CSIR NET Physical Sciences Quantum Mechanics Flashcards

24 sample cards from the Quantum Mechanics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

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

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

2. 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.

3. 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 $$\left(i\hbar\gamma^{\mu}\partial_\mu-mc\right)\psi=0$$

4. 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: $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}I$.

5. 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 ($\alpha,\beta$).

6. 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.

7. 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 $2\times2$.

8. 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.

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

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

10. 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.

11. 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).

12. 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.

13. 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.

14. 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 $p_\mu\to p_\mu-\frac{e}{c}A_\mu$.

15. 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.

16. 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.

17. 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.

18. 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.

19. 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).

20. 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 $\delta_l$.

21. 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 $\psi$.

22. 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.

23. 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 $E=+\sqrt{p^{2}c^{2}+m^{2}c^{4}}$.

24. 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.

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