"Header 2": "What is an 'elementary' particle?",
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This leads to the (successful) predictions given in Table [7.](#page-107-0)
#### 4.2 Elastic scattering
Much of what we know about proton and neutron substructure comes from scattering experiments, particularly those where the nucleon is probed using particles that seem to have
**Table 7**. Baryon magnetic moment... | {
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"Header 2": "What is an 'elementary' particle?",
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When electrons elastically collide with muons their collisions are well-described by point-particle scattering, and this is a large part of why we believe both the electron and the muon to be elementary. So far as their electromagnetic interactions are concerned, muons are pretty much identical to electrons except for ... | {
"Header 1": "4 Nucleon substructure",
"Header 2": "4.2.1 Elastic $e\\mu$ scattering",
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Instead there is a separate form factor for the electron's electric
and magnetic couplings, respectively called $G_E(q^2)$ and $G_M(q^2)$ , where both are Lorentz-invariant functions of the electron's 4-momentum transfer: $q^{\mu} = k^{\mu} - (k')^{\mu}$ . Because they are Lorentz-invariant the functions $G_E$ ... | {
"Header 1": "4 Nucleon substructure",
"Header 2": "4.2.1 Elastic $e\\mu$ scattering",
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Figure 23. Measured values for the electric and magnetic proton form factors for elastic *ep* scattering. (Figure source: http://www.mit.edu/~schmidta/olympus/guide.html).
Figure 23 shows the form factors obtained by fits to elastic scattering experiments, which for *ep* scattering are ... | {
"Header 1": "The Proton's Form Factors",
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"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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So $\hat{\sigma}_a$ is the cross section for elastic electron scattering from quark type 'a' while $\hat{s} \simeq -2\hat{p} \cdot k$ is the Mandelstam variable computed using the initial quark 4-momentum, $\hat{p}^{\mu}$ , rather than the 4-momentum of the entire target nucleon.
How is $\hat{s}$ related to s?... | {
"Header 1": "The Proton's Form Factors",
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Comparing the two expressions allows the determination of W<sup>1</sup> and W2, leading (in the limit ν = ω−ω <sup>0</sup> M) to the predictions W1(q 2 , ν) = F1(x) and νW2(q 2 , ν)/M = F2(x) with
$$2F_1(x) \simeq \mathcal{P}(x) + \mathcal{O}(M/\nu)$$
and $F_2(x) = x \mathcal{P}(x)$ . (4.33)
These predictions for ... | {
"Header 1": "The Proton's Form Factors",
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We now return to much lower energies (several MeV) than in the higher energies E >∼ 200 MeV we've seen to be associated with the substructure of the proton and neutron in previous section. The goal in this section is to discuss the properties of how nucleons organize themselves into nuclei and see what this can tell us... | {
"Header 1": "5 Nuclear structure",
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Much of what is known about inter-nucleon forces comes from the observed properties of the nuclei which they combine together to make. This section summarizes some of those properties and uses them to argue that inter-nucleon forces are:
Short-ranged inasmuch as they act only over distances of order a few fm or less;... | {
"Header 1": "5 Nuclear structure",
"Header 2": "5.1 Nuclear binding energies and nucleon forces",
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Short-range attractive forces have other implications, particularly when they act on identical particles (such as two protons or two neutrons). If these identical particles are fermions (as are nucleons) then short-ranged attraction can cause a preference for pairing: it is energetically preferable for pairs of particl... | {
"Header 1": "5.1.2 Short-range attractive forces and pairing",
"token_count": 2000,
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In practice a more precise way to measure binding energies is by creating the isotope by bombarding another isotope with an appropriate beam (perhaps neutrons) and looking for the created nucleus to de-excite by emitting a photon. For example Deuterium (also called a deuteron) can be made by bombarding protons with n... | {
"Header 1": "5.1.2 Short-range attractive forces and pairing",
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This quadratic dependence on the number of charges (Z) arises because the Coulomb interaction is long-ranged, and so every charge interacts with all the other charges. The factor Z(Z − 1) simply counts the number of such pairs of charges that can interact.
Other systems, like molecules in a fluid, also exhibit this k... | {
"Header 1": "5.1.2 Short-range attractive forces and pairing",
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In this case each type of nucleon (protons and neutrons) will have a set of single-particle energy levels available, each of which Fermi statistics implies is occupied by
Figure 33. Sketch of the energy levels for independent protons (right) and neutrons (left) within a potential well.... | {
"Header 1": "5.1.2 Short-range attractive forces and pairing",
"token_count": 1991,
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(Neglect the pairing energy when doing so.) Show that your result depends on both A and on x := c<sup>C</sup> /csym, and verify that Zopt → <sup>1</sup> <sup>2</sup> A if x → 0. Show that as A → ∞ the optimal value instead satisfies Zopt → 2A1/3/x. Using the best-fit values for x what is the prediction for Zopt when A ... | {
"Header 1": "5.1.2 Short-range attractive forces and pairing",
"token_count": 1978,
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In both cases total angular momentum is conserved since the potentials are spherically symmetric, so states will be labelled by quantum numbers (n, `, `z) with ` = 0, 1, 2, · · · and `<sup>z</sup> = −`, −` + 1, · · · , ` − 1, `, and n determined by solving the appropriate radial part of the Schr¨odinger equation.
Rot... | {
"Header 1": "5.1.2 Short-range attractive forces and pairing",
"token_count": 2036,
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But are nucleon-nucleon interactions really spin-dependent? There is good evidence they are, an example of which is given by the properties of the ground and first excited states of the isotope ${}_2^5\mathrm{He}$ . We know ${}_2^4\mathrm{He}$ is very tightly bound and spinless, as expected for a 'doubly magic' nu... | {
"Header 1": "5.1.2 Short-range attractive forces and pairing",
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We start by formulating more precisely the spin-dependence of the inter-nucleon forces, since this sets up the language with which to treat the charge-independence of the nucleonnucleon interaction.
There are really several nuclear potentials under discussion when talking about internucleon interactions, depending ... | {
"Header 1": "5.1.2 Short-range attractive forces and pairing",
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In the previous sections we saw that the inter-nucleon force is spin-dependent; how is this incorporated in the above expression? Although spin-dependence is not in itself pertinent to the issue of charge-independence of nuclear forces, it is worth digressing to discuss how to incorporate it since the tools used do pla... | {
"Header 1": "5.3.1 Spin-dependent and tensor interactions",
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Exercise 5.5: In the 4-dimensional basis of states given by [\(5.29\)](#page-140-0) show that the definitions [\(5.30\)](#page-140-1), [\(5.31\)](#page-140-2) and [\(5.32\)](#page-140-3) imply s (1) and s (2) are given by the following explicit 4-by-4 matrices:
$$\begin{split} s_x^{(1)} &= \frac{1}{2} \begin{pmatri... | {
"Header 1": "5.3.1 Spin-dependent and tensor interactions",
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With these tools in hand we can now state more precisely what is meant by the chargeindependence of the inter-nucleon force. The idea is to consider the proton and neutron as if they are two 'spin-like' components of the nucleon: N<sup>↑</sup> = p and N<sup>↓</sup> = n. This is not meant as anything to do with real spi... | {
"Header 1": "5.3.2 Isospin and charge-independence of nucleon forces",
"token_count": 1920,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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In particular, because
$$2\vec{T}^{(1)} \cdot \vec{T}^{(2)} = [\vec{T}]^2 - [\vec{T}^{(1)}]^2 - [\vec{T}^{(2)}]^2 = t(t+1) - 2\left(\frac{3}{4}\right) = \begin{cases} -3/2 & \text{if } t = 0\\ & , \\ +1/2 & \text{if } t = 1 \end{cases}$$
(5.42)
when acting on the isoscalar (t = 0) and isovector (t = 1) two-nucleon ... | {
"Header 1": "5.3.2 Isospin and charge-independence of nucleon forces",
"token_count": 1240,
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How can spin-dependent, tensor and exchange potentials arise from the underlying quark and gluon physics of the strong interactions? It happens that the longest-range part of the internucleon force can be regarded as arising due to the exchange of pions between nucleons. Such an exchange is quite likely to happen for n... | {
"Header 1": "5.3.3 Pions and inter-nucleon interactions",
"token_count": 1174,
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With the previous section's general picture of nuclear binding energy in hand, we can also now better describe radioactivity in terms of nuclear decays. The general picture is that nucleons are most tightly bound in nuclei like Fe for which EB/A is maximized. Decay options start to arise as one moves away from these sp... | {
"Header 1": "5.4 Radioactivity",
"token_count": 2020,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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(5.47)
Usually the very tight binding of the α particle means this criterion is satisfied first for Z <sup>0</sup> = 2 and A<sup>0</sup> = 4 and when this is true the nucleus becomes unstable towards α emission. A comparison of the energy released for several choices of (Z 0 , A<sup>0</sup> ) is shown in Table [12](#... | {
"Header 1": "5.4 Radioactivity",
"token_count": 1672,
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Whereas α and γ decays are well-described in terms of nucleons interacting through the electromagnetic and strong interactions, an understanding of β decays required both the proposal of a new type of particle (the neutrinos) and the discovery of new interactions: the weak interactions. An entirely new type of interact... | {
"Header 1": "5.4 Radioactivity",
"Header 2": "5.4.3 β decays and multiple neutrinos",
"token_count": 1039,
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This picture of multiple neutrinos and the separate conservation of L<sup>e</sup> and L<sup>µ</sup> provided a very good description of all neutrino measurements for more than 50 years, until it began to unravel not so long ago with the discovery of neutrino oscillations. In retrospect, the first signs of a problem aro... | {
"Header 1": "5.4 Radioactivity",
"Header 2": "5.4.4 Neutrino oscillations",
"token_count": 1690,
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The total probability for this process therefore is
$$P_{\mu \to \mu}(t, t_0) = |\mathcal{A}(t, t_0)|^2 = |\mathcal{A}_0|^2 \Big\{ \cos^4 \theta + \sin^4 \theta + 2\sin^2 \theta \cos^2 \theta \cos \Big[ (E_2 - E_1)(t - t_0) \Big] \Big\}$$
$$= |\mathcal{A}_0|^2 \Big\{ 1 - 2\sin^2 \theta \cos^2 \theta + 2\sin^2 \thet... | {
"Header 1": "5.4 Radioactivity",
"Header 2": "5.4.4 Neutrino oscillations",
"token_count": 2023,
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But another complication when considering hadronic (and in particular nuclear) decays is the uncertainty of their structure, since this is poorly understood and often cannot be computed reliably. An important exception where the nuclear structure does not interfere with β-decay calculations arises for decays between tw... | {
"Header 1": "5.4 Radioactivity",
"Header 2": "5.4.4 Neutrino oscillations",
"token_count": 2005,
"source_pdf": "datasets/websources/Physics_v1/Physics/PPNotes.pdf"
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**Exercise 5.11:** Neutron decay is the decay of the simplest 'nucleus.' Theory predicts the invariant differential rate for neutron decay — with 4-momentum assignments $n(r) \to p(p) + e(k) + \overline{\nu}(q)$ — is given to good approximation (in the neutron rest frame) by
$$\mathcal{M}(n \to p \, e \overline{\... | {
"Header 1": "5.4 Radioactivity",
"Header 2": "5.4.4 Neutrino oscillations",
"token_count": 2008,
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Heisenberg's treatment of this problem focusses on the ladder operator
$$A := \frac{1}{\sqrt{2m\omega}} \left( m\omega X + i P \right) = \frac{1}{\sqrt{2m\omega}} \left( m\omega x + \frac{\partial}{\partial x} \right), \tag{6.4}$$
and its adjoint,
$$A^{\star} := \frac{1}{\sqrt{2m\omega}} \left( m\omega X - i P ... | {
"Header 1": "5.4 Radioactivity",
"Header 2": "5.4.4 Neutrino oscillations",
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We now formalise this resemblance more explicitly. To this end suppose we consider a noninteracting particle whose single-particle states are labelled by momentum and a collection of other labels, |p σi, where σ denotes all of the other labels (spin, charge, baryon number, and so on) required to uniquely specify a give... | {
"Header 1": "6.2.1 Creation and annihilation for bosons",
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Keeping track of the density of states associated with the switch from discrete to continuum normalization, its eigenvalues count the number of particles in the following precise sense:
$$a_{p\sigma}^{\star}a_{p\sigma}|(\mathbf{q}_{1}\zeta_{1})_{n_{1}};\ldots;(\mathbf{q}_{r}\zeta_{r})_{n_{r}}\rangle = \sum_{j=1}^{r}n... | {
"Header 1": "6.2.1 Creation and annihilation for bosons",
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Consequently, because |q<sup>1</sup> ζ1; q<sup>2</sup> ζ2i = a ? q1ζ<sup>1</sup> a ? q2ζ<sup>2</sup> |0i we impose the following anticommutation relations for fermionic operators
$$\left\{ a_{q_1\zeta_1}^{\star}, a_{q_2\zeta_2} \right\} := a_{q_1\zeta_1}^{\star} a_{q_2\zeta_2} + a_{q_2\zeta_2} a_{q_1\zeta_1}^{\star} ... | {
"Header 1": "6.2.1 Creation and annihilation for bosons",
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To see how this works imagine a world containing only two kinds of particles: a heavy spinless boson, h, with mass M and a lighter spin-half fermion, f, with mass m < <sup>1</sup> <sup>2</sup> M. We assume relativistic single-particle dispersion relations, so
$$\epsilon(\mathbf{p}) = \sqrt{\mathbf{p}^2 + m^2} \quad \... | {
"Header 1": "6.2.1 Creation and annihilation for bosons",
"Header 2": "6.3.1 Interactions",
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There is one feature about interactions that the above discussion makes obscure: the locality of interactions. That is, we expect that if systems that are sufficiently far apart from one another at a given time and start off in uncorrelated states, then their evolution should preserve their lack of correlation. Since p... | {
"Header 1": "6.2.1 Creation and annihilation for bosons",
"Header 2": "6.3.2 Fields",
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Rather than expressions like (6.50) and (6.51), in relativistic theories one instead always finds the position-space field is given by expressions like
$$\psi(\mathbf{x}) = \int \frac{\mathrm{d}^3 \mathbf{p}}{\sqrt{(2\pi)^3 2\varepsilon(\mathbf{p})}} \left[ a_p e^{ipx} + \bar{a}_p^{\star} e^{-ipx} \right], \tag{6.62}... | {
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"Header 2": "6.3.2 Fields",
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The poster child of a relativistic quantum field theory is Quantum Electrodynamics, in which it is the electromagnetic field that gets expanded in terms of creation and annihilation operators for photons, as in [\(6.62\)](#page-171-2). That is, consider electric and magnetic fields describing an electromagnetic wave, w... | {
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This eigenvalue condition can be solved explicitly, leading to the following expression for |αi in terms of the occupation-number basis, |ni:
$$|\alpha\rangle = e^{-\frac{1}{2}|\alpha|^2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle = e^{-\frac{1}{2}|\alpha|^2} e^{\alpha a^*} |0\rangle.$$
(6.72)
Particl... | {
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This section gives a brief summary of the particle content and some of the properties and puzzles of the Standard Model, which is the quantum field theory that describes all but a very few relatively recent experiments and observations. (A list of the apparent failures of the Standard Model is given in the final subsec... | {
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Each generation contains two kinds of particles that do not take part in the strong interactions (more about which later), called leptons. Of these, the charged leptons (e, µ and τ ) must differ from their antiparticles because they carry electric charge. Because they have spin-half there is a total of 4 spin states fo... | {
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"Header 2": "7.1.1 Leptons",
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The other two kinds of fermions in each generation are quarks, which (unlike leptons) do participate in the strong interactions. There are two species of quarks in each generation: an up-type quark with charge +<sup>2</sup> 3 and a down-type one with charge − 1 3 . Since each of these carries charge they are distinct f... | {
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"Header 2": "7.1.2 Quarks and the hadronic zoo",
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As discussed above, fundamental bosons in the Standard Model tend to be associated with forces. Although not normally included in the Standard Model, this includes the spin-2 graviton which is the particle associated with waves in the gravitational field.
For the Standard Model proper there are a variety of bosons, e... | {
"Header 1": "7.2 Bosons and the four forces",
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The only elementary boson in the Standard Model that is not spin-1 is the Higgs boson, which is spinless. The Higgs boson was the last Standard Model particle to be found, being discovered only in 2013 with a mass m<sup>h</sup> = 125 GeV. The Higgs particle plays a special role in the Standard Model because it couples ... | {
"Header 1": "7.2.3 The Higgs boson",
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Although the Standard Model is an extremely successful synthesis of what we know about the structure of Nature, it gets a few things wrong and so these notes close with a brief summary of five of its known problems.
#### Neutrino Oscillations
The Standard Model predicts that neutrino masses vanish, and so cannot in... | {
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"Header 2": "7.3 Where the Standard Model fails",
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The ΛCDM description of cosmology is very successful, but only if the universe is started off in a very particular and unusual initial state. This is because the CMB is seen to have an almost uniform temperature in all directions in the sky, even though in the standard cosmology there has not yet been enough time in th... | {
"Header 1": "Dark Energy",
"Header 3": "Primordial Fluctuations",
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| 1<br>G/~c)<br>1/Mp<br>(=<br>2 | = | 10−29<br>8.1897<br>× | 2/eV<br>c | = | 10−35<br>1.6161<br>× | mc/~ |
|-------------------------------|---|----------------------|-----------|---|----------------------|------|
| 1/mp | = | 10−9<br>1.0658<br>× | 2/eV<br>c | = | 10−16<br>2.1031<br>× | mc/~ |... | {
"Header 1": "1. Length and Time",
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| 1 eV | = | 10−9 | GeV | = | 106<br>5.06773<br>× | ~c/m |
|------------------------------|---|-----------------------|--------|---|----------------------|------|
| 1 keV | = | 10−6 | GeV | = | 109<br>5.06773<br>× | ~c/m |
| 1 MeV ... | {
"Header 1": "2. Microscopic Energy and Mass",
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#### Physics of atoms and molecules
B. H. Bransden and C. J. Joachain
Copublished in the United States with John Wiley & Sons, Inc., New York
Longman Scientific & Technical
Longman Group UK Limited, Longman House, Burnt Mill, Harlow, Essex CM20 2JE, England and Associated Companies... | {
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| _ | | | |
|---|-------|----------------------------------------------------------------------|-----|
| 1 | Elec | trons, photons and atoms | ] |
| | 1.1 | The atomic nature of matter ... | {
"Header 3": "Contents",
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General features | 465 |
| | 11.3 The method of partial waves | 468 |
| | 11.4 The integral equ... | {
"Header 3": "Contents",
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The physics of atoms and molecules which constitutes the subject matter of this book rests on a long history of discoveries, both experimental and theoretical. A complete account of the historical development of atomic and molecular physics lies far outside the scope of this volume. Nevertheless, it is important to rec... | {
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"token_count": 1986,
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Since one Faraday (96484.6 C) liberates one mole of a monovalent substance and because one mole contains $N_{\rm A}$ atoms, where $N_{\rm A}$ is Avagadro's number, the 'natural unit of electricity', e, is given by
$$e = \frac{F}{N_{\rm A}} \tag{1.4}$$
Stoney suggested the word 'electron' for this unit, and he o... | {
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However, a brilliant modification of Wilson's method, by R. A. Millikan in 1909, gave the first accurate value for the magnitude, e, of the electronic charge.
In Millikan's experiments very small oil droplets a few microns in diameter were formed by spraying mechanically from a nozzle. The droplets became charged by ... | {
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We see that, for fixed $\lambda$ , $R(\lambda)$ increases with increasing T. At each temperature, there is a wavelength $\lambda_{\max}$ , for which $R(\lambda)$ has its maximum value. Using general thermodynamical arguments it had been predicted in 1893 by W. Wien that $\lambda_{\max}$ would vary inversely wit... | {
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In terms of frequency,
$$\rho(\nu) = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu/kT} - 1}$$
[1.31]
By expanding the denominator in [1.30], it is easy to show that at long wavelengths $\rho(\lambda) \to 8\pi kT/\lambda^4$ in agreement with the Rayleigh-Jeans formula [1.25]. On the other hand, for short wavelengths, ... | {
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It follows that the maximum kinetic energy of a photoelectron is given by
$$\frac{1}{2}mv_{\max}^2 = h\nu - W$$
[1.39]
This relation is called Einstein's equation. The threshold frequency $\nu_t$ is determined by the work function since in this case $v_{\text{max}} = 0$ , from which
$$h\nu_{\mathsf{t}} = W \ta... | {
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After the collision, the photon has an energy $E_1 = hc/\lambda_1$ and a momentum $\mathbf{p}_1(p_1 = E_1/c)$ in a direction making an angle $\theta$ with the direction of incidence, while the electron recoils with a momentum $\mathbf{p}_2$ making an angle $\phi$
1.12 A photon... | {
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Alpha particles produce scintillations in zinc sulphide and can be detected by observing a screen, coated with this substance, with a microscope. Most of the $\alpha$ particles are deflected through very small angles (<1°), but some are deflected through large angles; about 1 in 8000 being deflected through angles gr... | {
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Because of this a new scale has been introduced recently, in which the isotope of carbon $^{12}C$ is assigned a mass of 12 atomic mass units (a.m.u. or u). The absolute value of the atomic mass unit is
$$1 \text{ a.m.u.} = 1.661 \times 10^{-27} \text{ kg}$$
[1.60]
and differs very slightly from the previous unit ... | {
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As a result, atoms can only exist in certain allowed energy levels, with energies $E_a$ , $E_b$ , $E_c$ , . . .
Bohr further postulated that an electron in a stable orbit does not radiate electromagnetic energy, and that radiation can only take place when a transition is made between the allowed energy levels. To ... | {
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To complete the system the unit of charge is taken to be the absolute magnitude e of the electronic charge and the permittivity of free space $\varepsilon_0$ is $1/4\pi$ . In atomic units $(m = \hbar = e = 1, 4\pi\varepsilon_0 = 1)$ , we have
$$E_n = -\frac{1}{2}Z^2/n^2 \text{ a.u.}$$
[1.87]
The ground state en... | {
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**Table 1.2** Rydberg constants for some hydrogenic systems and the wavelengths of the resonance lines $n = 2 \rightarrow n = 1$
| | Wavelength of the transition (Å) | $\tilde{R}(M)(\mathrm{cm}^{-1})$ ... | {
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In the main the ordering of the elements of the table was given by their atomic weights $\mu$ , although Mendeleev found that certain pairs had to be reversed in order to preserve the periodicity of the chemical properties. For example in order of weights the 18th element is potassium ( $\mu = 39.102$ ) and the 19th a... | {
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We shall now discuss an experiment of fundamental importance, carried out by O. Stern and W. Gerlach in 1922, to measure the magnetic dipole moments of atoms. The results demonstrated, once more, the inability of classical mechanics to describe atomic phenomena and confirmed the necessity of a quantum theory of angular... | {
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If $L_z$ is also quantised in the form
$$L_z = m\hbar ag{1.119}$$
where m is a positive or negative integer or zero, then m must take on the values -l, -l+1, ..., l-1, l and the multiplicity $\alpha$ must be equal to (2l+1). The number m is known as a magnetic quantum number. In fact as we shall see in the next... | {
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In our brief historical survey, we have seen how, as knowledge of atomic structure increased, evidence accumulated that a description in terms of classical physics – Newton's laws of mechanics and Maxwell's electromagnetic equations – was inadequate. Electromagnetic radiation displays particle as well as wave character... | {
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Show that:
- (a) The number of photons per unit volume is $N = 2.029 \times 10^7 T^3$ photons/m<sup>3</sup>. [Hint: $\int_0^\infty x^2 (e^x 1)^{-1} dx = 2.40411$ ]
- (b) The average energy per photon is $\bar{E} = 3.73 \times 10^{-23} \ T$ joules = $2.33 \times 10^{-4} \ T$ eV.
- 1.7 A photon of wavelength $\la... | {
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In Chapter 1, we discussed some of the evidence for the atomic nature of matter. We also learned that the classical Newtonian form of mechanics could not describe phenomena on the atomic scale. In particular, we saw that experiments involving the diffraction of electrons or atoms by crystals demonstrate that particles ... | {
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The experiments on the corpuscular nature of the electromagnetic radiation, which we discussed in the previous chapter, require that with the electromagnetic field we must associate a particle, the photon, whose energy E and magnitude of momentum p are related to the frequency $\nu$ and wavelength $\lambda$ of the ... | {
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It has a maximum at x=0 and $|\psi(x)|$ falls to 1/e of its maximum value at $x=\pm 2\hbar/\gamma$ , so that the 'width' of the function $|\psi(x)|$ is $\Delta x=4\hbar/\gamma$ , as shown in Fig. 2.1. It is important to remark that if we increase $\gamma$ so that the width $\Delta x$ is decreased and the func... | {
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Since the probability of finding the particle somewhere must be unity, we deduce from [2.24] that the wave function $\Psi(\mathbf{r}, t)$ should be *normalised* so that
$$\int |\Psi(\mathbf{r}, t)|^2 d\mathbf{r} = 1$$
[2.26]
where the integral extends over all space. Wave functions satisfying this condition are... | {
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We first notice that the wave packet [2.19] for a free particle satisfies the differential equation $\frac{\partial}{\partial t} = \frac{\hbar^2}{2}$
$$i\hbar \frac{\partial}{\partial t} \Psi = -\frac{\hbar^2}{2m} \nabla^2 \Psi$$
[2.42]
because $E(p) = p^2/2m$ , and this is called the Schrödinger equation for a f... | {
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That is,
$$m\frac{\mathrm{d}\langle \mathbf{r}\rangle}{\mathrm{d}t} = \langle \mathbf{p}\rangle, \qquad \frac{\mathrm{d}\langle \mathbf{p}\rangle}{\mathrm{d}t} = -\langle \nabla V\rangle$$
[2.57]
#### Time-independent Schrödinger equation and energy eigenfunctions
When the potential does not depend on the time, t... | {
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#### 2.3 EXPANSIONS, OPERATORS AND OBSERVABLES
We shall assume that all the orthonormal eigenfunctions $\psi_n(\mathbf{r})$ of a given Hermitian operator A form a *complete set*, in the mathematical sense that an arbitrary normalised wave function $\Psi(\mathbf{r}, t)$ can be expanded in terms of them:
$$\Psi... | {
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Let us assume that the potential is independent of time, so that the time-dependent Schrödinger equation [2.47] admits stationary state solutions of the form [2.58]. Expanding the general solution of [2.47] in terms of energy eigenfunctions, we then write (see [2.71] and [2.73])
$$\Psi(\mathbf{r}, t) = \sum_{E} c_{E}... | {
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In particular, any 'old' basis vector may be expressed in the 'new' basis as
$$\psi_j = \sum_k U_{kj} \varphi_k \tag{2.96}$$
where
$$U_{ki} = \langle \varphi_k | \psi_i \rangle \tag{2.97}$$
In the same way, the reverse expansion is
$$\varphi_k = \sum_j \langle \psi_j | \varphi_k \rangle \psi_j = \sum_j U_{kj}... | {
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By taking suitable linear combinations of any degenerate eigenfunctions, this set can be made orthonormal,
$$\langle \psi_{i'j'} | \psi_{ij} \rangle = \delta_{ii'} \delta_{jj'}$$
[2.115]
From [2.114], we see that
$$[A, B]\psi_{ij} = (AB - BA)\psi_{ij} = 0$$
[2.116]
and since any wave function $\Psi$ can be ex... | {
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Indeed, the position uncertainty is roughly given by $\Delta x = a$ . The corresponding momentum uncertainty is therefore $\Delta p_x \sim \hbar/a$ , leading to a minimum kinetic energy of order $\hbar^2/ma^2$ , in qualitative agreement with the value of $E_1$ .
There is an important difference between the two cl... | {
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This is consistent with the following definition of the Hermite polynomials:
$$H_n(\xi) = (-1)^n e^{\xi^2} \frac{d^n e^{-\xi^2}}{d\xi^n}$$
[2.144]
The first few Hermite polynomials, obtained from [2.144], are
$$H_0(\xi) = 1$$
$$H_1(\xi) = 2\xi$$
$$H_2(\xi) = 4\xi^2 - 2$$
$$H_3(\xi) = 8\xi^3 - 12\xi$$
$$H_... | {
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For example,
$$[L_z, \mathbf{L}^2] = [L_z, L_x^2] + [L_z, L_y^2]$$
$$= [L_z, L_x] L_x + L_x[L_z, L_x] + [L_z, L_y] L_y + L_y[L_z, L_y]$$
$$= i\hbar(L_yL_x + L_xL_y) - i\hbar(L_xL_y + L_yL_x) = 0$$
[2.156]
Thus it is possible to construct simultaneous eigenfunctions of $L^2$ and one component of L, so that $L... | {
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The orthogonality relations read
$$\int_{-1}^{+1} P_l(w) P_{l'}(w) dw = \frac{2}{2l+1} \delta_{ll'}$$
[2.171]
One also has the closure relation
$$\frac{1}{2} \sum_{l=0}^{\infty} (2l+1) P_l(w) P_l(w') = \delta(w-w')$$
[2.172]
Important particular values of the Legendre polynomials are
$$P_l(1) = 1, P_l(-1) = (... | {
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In that appendix we also discuss *matrix representations* of angular momentum operators,
**Table 2.1** The first few spherical harmonics $Y_{lm}(\theta, \phi)$
| 1 | m | Spherical harmonic $Y_{lm}(heta, \phi)$ | 71 |
|---|----|--------... | {
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Table 2.2 The first few spherical harmonics in real form
| ı | m | Spherical harmonic in real form |
|---|---|---------------------------------------------------------------------------------|
| 0 | 0 | $s = \frac{1}{(4\pi)^{1/2}}$ ... | {
"Header 1": "2 The elements of quantum mechanics",
"Header 2": "General solution of the Schrödinger equation for a time-independent potential",
"token_count": 2015,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
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