Chapter 5 — Quantum Mechanics Review: The Tools You Need for Nuclear Physics

"The nucleus is a quantum object. There is no classical nuclear physics." — Aage Bohr

This chapter is not a course in quantum mechanics. You have already taken that course — or at least one semester of it. What this chapter does is gather the specific quantum mechanical tools you will need repeatedly throughout this book and sharpen them for nuclear physics applications. Think of it as the toolbox you set on the bench before starting work.

Some of these tools — angular momentum coupling, Clebsch-Gordan coefficients — you may have encountered only briefly in your quantum mechanics course, perhaps in a single lecture on addition of angular momenta. In nuclear physics, these are not side topics; they are the language in which nuclear states are described. A nuclear physicist who cannot couple angular momenta is like a carpenter who cannot measure.

Other tools — Fermi's golden rule, the WKB approximation — you likely derived once, applied to a textbook problem, and then set aside. Here, they become the workhorses that power nearly every calculation of decay rates, reaction cross sections, and tunneling probabilities in the chapters to come.

We begin with angular momentum, proceed through the coupling formalism and its Clebsch-Gordan coefficients, establish parity and selection rules, derive Fermi's golden rule from time-dependent perturbation theory, discuss identical particles and antisymmetrization, develop the WKB approximation for tunneling, and conclude with the density of states. Each section includes nuclear physics examples that preview how these tools will be applied later in the book.

Spaced Review (from earlier chapters): - Binding energy (Ch. 1): Recall that the binding energy $B(Z,N) = [Z m_p + N m_n - M(Z,N)]c^2$ quantifies how tightly a nucleus is held together. The tools in this chapter will explain why certain nuclei are more tightly bound than others. - Nuclear sizes (Ch. 2): The nuclear radius $R \approx r_0 A^{1/3}$ with $r_0 \approx 1.2$ fm sets the length scale for the quantum mechanical problems we will solve.

5.1 Angular Momentum in Nuclear Physics

In atomic physics, the electron's orbital angular momentum $\ell$ is often the dominant quantum number, with spin-orbit coupling treated as a perturbation. Nuclear physics reverses this hierarchy. The spin-orbit interaction in nuclei is strong — so strong that it reorders the single-particle energy levels and creates the magic numbers (as we shall see in Chapter 6). Understanding angular momentum in detail is therefore not optional; it is prerequisite.

5.1.1 Orbital Angular Momentum

The orbital angular momentum operator $\hat{\mathbf{L}}$ satisfies the fundamental commutation relations:

$$[\hat{L}_i, \hat{L}_j] = i\hbar \epsilon_{ijk} \hat{L}_k$$

The eigenstates $|l, m_l\rangle$ satisfy:

$$\hat{\mathbf{L}}^2 |l, m_l\rangle = \hbar^2 l(l+1)|l, m_l\rangle, \qquad \hat{L}_z |l, m_l\rangle = \hbar m_l |l, m_l\rangle$$

where $l = 0, 1, 2, \ldots$ and $m_l = -l, -l+1, \ldots, l$. In nuclear physics, nucleons occupy single-particle orbits characterized by the radial quantum number $n$, the orbital quantum number $l$, and the spectroscopic notation $s, p, d, f, g, h, \ldots$ for $l = 0, 1, 2, 3, 4, 5, \ldots$

The raising and lowering operators $\hat{L}_\pm = \hat{L}_x \pm i\hat{L}_y$ are indispensable for computing matrix elements and deriving Clebsch-Gordan coefficients:

$$\hat{L}_\pm |l, m_l\rangle = \hbar\sqrt{l(l+1) - m_l(m_l \pm 1)}\, |l, m_l \pm 1\rangle$$

A point often underappreciated in introductory courses: the orbital angular momentum quantum number $l$ not only determines the angular shape of the wavefunction (through the spherical harmonics $Y_l^m(\theta, \phi)$), but also controls the centrifugal barrier $\hbar^2 l(l+1)/(2mr^2)$ that appears in the radial Schrodinger equation. In nuclear physics, this centrifugal barrier is physically important — it modifies the tunneling probability for particle emission (Section 5.7) and contributes to the energy ordering of single-particle levels (Chapter 6).

The connection between angular momentum and the spatial wavefunction is made explicit through the spherical harmonics. The position-space wavefunction for a nucleon in orbit $nlm_l$ has the form:

$$\psi_{nlm_l}(\mathbf{r}) = R_{nl}(r) Y_l^{m_l}(\theta, \phi)$$

where $R_{nl}(r)$ is the radial wavefunction (determined by the nuclear potential — harmonic oscillator or Woods-Saxon) and $Y_l^{m_l}$ is the spherical harmonic. The product $|Y_l^{m_l}|^2$ gives the angular probability distribution, and its shape — isotropic for $l = 0$, dumbbell-shaped for $l = 1$, four-lobed for $l = 2$ — directly influences how nucleons interact with each other inside the nucleus.

5.1.2 Spin Angular Momentum

Both protons and neutrons are spin-$\frac{1}{2}$ fermions. The spin operator $\hat{\mathbf{S}}$ satisfies the same algebra:

$$[\hat{S}_i, \hat{S}_j] = i\hbar \epsilon_{ijk} \hat{S}_k$$

with eigenstates $|s, m_s\rangle$ where $s = \frac{1}{2}$ and $m_s = \pm\frac{1}{2}$. The spin operators are conveniently expressed in terms of the Pauli matrices:

$$\hat{\mathbf{S}} = \frac{\hbar}{2}\boldsymbol{\sigma}, \qquad \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

5.1.3 Total Angular Momentum

For a single nucleon, the total angular momentum is $\hat{\mathbf{j}} = \hat{\mathbf{L}} + \hat{\mathbf{S}}$. The quantum number $j$ takes the values:

$$j = l + \frac{1}{2} \quad \text{or} \quad j = l - \frac{1}{2} \quad (l \geq 1)$$

For $l = 0$, only $j = \frac{1}{2}$ is possible. The single-particle states are labeled $nl_j$, for example: $1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$, $1d_{5/2}$, etc. The degeneracy of each level is $2j + 1$.

A useful operator identity that appears frequently in nuclear physics is the relation between $\hat{\mathbf{L}} \cdot \hat{\mathbf{S}}$ and the quantum numbers $j$, $l$, $s$:

$$\hat{\mathbf{L}} \cdot \hat{\mathbf{S}} = \frac{1}{2}\left(\hat{\mathbf{j}}^2 - \hat{\mathbf{L}}^2 - \hat{\mathbf{S}}^2\right)$$

with eigenvalue:

$$\langle \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} \rangle = \frac{\hbar^2}{2}\left[j(j+1) - l(l+1) - s(s+1)\right]$$

For $j = l + \frac{1}{2}$: $\langle \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} \rangle = \frac{\hbar^2}{2}l$

For $j = l - \frac{1}{2}$: $\langle \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} \rangle = -\frac{\hbar^2}{2}(l+1)$

The energy splitting between the $j = l + 1/2$ and $j = l - 1/2$ partners due to the spin-orbit interaction $V_{ls}\hat{\mathbf{L}} \cdot \hat{\mathbf{S}}$ is therefore:

$$\Delta E_{ls} = V_{ls} \cdot \frac{\hbar^2}{2}(2l + 1)$$

This splitting increases with $l$, which is why the spin-orbit effect becomes dramatically more important for higher angular momentum orbits. For $l = 4$ (the $g$-shell), the splitting between $g_{9/2}$ and $g_{7/2}$ is large enough to push $g_{9/2}$ below the next major shell, creating the magic number $N = 50$. Understanding this energy splitting is the key to the nuclear shell model (Chapter 6).

💡 Why This Matters for Nuclear Physics The nuclear shell model (Chapter 6) classifies every nucleon in a nucleus by its $nl_j$ quantum numbers. The ground-state spin and parity of a nucleus are determined by the angular momenta of its valence nucleons — the nucleons outside the last closed shell. Getting this right requires mastery of the angular momentum coupling we develop in the next section.

📊 Example: The Nuclear Spin-Orbit Splitting

Consider the $1p$ shell. Without spin-orbit coupling, all $1p$ states ($l = 1$) would be degenerate. The spin-orbit interaction splits them into: - $1p_{3/2}$ ($j = 3/2$, degeneracy 4): pushed down in energy (for the attractive nuclear spin-orbit potential) - $1p_{1/2}$ ($j = 1/2$, degeneracy 2): pushed up in energy

The splitting between them is proportional to $(2l + 1) = 3$. For the $1d$ shell ($l = 2$), the splitting between $1d_{5/2}$ and $1d_{3/2}$ is proportional to 5 — almost twice as large. For the $1g$ shell ($l = 4$), it is proportional to 9. This escalating pattern is what generates the magic numbers 2, 8, 20, 28, 50, 82, 126 that organize the entire chart of nuclides.

5.2 Coupling Angular Momenta

When we combine two angular momenta — say the individual angular momenta $\hat{\mathbf{j}}_1$ and $\hat{\mathbf{j}}_2$ of two nucleons — the total angular momentum $\hat{\mathbf{J}} = \hat{\mathbf{j}}_1 + \hat{\mathbf{j}}_2$ has the quantum number:

$$J = |j_1 - j_2|, |j_1 - j_2| + 1, \ldots, j_1 + j_2$$

This is the triangle rule: $J$ ranges from $|j_1 - j_2|$ to $j_1 + j_2$ in integer steps. The total number of $M_J$ states is $(2j_1 + 1)(2j_2 + 1)$, which must equal $\sum_J (2J + 1)$ — a useful check.

5.2.1 L-S (Russell-Saunders) Coupling

In L-S coupling, the orbital angular momenta of all nucleons couple first to give a total orbital angular momentum $\mathbf{L} = \sum_i \mathbf{l}_i$, and all spins couple to give a total spin $\mathbf{S} = \sum_i \mathbf{s}_i$. Then $\mathbf{L}$ and $\mathbf{S}$ couple to give the total $\mathbf{J} = \mathbf{L} + \mathbf{S}$.

States are labeled $^{2S+1}L_J$ (spectroscopic notation). For example, two nucleons each in $p$-states ($l = 1$) with total $L = 2$, total $S = 1$, and total $J = 3$ would be written $^3D_3$.

L-S coupling is a good approximation when: - The residual interaction between nucleons is much stronger than the spin-orbit interaction - The system involves light nuclei ($A \lesssim 20$) where the spin-orbit force is relatively weak

5.2.2 j-j Coupling

In j-j coupling, each nucleon's orbital and spin angular momenta couple first to give the single-particle $j_i = l_i + s_i$, and then the individual $j$-values couple: $\mathbf{J} = \sum_i \mathbf{j}_i$. States are labeled by the individual $(j_1, j_2, \ldots)$ values and the total $J$.

j-j coupling is the appropriate scheme for nuclear physics in most cases because: - The spin-orbit interaction in nuclei is strong (comparable to the level spacing) - It is the natural basis for the nuclear shell model - It correctly reproduces the magic numbers

📊 Example: Two Nucleons in the $1d_{5/2}$ Shell

Consider two neutrons, each with $j = \frac{5}{2}$. The allowed total angular momenta are:

$$J = 0, 1, 2, 3, 4, 5$$

Wait — this needs a crucial correction. Because the two neutrons are identical fermions, antisymmetry of the wavefunction restricts the allowed $J$ values. For two identical nucleons in the same $j$-shell, only even $J$ values are allowed:

$$J = 0, 2, 4$$

This antisymmetry constraint is profound. It explains why even-even nuclei always have $J^\pi = 0^+$ ground states — the pairing interaction favors $J = 0$ for identical nucleon pairs. We return to this in Section 5.6 and in Chapter 7.

5.2.3 When to Use Which Scheme

In practice, nuclear physicists use j-j coupling as the default and invoke L-S coupling only for specific symmetry arguments, particularly when discussing isospin (Chapter 2) or low-energy nucleon-nucleon scattering (Chapter 3).

5.2.4 Worked Example: The Deuteron Angular Momentum

The deuteron ($^2$H) consists of one proton and one neutron. It has measured quantum numbers $J^\pi = 1^+$ and isospin $T = 0$. Let us see how these follow from angular momentum coupling.

Since $T = 0$ (the proton-neutron system is in the isospin singlet), and the total wavefunction must be antisymmetric under nucleon exchange, the spatial-spin part must be symmetric. The antisymmetry constraint $(-1)^{l+S+T} = -1$ with $T = 0$ gives $(-1)^{l+S} = -1$, so $l + S$ must be odd.

The possible states are: - $S = 1$ (spin triplet) with $l$ even: $l = 0, 2, 4, \ldots$ - $S = 0$ (spin singlet) with $l$ odd: $l = 1, 3, 5, \ldots$

The measured $J = 1$ with positive parity restricts the options further. Parity is $(-1)^l$, so $\pi = +$ requires $l$ even. This selects $S = 1$ with $l = 0$ or $l = 2$ (both give $\pi = +$):

• $l = 0$, $S = 1$: $J = 1$ (the $^3S_1$ state in spectroscopic notation)
• $l = 2$, $S = 1$: $J = 1, 2, 3$ — and we need $J = 1$, so this is the $^3D_1$ state

The deuteron ground state is predominantly $^3S_1$ with a small ($\sim$4--7%) admixture of $^3D_1$. The $D$-state admixture is the reason the deuteron has a nonzero electric quadrupole moment $Q_d = +0.2860$ fm$^2$ (a pure $S$-state would have $Q_d = 0$ by spherical symmetry). This is direct experimental evidence for the tensor component of the nuclear force (Chapter 3).

This example illustrates several key points: how coupling rules constrain the quantum numbers, how parity and antisymmetry work together, and how a measurable property (the quadrupole moment) reveals the angular momentum content of the wavefunction.

5.3 Clebsch-Gordan Coefficients

The Clebsch-Gordan (CG) coefficients are the expansion coefficients that connect the uncoupled basis $|j_1 m_1; j_2 m_2\rangle$ to the coupled basis $|J M\rangle$:

$$|J M\rangle = \sum_{m_1 m_2} \langle j_1 m_1; j_2 m_2 | J M \rangle |j_1 m_1; j_2 m_2 \rangle$$

The CG coefficient $\langle j_1 m_1; j_2 m_2 | J M \rangle$ is nonzero only when:

1. Triangle rule: $|j_1 - j_2| \leq J \leq j_1 + j_2$
2. Projection rule: $m_1 + m_2 = M$
3. Integer condition: $j_1 + j_2 + J$ is an integer

5.3.1 Symmetry Properties

The CG coefficients have important symmetry properties under exchange and sign reversal:

$$\langle j_1 m_1; j_2 m_2 | J M \rangle = (-1)^{j_1 + j_2 - J} \langle j_2 m_2; j_1 m_1 | J M \rangle$$

$$\langle j_1 m_1; j_2 m_2 | J M \rangle = (-1)^{j_1 + j_2 - J} \langle j_1, {-m_1}; j_2, {-m_2} | J, {-M} \rangle$$

These symmetries are not merely formal — they encode physical content. The phase $(-1)^{j_1 + j_2 - J}$ under particle exchange is precisely the factor that enforces the antisymmetry requirement for identical fermions.

5.3.2 Key CG Coefficients for Nuclear Physics

Certain CG coefficients appear so frequently that they should be committed to memory or readily accessible:

Coupling $j$ with $j$ to $J = 0$ (pairing):

$$\langle j\, m;\, j\, {-m} | 0\, 0 \rangle = \frac{(-1)^{j-m}}{\sqrt{2j+1}}$$

This coefficient governs nuclear pairing — the tendency of like nucleons to couple to $J = 0$.

Coupling spin-$\frac{1}{2}$ with orbital $l$:

$$\langle l\, m_l;\, \tfrac{1}{2}\, m_s | j\, m_j \rangle$$

For $j = l + \frac{1}{2}$, $m_j = m_l + m_s$:

$$\langle l\, m_l;\, \tfrac{1}{2}\, \tfrac{1}{2} | l+\tfrac{1}{2}\, m_l+\tfrac{1}{2} \rangle = \sqrt{\frac{l + m_l + 1}{2l + 1}}$$

$$\langle l\, m_l;\, \tfrac{1}{2}\, {-\tfrac{1}{2}} | l+\tfrac{1}{2}\, m_l-\tfrac{1}{2} \rangle = \sqrt{\frac{l - m_l + 1}{2l + 1}}$$

For $j = l - \frac{1}{2}$:

$$\langle l\, m_l;\, \tfrac{1}{2}\, \tfrac{1}{2} | l-\tfrac{1}{2}\, m_l+\tfrac{1}{2} \rangle = -\sqrt{\frac{l - m_l}{2l + 1}}$$

$$\langle l\, m_l;\, \tfrac{1}{2}\, {-\tfrac{1}{2}} | l-\tfrac{1}{2}\, m_l-\tfrac{1}{2} \rangle = \sqrt{\frac{l + m_l}{2l + 1}}$$

These are the CG coefficients you will use most often in the shell model (Chapter 6).

5.3.3 The Wigner-Eckart Theorem (Preview)

The Wigner-Eckart theorem is the single most important result connecting angular momentum theory to physical observables. It states that the matrix element of a tensor operator $\hat{T}^{(\lambda)}_\mu$ of rank $\lambda$ between angular momentum eigenstates can be factored into a geometric part (a CG coefficient) and a dynamical part (a reduced matrix element):

$$\langle j_f m_f | \hat{T}^{(\lambda)}_\mu | j_i m_i \rangle = \frac{(-1)^{j_f - m_f}}{\sqrt{2j_f + 1}} \begin{pmatrix} j_f & \lambda & j_i \\ -m_f & \mu & m_i \end{pmatrix} \langle j_f || \hat{T}^{(\lambda)} || j_i \rangle$$

The power of this result is that the $m$-dependence of the matrix element is entirely contained in the 3j symbol (or equivalently, a CG coefficient). The reduced matrix element $\langle j_f || \hat{T}^{(\lambda)} || j_i \rangle$ depends only on $j_f$, $\lambda$, $j_i$ and the dynamics of the operator — it is independent of the projections $m_f$, $\mu$, $m_i$.

In nuclear physics, the Wigner-Eckart theorem is used constantly: - Electromagnetic transitions: The reduced transition probability $B(\text{E}\lambda)$ is the square of the reduced matrix element of the electric multipole operator - Magnetic moments: The nuclear magnetic moment is the matrix element of the magnetic dipole operator, evaluated via the Wigner-Eckart theorem with $\lambda = 1$, $\mu = 0$, and $m = j$ (the "stretched" state) - Beta decay: The Fermi and Gamow-Teller matrix elements are reduced matrix elements of rank-0 and rank-1 operators, respectively

We will use the Wigner-Eckart theorem extensively in Chapters 6, 9, and 14. For now, the key takeaway is that angular momentum algebra (CG coefficients, 3j symbols) completely determines the geometry of nuclear transitions, while the nuclear dynamics enters only through the reduced matrix elements.

5.3.4 Wigner 3j Symbols

The Wigner 3j symbol is a symmetrized form of the CG coefficient:

$$\begin{pmatrix} j_1 & j_2 & J \\ m_1 & m_2 & -M \end{pmatrix} = \frac{(-1)^{j_1 - j_2 + M}}{\sqrt{2J+1}} \langle j_1 m_1; j_2 m_2 | J M \rangle$$

The 3j symbol has cleaner symmetry properties: it is invariant under even permutations of its columns and picks up a phase $(-1)^{j_1+j_2+J}$ under odd permutations. In the nuclear physics literature, 3j symbols appear alongside the related 6j and 9j symbols (Racah coefficients), which arise when coupling three or more angular momenta. We will introduce these as needed in Chapters 6 and 9.

💡 Practical Advice No one computes CG coefficients by hand for realistic nuclear physics problems. You will use tables (such as those in Appendix D of this book or the Particle Data Group tables) or computer routines (Python's sympy.physics.quantum.cg or the standalone code in this chapter's code/ directory). What matters is understanding when they are needed and what they mean physically.

5.3.4 Worked Example: Coupling Two $p_{3/2}$ Neutrons

Two neutrons in the $1p_{3/2}$ shell ($j_1 = j_2 = \frac{3}{2}$). The triangle rule gives $J = 0, 1, 2, 3$. Antisymmetry restricts this to $J = 0, 2$ (even values only, since the spatial and spin parts are tied together in the j-j scheme and the two neutrons are in the same orbit).

The $J = 0$ state is:

$$|J=0, M=0\rangle = \sum_{m} \langle \tfrac{3}{2}\, m;\, \tfrac{3}{2}\, {-m} | 0\, 0\rangle |\tfrac{3}{2}\, m;\, \tfrac{3}{2}\, {-m}\rangle$$

Using $\langle j\, m;\, j\, {-m} | 0\, 0\rangle = (-1)^{j-m}/\sqrt{2j+1}$:

$$= \frac{1}{\sqrt{4}}\left[|{\tfrac{3}{2}}\, {\tfrac{3}{2}}\rangle|{\tfrac{3}{2}}\, {-\tfrac{3}{2}}\rangle - |{\tfrac{3}{2}}\, {\tfrac{1}{2}}\rangle|{\tfrac{3}{2}}\, {-\tfrac{1}{2}}\rangle + |{\tfrac{3}{2}}\, {-\tfrac{1}{2}}\rangle|{\tfrac{3}{2}}\, {\tfrac{1}{2}}\rangle - |{\tfrac{3}{2}}\, {-\tfrac{3}{2}}\rangle|{\tfrac{3}{2}}\, {\tfrac{3}{2}}\rangle\right]$$

The properly antisymmetrized state includes a normalization factor $1/\sqrt{2}$ and looks like the above with the signs enforcing antisymmetry under particle exchange. This is the $J = 0$ "paired" configuration that is energetically favored by the pairing interaction — the reason even-even nuclei have $0^+$ ground states.

5.3.6 CG Coefficients in Practice: A Nuclear Magnetic Moment Calculation

To see CG coefficients in action, consider computing the magnetic moment of a nucleus with a single valence nucleon. The magnetic moment operator is:

$$\hat{\mu} = g_l \hat{L}_z + g_s \hat{S}_z$$

where $g_l$ is the orbital $g$-factor (1 for protons, 0 for neutrons, in nuclear magneton units) and $g_s$ is the spin $g$-factor ($g_s^p = 5.586$ for protons, $g_s^n = -3.826$ for neutrons — these differ from the Dirac values due to the internal quark structure of the nucleon).

For a nucleon in state $|l, j, m_j = j\rangle$ (the "stretched" state), the expectation value is:

$$\mu = \langle l, j, m_j = j | g_l \hat{L}_z + g_s \hat{S}_z | l, j, m_j = j \rangle$$

We need to expand $|j, m_j = j\rangle$ in terms of $|l, m_l; s, m_s\rangle$ using CG coefficients:

$$|j = l+\tfrac{1}{2}, m_j = l+\tfrac{1}{2}\rangle = |l, l; \tfrac{1}{2}, \tfrac{1}{2}\rangle$$

(only one term contributes because $m_j = l + 1/2$ requires $m_l = l$ and $m_s = 1/2$). So:

$$\mu(j = l + \tfrac{1}{2}) = g_l \cdot l + g_s \cdot \tfrac{1}{2} = l \cdot g_l + \tfrac{1}{2} g_s$$

For $j = l - 1/2$, the CG expansion of $|j, m_j = j\rangle = |l-1/2, l-1/2\rangle$ involves two terms:

$$|l-\tfrac{1}{2}, l-\tfrac{1}{2}\rangle = \langle l, l; \tfrac{1}{2}, -\tfrac{1}{2}|l-\tfrac{1}{2}, l-\tfrac{1}{2}\rangle |l, l; \tfrac{1}{2}, -\tfrac{1}{2}\rangle + \langle l, l-1; \tfrac{1}{2}, \tfrac{1}{2}|l-\tfrac{1}{2}, l-\tfrac{1}{2}\rangle |l, l-1; \tfrac{1}{2}, \tfrac{1}{2}\rangle$$

Using the CG coefficients from Section 5.3.2:

$$= \sqrt{\frac{1}{2l+1}} |l, l; \tfrac{1}{2}, -\tfrac{1}{2}\rangle - \sqrt{\frac{2l}{2l+1}} |l, l-1; \tfrac{1}{2}, \tfrac{1}{2}\rangle$$

Computing $\langle \hat{L}_z \rangle$ and $\langle \hat{S}_z \rangle$ and combining gives the Schmidt values for the magnetic moment:

$$\mu(j = l + \tfrac{1}{2}) = \left(l + \frac{1}{2}\right)g_l + \frac{1}{2}g_s = j \cdot g_l + \frac{1}{2}g_s$$

$$\mu(j = l - \tfrac{1}{2}) = \frac{j}{j+1}\left[\left(l + \frac{3}{2}\right)g_l - \frac{1}{2}g_s\right]$$

These Schmidt values define the Schmidt lines — the predictions for nuclear magnetic moments in the extreme single-particle model. Measured magnetic moments cluster between the Schmidt lines, confirming that the single-particle picture captures the qualitative physics, while deviations reveal the effects of configuration mixing, meson exchange currents, and nucleon substructure.

This calculation — seemingly a simple application of CG coefficients — connects directly to experimental data measured with extraordinary precision using nuclear magnetic resonance (NMR) and atomic beam techniques. The point is that the angular momentum formalism is not abstract mathematics; it is the language in which experimental nuclear data are interpreted.

5.4 Parity and Selection Rules

Parity is a discrete symmetry — unlike angular momentum, which takes continuous values, parity is either $+1$ or $-1$. In nuclear physics, parity is conserved by the strong and electromagnetic interactions (but violated by the weak interaction, as discovered by Wu in 1957). Because nuclear structure is dominated by the strong force, parity is an excellent quantum number for nuclear states, and parity selection rules are among the most powerful tools for classifying transitions.

5.4.1 Parity of Nuclear States

The parity operator $\hat{P}$ inverts spatial coordinates: $\hat{P}\psi(\mathbf{r}) = \psi(-\mathbf{r})$. Under parity, the spherical harmonics transform as $\hat{P}Y_l^m(\theta, \phi) = (-1)^l Y_l^m(\theta, \phi)$ — they are eigenstates of parity with eigenvalue $(-1)^l$. This is the fundamental connection between orbital angular momentum and parity.

For nuclear states, the total parity is a product of three contributions:

1. Intrinsic parities of the constituent nucleons (both proton and neutron have intrinsic parity $+1$ by convention)
2. Orbital parities from the spatial wavefunctions: a nucleon in an orbital with angular momentum $l$ contributes $(-1)^l$

For a single nucleon in orbit $nl_j$, the parity is $\pi = (-1)^l$. For a many-nucleon state, the total parity is:

$$\pi = \prod_i (-1)^{l_i}$$

where the product runs over all nucleons. In practice, only the valence nucleons matter — closed shells always contribute $\pi = +1$.

📊 Example: Ground State of $^{17}$O

$^{17}$O has $Z = 8$ (magic — closed proton shell) and $N = 9$. The first 8 neutrons fill the $1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$ shells. The 9th neutron occupies the $1d_{5/2}$ orbit. Therefore:

$$J^\pi = \frac{5}{2}^+$$

The $+$ parity comes from $(-1)^{l} = (-1)^2 = +1$ for a $d$-wave ($l = 2$) neutron. The measured value is indeed $\frac{5}{2}^+$. This is the shell model in action.

5.4.2 Selection Rules for Electromagnetic Transitions

Electromagnetic transitions between nuclear states obey strict selection rules arising from conservation of angular momentum and parity. For a transition of multipolarity $\lambda$ (meaning the photon carries angular momentum $\lambda$):

Electric multipole radiation (E$\lambda$): - $|J_i - J_f| \leq \lambda \leq J_i + J_f$ - Parity change: $\pi_i \cdot \pi_f = (-1)^\lambda$

Magnetic multipole radiation (M$\lambda$): - $|J_i - J_f| \leq \lambda \leq J_i + J_f$ - Parity change: $\pi_i \cdot \pi_f = (-1)^{\lambda+1}$

The lowest allowed multipolarity dominates because transition rates decrease rapidly with increasing $\lambda$ (roughly as $(R/\lambda_\gamma)^{2\lambda}$, where $R$ is the nuclear radius and $\lambda_\gamma$ is the photon wavelength — typically $\lambda_\gamma \gg R$ for nuclear gamma rays).

We develop the quantitative theory of electromagnetic transition rates in Chapter 9, including the Weisskopf estimates that provide benchmarks for whether an observed transition rate is "single-particle" or "collective."

📊 Worked Example: Transitions in $^{60}$Co Decay

The $^{60}$Co nucleus ($J^\pi = 5^+$) beta decays to excited states of $^{60}$Ni. The relevant gamma cascade in $^{60}$Ni is:

• $4^+ \to 2^+$: $\Delta J = 2$, no parity change $\Rightarrow$ E2 (electric quadrupole)
• $2^+ \to 0^+$: $\Delta J = 2$, no parity change $\Rightarrow$ E2 (electric quadrupole)

Both transitions are E2, and they produce the characteristic 1.173 MeV and 1.333 MeV gamma rays used in radiation calibration worldwide. The E2 character means the gamma rays have a characteristic angular distribution — they are not emitted isotropically, and this angular correlation between the two successive gammas is a standard technique in nuclear spectroscopy.

Note that M1 radiation is also allowed for the $4^+ \to 2^+$ transition ($\Delta J = 2$ and no parity change permits M1 only if $\Delta J \leq 1$; here $\Delta J = 2$, so M1 is actually forbidden and E2 is the lowest multipole). For the $2^+ \to 0^+$ transition, M1 is forbidden because $\Delta J = 2 > 1$. This illustrates why carefully checking both the $\Delta J$ and parity rules is essential.

5.4.3 The $0^+ \to 0^+$ Puzzle

A particularly important case: can a $0^+$ state decay to another $0^+$ state by gamma emission? The answer is no — a single photon must carry at least one unit of angular momentum ($\lambda \geq 1$), but the triangle rule for a $0 \to 0$ transition requires $\lambda = 0$, which is impossible for a real photon. (A virtual photon with $\lambda = 0$ corresponds to an E0, or electric monopole, transition.)

This prohibition has profound consequences. The $0^+$ first excited state of $^{16}$O at 6.05 MeV cannot emit a gamma ray to the $0^+$ ground state. Instead, it decays by: - Internal conversion (the nuclear electromagnetic field ejects an atomic electron) — the dominant mode - Internal pair creation ($e^+e^-$ pair production by the nuclear field) — possible because $6.05 > 2m_e c^2 = 1.022$ MeV

The E0 matrix element is a measure of the change in the nuclear charge radius between the two states, making $0^+ \to 0^+$ transitions a sensitive probe of nuclear shape changes. This will be important when we discuss shape coexistence in Chapter 8.

5.4.4 Selection Rules for Beta Decay (Preview)

Beta decay involves the weak interaction and has its own selection rules. For an allowed beta transition (the dominant type):

• $\Delta J = 0$ or $1$ (no parity change): Allowed transitions (Fermi: $\Delta J = 0$; Gamow-Teller: $\Delta J = 0, \pm 1$, no $0 \to 0$ for GT)
• $\Delta J > 1$ or wrong parity: Forbidden transitions ($n$th forbidden for $\Delta J = n + 1$ and specific parity relations)

The detailed treatment of beta decay selection rules appears in Chapter 14 (Weak Interactions and Beta Decay). The key point for now is that the degree of forbiddenness dramatically affects the half-life — each degree of forbiddenness typically suppresses the rate by a factor of $10^3$ to $10^4$.

5.5 Time-Dependent Perturbation Theory and Fermi's Golden Rule

Fermi's golden rule is arguably the single most important formula in nuclear physics. It gives the transition rate for virtually every process we will study: radioactive decay (alpha, beta, gamma), nuclear reactions, and scattering. It is worth deriving carefully.

5.5.1 Why This Derivation Matters

Before diving into the mathematics, let us be clear about why we derive Fermi's golden rule rather than simply stating it. The derivation reveals: 1. The assumptions under which it is valid (first-order perturbation theory, continuum of final states) 2. Where the density of states enters and why it has the physical meaning it does 3. How the energy-conserving delta function emerges from the time evolution 4. The connection between the transition rate and the matrix element that encodes all the nuclear physics

Students who understand this derivation will be able to identify when Fermi's golden rule applies, when it needs modification (resonances, strong coupling), and how to generalize it. Students who merely memorize the result will be lost when they encounter situations where it breaks down — such as narrow resonances in nuclear reactions (Chapter 18) or the breakdown of first-order theory for strong decays.

5.5.2 Setup: The Interaction Picture

Consider a system with unperturbed Hamiltonian $\hat{H}_0$ and a time-dependent perturbation $\hat{V}(t)$ that is "turned on" at $t = 0$. We expand the state in the eigenbasis of $\hat{H}_0$:

$$|\psi(t)\rangle = \sum_n c_n(t) e^{-iE_n t/\hbar} |n\rangle$$

where $\hat{H}_0|n\rangle = E_n|n\rangle$. Substituting into the Schrodinger equation $i\hbar \partial_t |\psi\rangle = (\hat{H}_0 + \hat{V})|\psi\rangle$ and projecting onto a final state $\langle f|$:

$$i\hbar \dot{c}_f(t) = \sum_n c_n(t) \langle f|\hat{V}(t)|n\rangle e^{i\omega_{fn}t}$$

where $\omega_{fn} = (E_f - E_n)/\hbar$.

5.5.2 First-Order Perturbation Theory

If the system starts in state $|i\rangle$ at $t = 0$, so that $c_i(0) = 1$ and $c_{f\neq i}(0) = 0$, then to first order in $\hat{V}$:

$$c_f^{(1)}(t) = \frac{1}{i\hbar}\int_0^t \langle f|\hat{V}(t')|i\rangle e^{i\omega_{fi}t'} dt'$$

For a constant perturbation $\hat{V}$ turned on at $t = 0$:

$$c_f^{(1)}(t) = \frac{\langle f|\hat{V}|i\rangle}{i\hbar} \int_0^t e^{i\omega_{fi}t'} dt' = \frac{\langle f|\hat{V}|i\rangle}{i\hbar} \cdot \frac{e^{i\omega_{fi}t} - 1}{i\omega_{fi}}$$

The transition probability is:

$$P_{i \to f}(t) = |c_f^{(1)}(t)|^2 = \frac{|\langle f|\hat{V}|i\rangle|^2}{\hbar^2} \cdot \frac{4\sin^2(\omega_{fi}t/2)}{\omega_{fi}^2}$$

5.5.3 The Long-Time Limit and the Transition Rate

The transition probability contains the factor:

$$D_t(\omega) \equiv \frac{4\sin^2(\omega t/2)}{\omega^2}$$

This function is peaked at $\omega = 0$ with a height of $t^2$ and a width of approximately $4\pi/t$. The area under the curve is $2\pi t$ (independent of the shape). As $t$ increases, $D_t(\omega)$ becomes taller and narrower, concentrating around $\omega = 0$ while preserving its area. In the mathematical limit $t \to \infty$, this is precisely the definition of a Dirac delta function:

$$\lim_{t\to\infty} D_t(\omega) = 2\pi t\, \delta(\omega)$$

The physical meaning is profound: transitions conserve energy. For short times, the function $D_t(\omega)$ has a finite width $\sim 2\pi/t$, allowing transitions to states with energies differing from $E_i$ by up to $\Delta E \sim \hbar/t$ — this is the energy-time uncertainty relation in action. For long times ($t \to \infty$), only transitions with $E_f = E_i$ (exact energy conservation, $\omega = 0$) survive.

Since the probability grows linearly in time for large $t$, we define the transition rate:

$$\Gamma_{i \to f} = \frac{dP_{i\to f}}{dt} = \frac{2\pi}{\hbar}|\langle f|\hat{V}|i\rangle|^2 \delta(E_f - E_i)$$

5.5.4 Fermi's Golden Rule

When the final state is part of a continuum (as it almost always is in nuclear physics — the emitted particle has a continuous energy spectrum, or the photon has a continuous direction), we sum over the density of final states $\rho(E_f)$:

$$\boxed{\Gamma_{i \to f} = \frac{2\pi}{\hbar}|\langle f|\hat{V}|i\rangle|^2 \rho(E_f)}$$

This is Fermi's golden rule. Every term has physical meaning:

• $|\langle f|\hat{V}|i\rangle|^2$ — the matrix element squared, encoding the dynamics of the interaction responsible for the transition. For gamma decay, $\hat{V}$ is the electromagnetic multipole operator. For beta decay, $\hat{V}$ is the weak interaction Hamiltonian. For nuclear reactions, $\hat{V}$ is the nuclear potential.
• $\rho(E_f)$ — the density of states at the final energy, counting how many final states are available. More available states means a faster transition. This is the "phase space" factor.
• $2\pi/\hbar$ — the fundamental constants that set the scale.

⚠️ Critical Assumption Fermi's golden rule is valid when: 1. The perturbation is weak enough that first-order perturbation theory applies 2. The final states form a continuum (or quasi-continuum) 3. The observation time is long compared to $\hbar/\Delta E$ where $\Delta E$ is the energy spread of the final states

In nuclear physics, condition (1) is well satisfied for electromagnetic and weak decays. For strong-interaction processes (nuclear reactions), higher-order effects can be important, and the $T$-matrix formalism (Chapter 17) extends Fermi's golden rule to all orders.

5.5.5 Nuclear Physics Applications (Preview)

Gamma decay rate (Chapter 9): The perturbation is the electromagnetic multipole operator $\hat{O}(\text{E}\lambda)$ or $\hat{O}(\text{M}\lambda)$. Fermi's golden rule gives:

$$\Gamma_\gamma(\text{E}\lambda) = \frac{2\pi}{\hbar} \frac{1}{2J_i + 1}\sum_{M_i, M_f, \mu}|\langle J_f M_f | \hat{O}_\mu(\text{E}\lambda)|J_i M_i\rangle|^2 \rho_\gamma(E_\gamma)$$

The density of states for photons gives $\rho_\gamma = E_\gamma^2 V / (\pi^2 \hbar^3 c^3)$. The matrix element determines whether the transition is fast (collective) or slow (single-particle).

Beta decay rate (Chapter 14): The weak interaction matrix element involves the Fermi ($\hat{1}$) and Gamow-Teller ($\hat{\boldsymbol{\sigma}}\hat{\boldsymbol{\tau}}$) operators. The density of states includes both the electron and neutrino continua.

Nuclear reaction cross section (Chapter 17): The cross section is related to the transition rate by $\sigma = \Gamma/\Phi$, where $\Phi$ is the incident flux. This connects Fermi's golden rule to measurable quantities.

5.5.6 The Lifetime-Width Relation

The transition rate $\Gamma$ (in units of s$^{-1}$) is the reciprocal of the mean lifetime $\tau$:

$$\tau = \frac{1}{\Gamma}, \qquad t_{1/2} = \tau \ln 2 = \frac{\ln 2}{\Gamma}$$

When multiple decay modes compete, the total decay rate is the sum of the partial rates:

$$\Gamma_\text{total} = \Gamma_1 + \Gamma_2 + \Gamma_3 + \cdots$$

and the branching ratio for mode $i$ is $\text{BR}_i = \Gamma_i / \Gamma_\text{total}$.

The energy-time uncertainty relation connects the lifetime to the natural energy width of the state:

$$\Delta E = \hbar \Gamma = \frac{\hbar}{\tau}$$

For a typical nuclear excited state with $\tau \sim 10^{-12}$ s (a picosecond), $\Delta E \sim 10^{-3}$ eV — far too narrow to resolve with any detector. For particle-unstable states ($\tau \sim 10^{-22}$ s), the width $\Delta E \sim$ MeV is experimentally measurable as a resonance width. This distinction between "narrow" and "broad" states will be central to our discussion of nuclear reactions in Chapters 17--19.

📊 Numerical Example: Competing Decay Modes in $^{152}$Eu

$^{152}$Eu ($J^\pi = 3^-$, $t_{1/2} = 13.5$ years) decays by three competing modes: - Electron capture (EC): 72.1% - Beta-minus ($\beta^-$): 27.9% - Beta-plus ($\beta^+$): 0.027%

The total decay rate is $\Gamma_\text{total} = \ln 2 / t_{1/2} = 1.63 \times 10^{-9}$ s$^{-1}$. The partial rates are: - $\Gamma_\text{EC} = 0.721 \times 1.63 \times 10^{-9} = 1.17 \times 10^{-9}$ s$^{-1}$ - $\Gamma_{\beta^-} = 0.279 \times 1.63 \times 10^{-9} = 4.55 \times 10^{-10}$ s$^{-1}$

Each partial rate is independently determined by Fermi's golden rule with the appropriate matrix element and density of states. The total rate is their sum, and the branching ratios follow.

5.5.7 From the Golden Rule to Measurable Quantities

It is worth pausing to emphasize how Fermi's golden rule connects to what experimentalists actually measure. The golden rule gives a rate $\Gamma$ (inverse seconds). This connects to:

• Half-life: $t_{1/2} = \ln 2 / \Gamma$. Measured by counting decays as a function of time.
• Cross section: $\sigma = \Gamma / (n_\text{target} v_\text{rel})$ where $n_\text{target}$ is the areal density and $v_\text{rel}$ is the beam velocity. Measured by counting reaction products relative to incident flux.
• Branching ratio: $\text{BR} = \Gamma_i / \Gamma_\text{total}$. Measured by comparing the number of decays through different channels.
• Resonance width: $\Gamma_\text{total} = \hbar / \tau$. Measured from the energy width of a resonance peak in a cross-section plot.

Every measured nuclear property listed above traces back, through Fermi's golden rule, to a matrix element and a density of states. The theoretical challenge in nuclear physics is computing those matrix elements accurately — and that requires the nuclear wavefunctions, which require the shell model (Chapter 6), collective models (Chapter 8), or more sophisticated approaches.

5.6 Identical Particles and Antisymmetrization

5.6.1 The Symmetrization Postulate

Protons are identical to other protons. Neutrons are identical to other neutrons. As fermions (spin-$\frac{1}{2}$), the total wavefunction must be antisymmetric under exchange of any two identical nucleons:

$$\Psi(\ldots, \mathbf{r}_i, \sigma_i, \tau_i, \ldots, \mathbf{r}_j, \sigma_j, \tau_j, \ldots) = -\Psi(\ldots, \mathbf{r}_j, \sigma_j, \tau_j, \ldots, \mathbf{r}_i, \sigma_i, \tau_i, \ldots)$$

where $\sigma$ denotes spin and $\tau$ denotes isospin (proton or neutron identity). This antisymmetry requirement — the Pauli exclusion principle — is the foundation of nuclear structure. Without it, all nucleons would collapse into the lowest energy orbit, and the shell structure that gives nuclei their rich variety of properties would not exist.

5.6.2 Two-Nucleon Wavefunctions

For two nucleons in orbits $a$ and $b$ (where $a$ and $b$ label all quantum numbers $n, l, j, m_j$), the antisymmetrized two-particle state is:

$$|\Psi_{ab}\rangle = \frac{1}{\sqrt{2}}\left(|a\rangle_1 |b\rangle_2 - |b\rangle_1 |a\rangle_2\right)$$

If $a = b$ (both nucleons in the same quantum state), then $|\Psi_{aa}\rangle = 0$ — the state vanishes. This is the Pauli exclusion principle: no two identical fermions can occupy the same quantum state.

For two identical nucleons in the same $j$-shell but different $m_j$ values, coupling to total angular momentum $J$:

$$|j^2; J M\rangle = \sum_{m_1 m_2} \langle j\, m_1;\, j\, m_2 | J\, M\rangle \frac{1}{\sqrt{2}}\left(|j\, m_1\rangle_1 |j\, m_2\rangle_2 - |j\, m_2\rangle_1 |j\, m_1\rangle_2\right)$$

Using the symmetry of the CG coefficient under exchange of $m_1$ and $m_2$:

$$\langle j\, m_2;\, j\, m_1 | J\, M\rangle = (-1)^{2j-J}\langle j\, m_1;\, j\, m_2 | J\, M\rangle$$

The antisymmetrized state vanishes unless $(-1)^{2j-J} = -1$, i.e., $2j - J$ is odd, i.e., $J$ is even (since $2j$ is odd for half-integer $j$). This is the formal proof of the result stated in Section 5.2: two identical nucleons in the same $j$-shell can only couple to even $J$.

5.6.3 Slater Determinants

For a system of $A$ nucleons, the antisymmetrized product state (assuming independent particles) is written as a Slater determinant:

$$\Psi(\mathbf{r}_1, \ldots, \mathbf{r}_A) = \frac{1}{\sqrt{A!}} \begin{vmatrix} \phi_a(\mathbf{r}_1) & \phi_a(\mathbf{r}_2) & \cdots & \phi_a(\mathbf{r}_A) \\ \phi_b(\mathbf{r}_1) & \phi_b(\mathbf{r}_2) & \cdots & \phi_b(\mathbf{r}_A) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_z(\mathbf{r}_1) & \phi_z(\mathbf{r}_2) & \cdots & \phi_z(\mathbf{r}_A) \end{vmatrix}$$

where $\phi_a, \phi_b, \ldots, \phi_z$ are $A$ distinct single-particle states. The determinant automatically ensures antisymmetry: exchanging two particles (columns) changes the sign; placing two particles in the same state (two identical rows) gives zero.

In the nuclear shell model, the ground state of a closed-shell nucleus is a single Slater determinant. States near closed shells involve one or a few Slater determinants. States far from closed shells — in the middle of a shell — require many Slater determinants (configuration mixing), and this is where the computational challenge of nuclear structure lies.

📊 Why Slater Determinants Matter: The Scale of the Problem

Consider $^{28}$Si ($Z = 14$, $N = 14$). In the $sd$-shell model space ($1d_{5/2}$, $2s_{1/2}$, $1d_{3/2}$), there are 6 valence protons and 6 valence neutrons (beyond the $^{16}$O core). The number of ways to distribute 6 protons among the 12 available single-particle states is $\binom{12}{6} = 924$. Similarly for neutrons. The total number of Slater determinants (the dimension of the shell model matrix) is $924 \times 924 = 854,\!016$.

For heavier nuclei, these dimensions explode. $^{56}$Fe in the $pf$-shell has a matrix dimension exceeding $10^9$. Diagonalizing matrices of this size requires the most advanced computational techniques, and this is the frontier of modern nuclear structure theory (Chapter 10, Exotic Nuclei).

Despite this complexity, the underlying principle is simple: build antisymmetrized many-body states from single-particle orbits, and diagonalize the nuclear Hamiltonian in that basis.

5.6.4 Isospin Formalism

If we treat the proton and neutron as two states of the same particle (the nucleon), with isospin $t = \frac{1}{2}$ and projections $t_z = +\frac{1}{2}$ (proton) and $t_z = -\frac{1}{2}$ (neutron), then the antisymmetrization requirement applies to all $A$ nucleons simultaneously, including the isospin degree of freedom. This is particularly useful for light nuclei where isospin is a good quantum number.

For the two-nucleon system, the antisymmetry requirement becomes:

$$(-1)^{l+S+T} = -1$$

where $T$ is the total isospin ($T = 0$ for the proton-neutron $np$ system, $T = 1$ for $pp$, $nn$, or the $T = 1$ component of $np$). This constraint is powerful: it links the spatial, spin, and isospin quantum numbers. For example, the deuteron ($T = 0$, $S = 1$) must have $l$ even, which explains why it is predominantly an $S$-wave ($l = 0$) state with a small $D$-wave ($l = 2$) admixture.

The table below summarizes the allowed two-nucleon states:

The fact that the deuteron exists ($T = 0$, $S = 1$, $l = 0$) but the dineutron does not ($T = 1$, $S = 0$, $l = 0$ — the $^1S_0$ state is nearly bound but not quite) tells us that the nuclear force is stronger in the $S = 1$ (spin-triplet) channel than in the $S = 0$ (spin-singlet) channel. This spin dependence of the nuclear force is a fundamental property that must be built into any realistic nuclear potential (Chapter 3).

5.7 The WKB Approximation and Quantum Tunneling

The Wentzel-Kramers-Brillouin (WKB) approximation is the essential tool for calculating tunneling probabilities through potential barriers. In nuclear physics, it governs:

• Alpha decay (Chapter 13): the alpha particle tunnels through the Coulomb barrier
• Proton radioactivity (Chapter 10): the proton tunnels through the Coulomb barrier
• Fission (Chapter 20): the nucleus tunnels through the fission barrier
• Stellar fusion (Chapter 21): light nuclei tunnel through the Coulomb barrier at stellar temperatures

5.7.1 The WKB Wavefunction

For a particle of mass $m$ and energy $E$ in a one-dimensional potential $V(x)$, the WKB approximation to the wavefunction in a classically allowed region ($E > V$) is:

$$\psi(x) \approx \frac{C}{\sqrt{p(x)}}\exp\left(\pm \frac{i}{\hbar}\int^x p(x')\, dx'\right)$$

where $p(x) = \sqrt{2m[E - V(x)]}$ is the local classical momentum. In a classically forbidden region ($E < V$):

$$\psi(x) \approx \frac{C}{\sqrt{\kappa(x)}}\exp\left(\pm \frac{1}{\hbar}\int^x \kappa(x')\, dx'\right)$$

where $\kappa(x) = \sqrt{2m[V(x) - E]}$.

The WKB approximation is valid when the potential varies slowly on the scale of the de Broglie wavelength: $|dp/dx| \ll p^2/\hbar$, or equivalently, when the wavelength changes little over one oscillation. This is a semiclassical approximation — it works best when the action integral $\int p\, dx \gg \hbar$, i.e., when quantum effects are not too dramatic. Paradoxically, we use it precisely to calculate tunneling, which is a purely quantum effect. The WKB approximation handles this by giving an exponentially small transmission probability — the classical result (zero transmission) modified by a quantum correction.

Connection formulas. At the classical turning points $a$ and $b$ (where $E = V(x)$ and $p(x) = 0$), the WKB wavefunction diverges. The solution is to match the WKB wavefunctions on either side of the turning point through an exact solution (typically an Airy function) in a narrow region around the turning point. This matching gives the connection formulas that relate the amplitudes and phases of the WKB wavefunctions across the turning point. The details are found in advanced quantum mechanics texts (Landau & Lifshitz, Chapter 7, gives the definitive treatment), but the final result for the tunneling probability depends only on the exponential factor, which is insensitive to the connection formula details.

5.7.2 Tunneling Through a Barrier

Consider a barrier extending from $x = a$ to $x = b$, where $V(x) > E$ in this region. The WKB transmission coefficient (tunneling probability) is:

$$\boxed{T \approx \exp\left(-\frac{2}{\hbar}\int_a^b \sqrt{2m[V(x) - E]}\, dx\right) \equiv e^{-2\gamma}}$$

where we define the Gamow factor:

$$\gamma = \frac{1}{\hbar}\int_a^b \sqrt{2m[V(x) - E]}\, dx$$

The exponential sensitivity of $T$ to the integral in the exponent is the key physical result. Small changes in the barrier height, width, or the particle's energy produce enormous changes in the tunneling probability. This exponential sensitivity explains:

• Why alpha decay half-lives span 25 orders of magnitude (from $10^{-7}$ s for $^{212}$Po to $10^{17}$ years for $^{209}$Bi) while alpha particle energies vary by only a factor of two
• Why stellar fusion rates are so sensitive to temperature (the Gamow peak)
• Why the fission barrier height determines whether a nucleus is fissile, fissionable, or stable against fission

5.7.3 Application to the Coulomb Barrier

For a charged particle (charge $z_1 e$) tunneling through the Coulomb barrier of a nucleus (charge $z_2 e$), the potential is:

$$V(r) = \frac{z_1 z_2 e^2}{4\pi\epsilon_0 r} \quad (r > R)$$

where $R$ is the nuclear radius. The classical turning points are $r = R$ (the nuclear surface, where the nuclear potential takes over) and $r = R_c = z_1 z_2 e^2 / (4\pi\epsilon_0 E)$ (where $V(R_c) = E$). The Gamow factor becomes:

$$\gamma = \frac{1}{\hbar}\int_R^{R_c} \sqrt{2\mu\left(\frac{z_1 z_2 e^2}{4\pi\epsilon_0 r} - E\right)}\, dr$$

where $\mu$ is the reduced mass. This integral can be evaluated analytically. Defining $\rho = R/R_c$ and $\eta = z_1 z_2 e^2 / (4\pi\epsilon_0 \hbar v)$ (the Sommerfeld parameter, where $v = \sqrt{2E/\mu}$):

$$\gamma = \eta\left[\arccos\sqrt{\rho} - \sqrt{\rho(1-\rho)}\right]$$

In the limit $R \ll R_c$ (low energy, thick barrier), this simplifies to:

$$\gamma \approx \pi\eta = \frac{\pi z_1 z_2 e^2}{4\pi\epsilon_0 \hbar v}$$

and the tunneling probability is:

$$T \approx e^{-2\pi\eta}$$

The centrifugal barrier. When the emitted particle carries orbital angular momentum $l > 0$, the effective potential includes the centrifugal term:

$$V_\text{eff}(r) = \frac{z_1 z_2 e^2}{4\pi\epsilon_0 r} + \frac{\hbar^2 l(l+1)}{2\mu r^2}$$

The centrifugal barrier raises the effective barrier height and shifts the inner turning point outward, reducing the tunneling probability. For alpha decay, the dominant transition is usually $l = 0$ (the alpha particle carries away no orbital angular momentum when the parent and daughter have the same spin-parity). But for transitions between states with different spins, $l > 0$ is required, and the angular momentum barrier significantly suppresses the decay rate. Each unit of angular momentum reduces the penetrability by roughly a factor of 3--10 (depending on the energy and the Coulomb parameter), which is why alpha decay "fine structure" (decay to excited states of the daughter nucleus) typically involves only the lowest few $l$ values.

📊 Numerical Example: Alpha Decay of $^{238}$U

For alpha decay of $^{238}$U ($E_\alpha \approx 4.27$ MeV, $z_1 = 2$, $z_2 = 90$, $R \approx 7.4$ fm):

• Coulomb barrier height: $V_C = z_1 z_2 e^2 / (4\pi\epsilon_0 R) \approx 35$ MeV
• The alpha particle must tunnel through a barrier that is ~31 MeV higher than its energy
• The Gamow factor: $2\gamma \approx 86$
• Tunneling probability: $T \approx e^{-86} \approx 10^{-37}$

Yet the alpha particle "attempts" to escape with a frequency $\nu \approx v/2R \approx 10^{21}$ s$^{-1}$ (from the internal velocity of the alpha particle bouncing inside the nucleus). So the decay rate is $\lambda \approx \nu T \approx 10^{21} \times 10^{-37} = 10^{-16}$ s$^{-1}$, giving a half-life $t_{1/2} = \ln 2/\lambda \approx 10^{16}$ s $\approx 3 \times 10^8$ years. The measured half-life of $^{238}$U is $4.47 \times 10^9$ years — the right order of magnitude. Chapter 13 refines this estimate with proper treatment of the nuclear interior and centrifugal barrier.

5.7.4 The Gamow Peak in Stellar Fusion (Preview)

In stellar fusion (Chapter 21), the tunneling probability $T \propto e^{-2\pi\eta} \propto \exp(-b/\sqrt{E})$ decreases with decreasing energy, while the Maxwell-Boltzmann distribution $\propto \exp(-E/k_BT)$ decreases with increasing energy. The product peaks at the Gamow energy:

$$E_G = \left(\frac{b k_B T}{2}\right)^{2/3}$$

where $b = \pi z_1 z_2 e^2 \sqrt{2\mu} / (4\pi\epsilon_0 \hbar)$. For proton-proton fusion at the solar core temperature ($T \approx 1.5 \times 10^7$ K), $E_G \approx 6$ keV — far below the Coulomb barrier of $\sim$550 keV. Stellar fusion proceeds entirely by quantum tunneling.

The width of the Gamow peak is:

$$\Delta = \frac{4}{\sqrt{3}}\sqrt{E_G k_B T} \approx 6\ \text{keV}$$

for $pp$ fusion in the Sun. The narrowness of this peak (relative to $k_BT$) justifies treating $S(E)$ as approximately constant over the peak, which enormously simplifies the evaluation of stellar reaction rates.

The temperature sensitivity of the reaction rate follows from the Gamow peak analysis. Writing $\langle\sigma v\rangle \propto T^n$, the exponent is:

$$n = \frac{E_G}{3k_BT} - \frac{2}{3}$$

For $pp$ fusion: $n \approx 5.9/(3 \times 1.3) - 2/3 \approx 3.9$. For the $^{12}$C + $^{12}$C reaction (important in carbon-burning massive stars), the much higher Coulomb barrier gives $n \approx 30$ — an extraordinary sensitivity that makes carbon burning explosive once it ignites.

This discussion of the Gamow peak illustrates a recurring theme: quantum mechanics (tunneling) and statistical mechanics (the thermal distribution) combine to determine the rates of nuclear processes in astrophysical environments. Chapter 21 develops this theme fully.

5.8 Density of States

The density of states $\rho(E)$ counts the number of quantum states per unit energy interval. It appears in Fermi's golden rule and throughout nuclear physics: in decay rates, reaction cross sections, statistical nuclear properties, and the nuclear level density. Despite its apparently simple definition, the density of states is the quantity that often determines the qualitative behavior of nuclear processes — it controls whether a transition is fast or slow, whether resonances are isolated or overlapping, and whether statistical approaches to nuclear reactions are valid.

5.8.1 Free-Particle Density of States

For a free particle of mass $m$ in a box of volume $V$, the allowed momenta form a grid with spacing $\Delta p = 2\pi\hbar/L$ (for a cube of side $L$, so $V = L^3$). The number of states with momentum between $p$ and $p + dp$ is the number of grid points in a spherical shell:

$$dn = \frac{V \cdot 4\pi p^2\, dp}{(2\pi\hbar)^3}$$

This is one of the most fundamental results in quantum statistical mechanics. The factor $1/(2\pi\hbar)^3$ is the volume of a single quantum state in phase space — a direct consequence of the uncertainty principle.

Converting to energy $E = p^2/(2m)$, we use $p = \sqrt{2mE}$ and $dp = \sqrt{m/(2E)}\, dE$:

$$\rho(E) = \frac{dn}{dE} = \frac{V}{(2\pi)^2}\left(\frac{2m}{\hbar^2}\right)^{3/2} \sqrt{E}$$

The $\sqrt{E}$ dependence means that higher-energy final states have more available states. This has an immediate physical consequence: for processes where the matrix element is approximately constant (the allowed approximation in beta decay), the energy spectrum of emitted particles is proportional to $\sqrt{E}$ — a prediction that is directly tested experimentally.

For a relativistic particle ($E = pc$ for massless photons):

$$\rho_\gamma(E) = \frac{V \cdot E^2}{\pi^2 \hbar^3 c^3}$$

including a factor of 2 for the two independent polarization states. The $E^2$ dependence is stronger than the $\sqrt{E}$ for non-relativistic particles, which is one reason why gamma decay rates increase strongly with photon energy: $\Gamma_\gamma \propto E_\gamma^{2\lambda+1}$ for multipole $\lambda$, where the $E^2$ from the density of states combines with additional $E$-dependent factors from the matrix element.

5.8.2 Density of States in Nuclear Physics

The density of states enters nuclear physics in several distinct ways:

1. Continuum states (decay products and reaction products): The outgoing alpha particle, electron, neutrino, or photon from a nuclear decay occupies continuum states described by the free-particle formula above (with appropriate modifications for the Coulomb field).

2. Nuclear level density: The density of bound excited states in a nucleus increases approximately exponentially with excitation energy. For a nucleus at excitation energy $E^*$, the level density is approximately:

$$\rho(E^*) \approx \frac{\sqrt{\pi}}{12} \frac{\exp(2\sqrt{aE^*})}{a^{1/4}(E^*)^{5/4}}$$

where $a \approx A/8$ MeV$^{-1}$ is the level density parameter, and $A$ is the mass number. This is the Bethe formula, derived by treating the nucleus as a Fermi gas of non-interacting nucleons. The exponential growth is dramatic: for $^{56}$Fe ($a \approx 7$ MeV$^{-1}$) at $E^* = 10$ MeV, the exponent is $2\sqrt{70} \approx 16.7$, giving $\rho \sim e^{16.7} \sim 10^7$ states per MeV. At $E^* = 20$ MeV, the exponent doubles to $2\sqrt{140} \approx 23.7$, and $\rho \sim 10^{10}$ states per MeV — a thousand-fold increase for a mere doubling of the excitation energy.

At low excitation energies, individual states can be resolved experimentally — gamma-ray spectroscopy with high-purity germanium detectors can identify states separated by a few keV. At high excitation energies ($E^* \gtrsim 8$--10 MeV in medium-mass nuclei), the average level spacing becomes smaller than the experimental resolution, and the level density becomes the appropriate statistical description. This transition from discrete spectroscopy to statistical behavior is one of the most important conceptual thresholds in nuclear physics.

The nuclear level density is crucial for: - Compound nucleus reactions (Chapter 18): The formation and decay of the compound nucleus are governed by the level density at the compound nucleus excitation energy - Statistical gamma decay (Chapter 9): High-lying states decay through a cascade of gamma rays, and the cascade pattern depends on the level density - Neutron capture cross sections (Chapter 19): Resonances in neutron capture correspond to individual levels of the compound nucleus; their spacing is determined by the level density

3. Phase space for multi-particle final states: When a decay or reaction produces multiple particles, the available phase space is an integral over the density of states for each particle, subject to energy and momentum conservation. The larger the phase space, the faster the decay.

🔗 Connection to Fermi's Golden Rule

Every time we apply Fermi's golden rule $\Gamma = (2\pi/\hbar)|V_{fi}|^2 \rho(E_f)$, we must identify the appropriate $\rho(E_f)$:

Process	Matrix element $V_{fi}$	Density of states $\rho$
Gamma decay	EM multipole operator	Photon: $\rho_\gamma = E_\gamma^2 V/(\pi^2\hbar^3 c^3)$
Alpha decay	Nuclear + Coulomb potential	Alpha continuum states (with Coulomb correction)
Beta decay	Weak interaction operator	Electron $\times$ neutrino phase space
Nuclear reaction	Nuclear potential ($T$-matrix)	Outgoing particle continuum
Compound nucleus decay	Average matrix element	Nuclear level density

5.8.3 Counting States: A Practical Rule

A useful rule for counting the number of magnetic substates: a level with angular momentum $J$ has $2J + 1$ substates. When we compute a decay rate for an unpolarized initial state (the usual case in nuclear physics), we average over initial substates and sum over final substates:

$$\Gamma = \frac{2\pi}{\hbar} \frac{1}{2J_i + 1}\sum_{M_i, M_f} |\langle f; J_f M_f|\hat{V}|i; J_i M_i\rangle|^2 \rho(E_f)$$

The factor $1/(2J_i + 1)$ is the averaging over initial substates. The Wigner-Eckart theorem (which relates the $M$-dependent matrix elements to a single reduced matrix element) then simplifies this sum enormously — a technique we develop in Chapter 9.

5.8.4 The Fermi Gas Model of the Nucleus

The simplest model for the nuclear level density treats the nucleus as a gas of non-interacting fermions (the Fermi gas model). This model, despite its crude approximation (nucleons in a nucleus interact strongly), captures the essential physics remarkably well.

In a Fermi gas at zero temperature, all single-particle states are filled up to the Fermi energy $E_F$, and none above. As the nucleus is excited, nucleons near the Fermi surface are promoted to states above $E_F$, creating particle-hole excitations. The number of ways to create excitations with total energy $E^*$ grows exponentially because the combinatorics of distributing $E^*$ among many possible particle-hole pairs explodes rapidly.

The Fermi gas model gives the level density parameter:

$$a = \frac{\pi^2}{6} g(E_F)$$

where $g(E_F)$ is the single-particle level density at the Fermi energy. For a nuclear potential with level spacing $\epsilon_0 \approx 41 A^{-1/3}$ MeV (the harmonic oscillator estimate), this gives $a \approx A/15$ MeV$^{-1}$. The experimental value $a \approx A/8$ MeV$^{-1}$ is somewhat larger, reflecting the fact that the nucleon effective mass in the nuclear medium exceeds the free nucleon mass — an important many-body effect.

The level density also depends on the nuclear angular momentum $J$ through the spin-cutoff parameter $\sigma$:

$$\rho(E^*, J) \propto (2J + 1) \exp\left(-\frac{J(J+1)}{2\sigma^2}\right) \rho(E^*)$$

This Gaussian suppression of high-spin states reflects the fact that aligning many nucleon spins requires energy. High-spin states at a given excitation energy are rarer than low-spin states, a fact that governs the gamma-ray cascade patterns observed in nuclear spectroscopy experiments and is essential for the statistical model of compound nucleus reactions (Chapter 18).

Chapter Summary

"I heard Rabi once say that the real trick in physics is knowing which approximation to use."

This chapter has assembled the quantum mechanical toolkit for nuclear physics. More than just a collection of formulas, it provides a way of thinking about nuclear states and transitions: every state is characterized by angular momentum and parity quantum numbers $J^\pi$, every transition is governed by Fermi's golden rule with an appropriate matrix element and density of states, and every tunneling process is controlled by the exponential Gamow factor. Let us gather the key results:

Angular momentum: Nuclear states are characterized by total angular momentum $J$ and parity $\pi$, written $J^\pi$. In the shell model, each nucleon carries $j = l \pm \frac{1}{2}$, and nucleons couple primarily via the j-j coupling scheme.

Clebsch-Gordan coefficients: The CG coefficients $\langle j_1 m_1; j_2 m_2 | J M\rangle$ connect uncoupled and coupled bases. They enforce the triangle rule and projection rule, and their symmetry under particle exchange generates the antisymmetry constraints for identical nucleons.

Parity and selection rules: The parity of a nuclear state is $\pi = \prod_i (-1)^{l_i}$. Selection rules for electromagnetic transitions depend on the multipolarity and type (E or M). Selection rules for weak transitions depend on the degree of forbiddenness.

Fermi's golden rule: $\Gamma = (2\pi/\hbar)|V_{fi}|^2 \rho(E_f)$ — the master formula for all transition rates and cross sections.

Identical particles: The antisymmetry requirement for fermions, enforced through Slater determinants, is the foundation of nuclear shell structure. Two identical nucleons in the same $j$-shell can only couple to even $J$.

WKB tunneling: $T \approx \exp(-2\gamma)$ where $\gamma$ is the integral of $\kappa(x)$ through the barrier. The exponential sensitivity of tunneling to barrier parameters explains the vast range of alpha decay half-lives and the possibility of stellar fusion. The Sommerfeld parameter $\eta$ characterizes the Coulomb barrier strength, and the Gamow peak determines the most probable energy for stellar fusion reactions.

Density of states: $\rho(E)$ counts states per unit energy and appears in every application of Fermi's golden rule. The free-particle density of states scales as $\sqrt{E}$ (non-relativistic) or $E^2$ (photons). The nuclear level density increases approximately exponentially with excitation energy according to the Bethe formula, transitioning from discrete, resolvable levels at low excitation to a statistical continuum at high excitation.

The connections between these tools are as important as the tools themselves. Angular momentum coupling and CG coefficients enter every matrix element calculation. The Wigner-Eckart theorem factorizes matrix elements into geometric (CG) and dynamical (reduced matrix element) parts. Fermi's golden rule combines the matrix element with the density of states to give transition rates. The WKB approximation provides the tunneling matrix element for barrier penetration problems. And the density of states — whether for continuum particles or bound nuclear levels — determines the phase space available for any process.

These tools will be used in virtually every chapter that follows. You may find yourself returning to this chapter often — it is designed as a reference as much as a narrative. The machinery of angular momentum coupling will be exercised immediately in Chapters 6 and 7 (the shell model), Fermi's golden rule reappears in Chapters 9, 13, 14, and 17-19, and the WKB approximation is the heart of the alpha decay theory in Chapter 13 and the stellar fusion rate in Chapter 21.

Key Equations Reference

For quick reference, the essential equations of this chapter: