CSIR NET Physical Sciences Atomic & Molecular Physics Syllabus

Classify natural, Doppler, and collision broadening as homogeneous or inhomogeneous, and give their line shapes.

Natural broadening: homogeneous, Lorentzian. Collision (pressure) broadening: homogeneous, Lorentzian. Doppler broadening: inhomogeneous, Gaussian. (A convolution of Gaussian and Lorentzian gives a Voigt profile.)

What is the chemical shift in NMR spectroscopy, and why does it occur?

The chemical shift is the small change in the resonance frequency of a nucleus due to shielding of the external magnetic field by surrounding electrons. The effective field is $B_{eff} = B_0(1-\sigma)$, where $\sigma$ is the shielding constant, so chemically inequivalent nuclei resonate at different frequencies.

Write the definition of the NMR chemical shift $\delta$ in ppm.

$$\delta = \frac{\nu_{sample} - \nu_{ref}}{\nu_{ref}}\times 10^{6}\ \text{(ppm)}$$ usually referenced to tetramethylsilane (TMS) for $^{1}\text{H}$ and $^{13}\text{C}$.

State the electric-dipole selection rules for $l$, $m_l$, and $s$ in a one-electron atom.

$$\Delta l = \pm 1,\qquad \Delta m_l = 0, \pm 1,\qquad \Delta s = 0$$

State the electric-dipole selection rules for the total angular momentum quantum numbers $j$ and $m_j$.

$$\Delta j = 0, \pm 1\ \ (\text{but } j=0 \nrightarrow j=0),\qquad \Delta m_j = 0, \pm 1$$

State the electric-dipole selection rules for $L$, $S$, and $J$ in LS (Russell–Saunders) coupling for many-electron atoms.

$$\Delta L = 0, \pm 1\ (\text{but } L=0\nrightarrow L=0),\quad \Delta S = 0,\quad \Delta J = 0, \pm 1\ (J=0\nrightarrow J=0).$$ Also the parity must change.

What is the parity (Laporte) selection rule for electric-dipole transitions?

Electric-dipole transitions are allowed only between states of opposite parity. Since dipole transitions require $\Delta l = \pm 1$, the parity $(-1)^{l}$ must change.

Define spontaneous emission and stimulated emission.

Spontaneous emission: an excited atom decays to a lower level emitting a photon randomly in phase, direction, and time, with no external field. Stimulated emission: an incident photon induces an excited atom to emit a second photon identical in frequency, phase, direction, and polarization (coherent).

How do spontaneous and stimulated emission compare in terms of coherence and rate dependence?

Spontaneous emission is incoherent (random phase/direction) and its rate is independent of the radiation field. Stimulated emission is coherent (identical to the inducing photon) and its rate is proportional to the spectral energy density of the field. Lasers rely on stimulated emission.

Write Einstein's three rate equations relating the $A$ and $B$ coefficients for the populations $N_1$ (lower) and $N_2$ (upper).

Spontaneous emission rate $= A_{21}N_2$; stimulated emission rate $= B_{21}N_2\,\rho(\nu)$; absorption rate $= B_{12}N_1\,\rho(\nu)$, where $\rho(\nu)$ is the spectral energy density.

State the two relations between the Einstein coefficients $A_{21}$, $B_{21}$, and $B_{12}$ for degeneracies $g_1$, $g_2$.

$$g_1 B_{12} = g_2 B_{21},\qquad \frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^{3}}{c^{3}}$$

How does the ratio of spontaneous to stimulated emission rate vary with frequency, and what does this imply for masers vs lasers?

$$\frac{A_{21}}{B_{21}\rho(\nu)} = \frac{8\pi h \nu^{3}}{c^{3}\,\rho(\nu)} = e^{h\nu/k_B T}-1$$ Spontaneous emission grows as $\nu^{3}$, so it dominates at optical/higher frequencies, making population inversion harder for lasers than for microwave masers.

What is optical pumping?

Optical pumping is the use of light (resonant photons) to excite atoms from a lower energy level to a higher one, redistributing populations among energy/sublevels to create a non-thermal distribution and achieve population inversion needed for laser action.

What is population inversion, and why is it necessary for laser action?

Population inversion is a non-equilibrium condition where a higher energy level has a greater population than a lower one ($N_2 > N_1$, accounting for degeneracy $\frac{N_2}{g_2} > \frac{N_1}{g_1}$). It is required so that stimulated emission exceeds absorption, giving net optical gain.

Why is population inversion impossible in a strict two-level system under optical pumping?

In a two-level system, the same pump field that excites atoms also drives stimulated emission with equal $B$-coefficient rates. At best the populations equalize ($N_2 = N_1$, saturation), so a true inversion $N_2 > N_1$ cannot be achieved; three- or four-level schemes are needed.

Compare three-level and four-level laser systems regarding the threshold for population inversion.

In a three-level laser the lower laser level is the ground state, which is highly populated, so more than half the atoms must be pumped up—a high threshold (e.g., ruby laser). In a four-level laser the lower laser level is an excited state that empties rapidly, so even a small pump gives inversion—a low threshold (e.g., Nd:YAG, He-Ne).

Write the rate equation for the upper-level population $N_2$ in a laser, including pumping, spontaneous and stimulated terms.

$$\frac{dN_2}{dt} = R_2 - \frac{N_2}{\tau_2} - B_{21}\,N_2\,\rho(\nu) + B_{12}\,N_1\,\rho(\nu)$$ where $R_2$ is the pump rate and $\tau_2$ the upper-level lifetime.

In the rate-equation approach, write the photon-number rate equation for a laser cavity.

$$\frac{d\phi}{dt} = B\,\Delta N\,\phi - \frac{\phi}{\tau_c} + (\text{spontaneous})$$ where $\phi$ is photon number, $\Delta N = N_2 - N_1$ the inversion, and $\tau_c$ the cavity photon lifetime; gain must exceed cavity loss for lasing.

What is the threshold condition for laser oscillation in terms of gain and loss?

Lasing begins when the round-trip gain equals the round-trip loss: $$R_1 R_2\, e^{2(g-\alpha)L} = 1$$ where $g$ is the gain coefficient, $\alpha$ the loss coefficient, $L$ the cavity length, and $R_1,R_2$ the mirror reflectivities.

What are the modes of an optical resonator, and what two categories exist?

Resonator modes are the stable standing-wave field distributions supported by the cavity. They divide into longitudinal (axial) modes, which differ in the number of half-wavelengths along the axis, and transverse modes (TEM$_{mn}$), which describe the transverse intensity pattern.

Write the resonance condition for longitudinal modes of a cavity of length $L$ and the frequency spacing between adjacent modes.

$$L = q\,\frac{\lambda}{2}\ \Rightarrow\ \nu_q = \frac{q c}{2L},\qquad \Delta\nu = \frac{c}{2L}$$ where $q$ is a positive integer; $\Delta\nu = \frac{c}{2L}$ is the free spectral range.

What does the designation TEM$_{mn}$ mean for transverse cavity modes?

TEM$_{mn}$ denotes a transverse electromagnetic mode where $m$ and $n$ are the number of intensity nodes in the two transverse directions. TEM$_{00}$ is the fundamental mode with a single Gaussian spot (lowest loss, best beam quality).

Define the quality factor $Q$ of an optical resonator and relate it to the cavity photon lifetime.

$$Q = 2\pi\,\frac{\text{energy stored}}{\text{energy lost per cycle}} = \omega\,\tau_c = 2\pi\nu\,\tau_c$$ A high $Q$ means low loss and a long photon lifetime $\tau_c$.

Define coherence length and write its relation to spectral line width.

Coherence length $l_c$ is the propagation distance over which a wave maintains a fixed phase relationship. $$l_c = c\,\tau_c = \frac{c}{\Delta\nu} = \frac{\lambda^{2}}{\Delta\lambda}$$ where $\tau_c$ is the coherence time and $\Delta\nu$ the source bandwidth.

Define coherence time and state its relation to coherence length and bandwidth.

Coherence time $\tau_c$ is the time over which the phase of a wave remains predictable. $$\tau_c = \frac{1}{\Delta\nu},\qquad l_c = c\,\tau_c$$ A narrower bandwidth $\Delta\nu$ gives a longer coherence time and length.

Distinguish temporal coherence from spatial coherence.

Temporal (longitudinal) coherence relates to the correlation of the wave's phase at the same point at different times; it is set by the bandwidth/coherence length. Spatial (transverse) coherence relates to phase correlation between different points across the wavefront; it is set by the source size/angular extent.

Why does a single excited state's finite lifetime give rise to natural broadening (link to the uncertainty principle)?

A state with lifetime $\tau$ has an energy uncertainty $\Delta E \approx \hbar/\tau$ by the energy–time uncertainty relation. This energy spread converts to a frequency spread $\Delta\nu \approx \frac{1}{2\pi\tau}$, the natural line width.

What is the Lamb shift, and how does it differ from ordinary fine structure?

The Lamb shift is the small energy difference between the $2S_{1/2}$ and $2P_{1/2}$ levels of hydrogen, which the Dirac (fine-structure) theory predicts to be degenerate. It arises from quantum electrodynamic effects (vacuum fluctuations, self-energy), beyond the relativistic fine-structure corrections.

Using the relativistic/fine-structure result, which hydrogen levels with the same $n$ and $j$ but different $l$ are degenerate, and what lifts this degeneracy?

Levels with the same $n$ and $j$ but different $l$ (e.g. $2S_{1/2}$ and $2P_{1/2}$) are degenerate in the Dirac fine-structure result. This degeneracy is lifted by the Lamb shift (QED) and, including nuclear spin, by hyperfine structure.

State the spectroscopic term symbol notation and give the ground-state term symbol of hydrogen.

The term symbol is written $^{2S+1}L_J$, where $2S+1$ is the multiplicity, $L$ the orbital letter ($S,P,D,\dots$), and $J$ the total angular momentum. Hydrogen's ground state ($1s$, $S=\tfrac12$, $L=0$, $J=\tfrac12$) is $^{2}S_{1/2}$.

Compare the spectral output of laser light with that of an ordinary thermal source in terms of coherence, directionality, and emission mechanism.

Laser light is highly coherent (long coherence length), monochromatic, highly directional, and produced mainly by stimulated emission with population inversion. Thermal (ordinary) light is incoherent, broadband, emitted in all directions, and produced by spontaneous emission from atoms in thermal equilibrium.