CSIR NET Physical Sciences Quantum Mechanics Syllabus

Write the Lippmann-Schwinger integral equation whose iteration generates the Born series.

$$\psi(\vec r)=\phi(\vec r)+\int G_0(\vec r,\vec r')\,V(\vec r')\,\psi(\vec r')\,d^{3}r'$$ where $\phi$ is the incident wave and $G_0=-\frac{m}{2\pi\hbar^{2}}\frac{e^{ik|\vec r-\vec r'|}}{|\vec r-\vec r'|}$ is the free Green's function. The first Born approximation replaces $\psi\to\phi$ inside the integral.

What is the Born-approximation relation between phase shift $\delta_l$ and the potential for small phase shifts?

$$\delta_l\approx-\frac{2m k}{\hbar^{2}}\int_{0}^{\infty}V(r)\,[\,j_l(kr)\,]^{2}\,r^{2}\,dr$$ valid when $\delta_l\ll1$; $j_l$ is the spherical Bessel function.

Write the free (time-dependent) Klein-Gordon equation for a scalar field $\phi$.

$$\left(\Box+\frac{m^{2}c^{2}}{\hbar^{2}}\right)\phi=0,\qquad \Box=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}$$ i.e. $$\left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}+\frac{m^{2}c^{2}}{\hbar^{2}}\right)\phi=0$$

From which classical relation is the Klein-Gordon equation obtained, and via what substitution?

From the relativistic energy-momentum relation $E^{2}=p^{2}c^{2}+m^{2}c^{4}$, using the operator substitutions $E\to i\hbar\frac{\partial}{\partial t}$ and $\vec p\to-i\hbar\nabla$ acting on $\phi$.

What is the dispersion (energy) relation for plane-wave solutions of the Klein-Gordon equation, and why is it problematic?

$$E=\pm\sqrt{p^{2}c^{2}+m^{2}c^{4}}$$ It admits negative-energy solutions, which seem unbounded below for a single particle.

What is the conserved probability/charge density for the Klein-Gordon equation, and what is its key difficulty?

$$\rho=\frac{i\hbar}{2mc^{2}}\left(\phi^{*}\frac{\partial\phi}{\partial t}-\phi\frac{\partial\phi^{*}}{\partial t}\right)$$ It is not positive-definite, so it cannot serve as a probability density (later reinterpreted as charge density).

What type of particles does the Klein-Gordon equation describe?

Spin-0 (scalar) particles, such as pions and the Higgs boson.

Why was the Klein-Gordon equation initially abandoned as a single-particle wave equation?

Because it gave a non-positive-definite probability density and negative-energy solutions, both of which are inconsistent with a one-particle probabilistic interpretation. It was later revived as a field equation for spin-0 bosons.

Write the free Dirac equation in its standard (covariant) Hamiltonian form.

$$i\hbar\frac{\partial\psi}{\partial t}=\left(c\,\vec\alpha\cdot\vec p+\beta mc^{2}\right)\psi$$ or covariantly $$\left(i\hbar\gamma^{\mu}\partial_\mu-mc\right)\psi=0$$

What algebraic (anticommutation) relations must the Dirac matrices $\alpha_i$ and $\beta$ satisfy?

$$\{\alpha_i,\alpha_j\}=2\delta_{ij}I,\quad \{\alpha_i,\beta\}=0,\quad \alpha_i^{2}=\beta^{2}=I$$ Equivalently for gamma matrices: $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}I$.

Why did Dirac seek a wave equation first-order in time derivatives?

To obtain a positive-definite probability density (avoiding the Klein-Gordon problem). A first-order time equation, combined with relativistic covariance, requires first-order space derivatives too, forcing the introduction of matrix coefficients ($\alpha,\beta$).

What is the smallest dimension of the Dirac matrices, and what does this imply about the wave function?

They are $4\times4$ matrices, so the Dirac wave function $\psi$ is a four-component spinor.

Give the standard (Dirac-Pauli) representation of $\beta$ and $\vec\alpha$ in terms of Pauli matrices.

$$\beta=\begin{pmatrix}I&0\\0&-I\end{pmatrix},\qquad \alpha_i=\begin{pmatrix}0&\sigma_i\\\sigma_i&0\end{pmatrix}$$ where $\sigma_i$ are the $2\times2$ Pauli matrices and $I$ is $2\times2$.

What is the conserved probability density for the Dirac equation, and why is it satisfactory?

$$\rho=\psi^{\dagger}\psi=\sum_{i=1}^{4}|\psi_i|^{2}$$ It is positive-definite, so it is a valid probability density, resolving the Klein-Gordon problem.

Define the gamma matrices $\gamma^{\mu}$ in terms of $\beta$ and $\alpha_i$.

$$\gamma^{0}=\beta,\qquad \gamma^{i}=\beta\alpha_i\quad(i=1,2,3)$$

How does the Dirac equation naturally account for electron spin and the gyromagnetic ratio?

The non-relativistic reduction (Pauli equation) yields a magnetic moment $\vec\mu=-g\frac{e}{2m}\vec S$ with $g=2$ automatically, correctly predicting the electron's spin magnetic moment without ad hoc assumptions.

What is Dirac's interpretation of the negative-energy solutions (the 'Dirac sea')?

The negative-energy states are all filled (forming the 'sea'); the Pauli exclusion principle forbids transitions into them. A hole in this sea behaves as a positive-energy, positive-charge particle: the antiparticle (positron).

What antiparticle did the Dirac equation predict, and when was it experimentally discovered?

The positron (anti-electron), predicted by Dirac in 1928/1931 and discovered by Carl Anderson in 1932 in cosmic rays.

What is the spin and class of particles described by the Dirac equation?

Spin-$\tfrac{1}{2}$ fermions, such as electrons, protons, neutrons, and quarks.

Write the covariant Dirac equation in the presence of an electromagnetic field (minimal coupling).

$$\left[\gamma^{\mu}\left(i\hbar\partial_\mu-\frac{e}{c}A_\mu\right)-mc\right]\psi=0$$ obtained by the minimal substitution $p_\mu\to p_\mu-\frac{e}{c}A_\mu$.

Define the chirality matrix $\gamma^{5}$ and give its defining properties.

$$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$$ with $(\gamma^{5})^{2}=I$, $\{\gamma^{5},\gamma^{\mu}\}=0$, and $(\gamma^{5})^{\dagger}=\gamma^{5}$. Its eigenvalues $\pm1$ define right/left chirality.

Compare the Klein-Gordon and Dirac equations: order, spin, and probability density.

Klein-Gordon: 2nd order in time, spin-0, non-positive-definite density. Dirac: 1st order in time, spin-$\tfrac12$, positive-definite density $\psi^\dagger\psi$. Both are Lorentz covariant and have negative-energy solutions interpreted via antiparticles.

What is the velocity operator in the Dirac theory, and what is the eigenvalue puzzle (Zitterbewegung)?

The velocity operator is $\hat{\vec v}=c\,\vec\alpha$, whose eigenvalues are $\pm c$. The interference of positive- and negative-energy components produces a rapid trembling motion called Zitterbewegung.

In partial-wave scattering, what is the asymptotic radial wave function and how does $\delta_l$ appear?

$$R_l(r)\xrightarrow{r\to\infty}\frac{1}{kr}\sin\!\left(kr-\frac{l\pi}{2}+\delta_l\right)$$ The phase shift $\delta_l$ measures the asymptotic displacement of the radial wave relative to the free solution.

How does the hard-sphere potential of radius $a$ determine the $s$-wave phase shift at all energies?

For a hard sphere, the $l=0$ phase shift is $\delta_0=-ka$, giving a low-energy cross-section $\sigma=4\pi a^{2}$ (four times the geometric cross-section).

What is the relationship between the spherical Bessel functions and the free-particle partial waves used in scattering?

For $V=0$ the regular radial solution is $R_l(r)=j_l(kr)$ (spherical Bessel function), with asymptotic form $j_l(kr)\to\frac{1}{kr}\sin\!\left(kr-\frac{l\pi}{2}\right)$; turning on the potential adds the phase $\delta_l$.

Why does the second-order-in-time nature of the Klein-Gordon equation require specifying both $\phi$ and $\partial_t\phi$ initially?

Because it is second order in $\partial/\partial t$, the initial value problem needs both the field $\phi$ and its time derivative $\partial\phi/\partial t$ at $t=0$ to determine the solution — unlike the first-order Schrödinger or Dirac equations which need only $\psi$.

What conserved four-current density is associated with the Klein-Gordon equation (covariant form)?

$$j^{\mu}=\frac{i\hbar}{2m}\left(\phi^{*}\partial^{\mu}\phi-\phi\,\partial^{\mu}\phi^{*}\right),\qquad \partial_\mu j^{\mu}=0$$ The time component $j^{0}$ is the (non-positive-definite) density.

What is the form of the plane-wave (spinor) solution of the free Dirac equation for positive energy?

$$\psi(x)=u(\vec p)\,e^{-i(Et-\vec p\cdot\vec r)/\hbar}$$ where $u(\vec p)$ is a four-component spinor satisfying $(\gamma^{\mu}p_\mu-mc)u=0$, with $E=+\sqrt{p^{2}c^{2}+m^{2}c^{4}}$.

Summarize the historical/conceptual progression from Klein-Gordon to Dirac to antiparticles.

Klein-Gordon (1926) gave a relativistic spin-0 equation but with negative-energy states and non-positive probability density. Dirac (1928) built a first-order spin-$\tfrac12$ equation with positive density, still containing negative-energy states. Interpreting these via the Dirac sea predicted antiparticles (positron, 1932), launching relativistic quantum field theory.