Atomic Physics

Question 1

Given an electron with linear momentum p, what is its de Broglie wavelength?

Answer 1

• The de Broglie wavelength λ is given by λ = h/p
• h is Planck’s constant and p is the momentum of the electron
• For an electron with velocity v, p = mv

The de Broglie wavelength allows us to access the wave nature of electrons. It gives an intuitive picture of quantization in systems such as atoms and the infinite potential well.

Question 2

Derive the expression for the de Broglie wavelength of an electron accelerated through a potential difference V.

Answer 2

Use conservation of energy to give eV= (1/2) mv². Then apply λ=h/p = h/(mv).

Question 3

State the Pauli exclusion principle for electrons in atoms.

Answer 3

Two electrons in an atom cannot have the same set of quantum numbers.

The Pauli exclusion principle comes about because electrons are identical fermions. This principle explains many observations related to multielectron systems.

For example, it helps to explain the electronic configuration of atoms as well as the structure of the periodic table. It is critical to explain the properties of metals.

Question 4

True or False:

An electron’s spin is a classical angular momentum.

Answer 4

False

Electron spin is an intrinsic quantum property without classical analog; it cannot be described as classical angular momentum. It is intrinsic and quantized, with possible values of spin measurement along a given axis being +(1/2)ℏ and -(1/2) ℏ.

The spin angular momentum and orbital angular momentum live in different Hilbert spaces. The electron’s spin contributes to the magnetic moment of the electron.

Question 5

Discuss the role of electron tunneling in the operation of a Scanning Tunneling Microscope (STM).

Answer 5

Quantum tunneling allows electrons to pass through the barrier formed by the vacuum gap with a non-zero probability (classically, electrons cannot pass).

This vacuum gap is between the sharp metal tip in the STM and the sample. Due to this tunneling, there is a tunneling current.

This current is highly sensitive to the distance between the tip and the sample, enabling atomic resolution imaging as the tip is scanned across the surface, the distance between the tip and the sample is varied so that the tunneling current remains constant. In this way, the features of the surface are captured at the atomic scale.

STM measures electron tunneling current between a sharp tip and a conductive surface.

Question 6

Under what condition can electrons in a metal be approximated as a free electron gas?

Answer 6

When electron-electron and electron-ion interactions are negligible and the mean free path is much greater than the lattice spacing, allowing electrons to move freely through the crystal.

In this regime, the periodic potential from the lattice and interactions between electrons have minimal effect on motion, so electrons can be modeled as free particles.

This is the basis of the free electron gas model, which leads to the Drude model for conductivity. It works well in simple metals where scattering is infrequent and the potential is nearly uniform over the electron’s path.

Question 7

Fill in the blank:

The ________- _______ distribution describes the distribution of electrons in a solid at thermal equilibrium and is essential for understanding semiconductor physics.

Answer 7

Fermi-Dirac

It determines the average number of electrons occupying a single particle energy state at a given temperature (it also depends on the chemical potential).

Critical for predicting electrical and thermal properties of materials, particularly at low temperatures.

Question 8

Derive the expression for the radius of the nth orbit in the Bohr model for a hydrogen atom.

Answer 8

• The Coulomb force equals the centripetal force.
• This gives one relationship between v and r.
• Then quantize the angular momentum as L=mvr=nℏ.
• So v = nℏ/mr.
• Plug this into the relationship between v and r derived previously and solve for r.

Question 9

Explain why the Bohr model is able to predict the Rydberg formula for hydrogen spectral lines.

Answer 9

The model allows us to find the quantized energy levels in hydrogen by assuming that the angular momentum is quantized.

The difference in the energy levels then come out to be in correspondence with the frequencies of the observed spectral lines.

Energy difference ΔE=E_n−E_m =hν leads directly to 1/λ=R_H(1/m²−1/n²). Here R_H is the Rydberg constant.

Question 10

The hydrogen atom is often imagined as a proton around which an electron is moving in a circular orbit.

If we now apply classical physics to this system, we arrive at the conclusion that the hydrogen atom will not be stable, and that the electron will inevitably crash into the proton.

Why?

Answer 10

• As the electron moves in a circle about the proton, it accelerates.
• If it accelerates, it emits electromagnetic radiation since it is charged.
• Therefore, it loses energy continuously.
• Consequently, the distance between the electron and the proton keeps on decreasing and the electron spirals inward toward the nucleus, eventually collapsing into the proton.

If we have quantized energy levels, this problem goes away. Since there are only a discrete set of energies an electron can have, it cannot radiate continuously.

Question 11

In the Bohr model, what assumptions are made about electron orbits and radiation emission?

Answer 11

• Electrons move in circular orbits under the Coulomb force, with no radiation emitted in stable orbits.
• Radiation is emitted only when an electron transitions between discrete energy levels.
• The angular momentum is quantized. This leads to energy quantization.

These assumptions allow the model to predict discrete spectral lines but limit its applicability beyond hydrogen-like systems (especially multielectron atoms).

Question 12

Fill in the blank:

The quantization condition for angular momentum in the Bohr model is given by the formula _______.

Answer 12

This quantization condition is fundamental to the model. It ensures that only certain orbits are allowed, leading to discrete energy levels.

Question 13

What are the main limitations of the Bohr model of the atom?

Answer 13

• Fails to include electron-electron interactions in multielectron atoms (for example, lithium).
• Limited to the hydrogen atom and cannot be applied to other quantum systems.

The Bohr model treats electrons as orbiting in fixed circular paths and does not incorporate the probabilistic nature of quantum mechanics or electron spin, so it only works for hydrogen-like atoms (one electron).

It cannot accurately explain spectral lines of multi-electron atoms, fine structure, or quantum phenomena like Zeeman splitting and spin-orbit coupling, which require quantum mechanics and wavefunctions.

Question 14

Calculate the velocity of an electron in the first orbit of a hydrogen atom using the Bohr model.

Answer 14

• Set the Coulomb force equal to the centripetal force.
• From the quantization condition, mvr =ℏ (since n = 1).
• Therefore, r = ℏ/mv.
• Plug this into the equation relating the Coulomb force to the centripetal force.
• Solve for v.

Question 15

Explain how the concept of energy quantization in the Bohr model leads to the stability of atoms.

Answer 15

Electrons cannot spiral into the nucleus because only discrete orbits with fixed energy are allowed.

Classical EM theory predicts collapse; quantization removes intermediate energy states.

Question 16

Derive the energy levels of a particle in a 1D infinite potential well of length L.

Answer 16

where n is a positive integer n = 1, 2, 3, …

Solving the time-independent Schrodinger equation with zero potential inside and infinite walls at boundaries. The solutions of the Schrodinger equation are sinusoidal. The boundary conditions for the potential imposes the condition that the wavefunction must be zero at the walls (since the particle cannot cross these boundaries and the wavefunction is related to the probability density). This directly leads to the quantization of the energy levels.

Question 17

True or False:

The quantization of energy levels in atoms is a direct consequence of the wave nature of electrons.

Answer 17

True

The wave-particle duality of electrons leads to standing wave conditions in atomic orbitals, resulting in discrete energy levels. We can also consider the quantization as coming about once we solve the time-independent Schrodinger equation for the energy eigenfunctions.

Question 18

Identify the condition under which the energy quantization in a harmonic oscillator becomes significant.

Answer 18

It becomes significant when the quantum of energy ħω is comparable to or larger than the thermal energy k_BT.

This is typically observed at low temperatures or in systems with very high frequencies, where quantum effects dominate over classical behavior.

The discreteness of the energy levels then becomes important. It is just like looking at sand from different distances. Close up, we can observe the discreteness of the grains of sand, but from further away, the sand appears to continuous.

Question 19

What are the energy levels of a particle in a one-dimensional quantum harmonic oscillator potential?

Answer 19

Quantized energies arise from solutions to the time-independent Schrödinger equation in a quadratic potential.

The boundary conditions now are that the wavefunctions must vanish as we go far away from the equilibrium point (typically x = 0).

If E_n is not equal to (n+ 1/2)ℏω, the energy eigenfunctions blow up far away from the equilibrium point.

Question 20

For multielectron atoms, what is the central field approximation?

Answer 20

We assume that each electron in a multi-electron atom moves in an average, spherically symmetric potential created by the nucleus and the average effect of all the other electrons.

This total potential is called the central field. It is important because then, via the spherical symmetry of the central field, the energy eigenfunctions are radial functions times the spherical harmonics. The energy states have well-defined values of l and m_l. We have hydrogen-like energy levels.

Note that the energy levels will depend on l in general. The central field means that the potential is no longer exactly 1/r. It is the 1/r potential that leads to the additional degeneracy in l.

Question 21

What is the quantum defect in alkali atoms?

Answer 21

It’s a correction to energy levels because the valence electron feels a non-1/r potential. The energy becomes:

• δ₁ depends on l
• For small l, the electron penetrates the core, increasing the defect
• For large l, δ₁ is effectively 0

Question 22

Explain why the wave function of a quantum particle in a bound state must be normalizable.

Answer 22

• A bound state wave function must be normalizable to ensure a finite probability of finding the particle in space.
• Note that the absolute value square of the wavefunction gives us the probability density.
• This implies that the wave function must approach zero as ∣𝑥∣ approaches infinity.

Normalizability is a requirement for physical acceptability of quantum states and ensures that the total probability is unity.

Question 23

A hydrogen atom transitions from level n=3 to n=2.

How would you derive the emitted photon’s wavelength?

Answer 23

Derived from the Bohr formula for the energy levels and E=hc/λ.

Question 24

Describe the role of selection rules in atomic transitions and how they affect spectral lines.

Answer 24

• They determine allowed transitions between quantum states.
• For electric dipole transitions: Δl = ±1, Δm = 0, ±1.
• These rules arise from the conservation of angular momentum and parity considerations. To derive them rigorously, the Wigner-Eckart theorem can be used.

Selection rules explain why certain transitions are observed in spectra while others are forbidden, thereby influencing the intensity and presence of spectral lines.

Question 25

A transition from **2s→1s in hydrogen** is not observed in electric dipole spectra. Why?

Answer 25

Δl=0, **violating the selection rule** Δl=±1 for electric dipole transitions. ## Footnote In particular, Δl=0 is excluded via parity considerations. The electric dipole operator changes sign under a parity transformation; the 1s and 2s states are unchanged under this transformation since the parity is given by(−1)ˡ. This means that the matrix element of the electric dipole operator between the 1s state and the 2s state must be zero. **Electric dipole** selection rules arise from angular momentum conservation and parity considerations.

Question 26

In atomic physics, what is the selection **rule for spin-flip transitions** in electron spin resonance?

Answer 26

Δm_s = ±1 (where m_s is the spin magnetic quantum number). A spin up electron becomes spin down, or vice versa. So the Δm_s = ±1. ## Footnote Spin-flip transitions involve changes in the **electron spin orientation** and are probed in techniques like ESR (electron spin resonance) or EPR (electron paramagnetic resonance).

Question 27

# True or False: A transition from a **d** orbital to an **s** orbital is **forbidden** under electric dipole selection rules.

Answer 27

False ## Footnote Δl=−2, but electric dipole allows Δl=±1; thus it's forbidden. Dipole transitions must involve l changing by ±1. Note that 'forbidden transitions' are not really forbidden - they generally do take place (due to, for example, two photon emission), just at a much slower rate as compared to electric dipole transitions.

Question 28

# True or False: The **Zeeman effect** results in the splitting of atomic energy levels in the presence of an electric field.

Answer 28

False ## Footnote The Zeeman effect refers to the splitting of atomic energy levels due to an external magnetic field, while the Stark effect involves electric fields.

Question 29

How are the **hydrogen atom’s energy levels quantized**?

Answer 29

## Footnote The time-independent Schrodinger equation in 3D can be converted to an effective 1D Schrodinger equation by using the spherical symmetry in the hydrogen atom (and thus looking for simultaneous eigenfunctions of the Hamiltonian and the angular momentum). Solving this effective 1D Schrodinger equation with the Coulomb potential (and a centrifugal term added that depends on the angular momentum) leads to quantized negative energies.

Question 30

Identify the concept of **fine structure** in the atomic spectrum of hydrogen.

Answer 30

It refers to small **splittings in spectral lines** within the same **principal quantum number**. It is caused by relativistic corrections and spin-orbit coupling. ## Footnote Fine structure is a crucial aspect of **precision spectroscopy** and can be observed in high-resolution spectroscopic techniques.

Question 31

# True or False: The **Bohr model** accurately predicts the fine structure and hyperfine structure of hydrogen spectra.

Answer 31

False ## Footnote **Fine structure and hyperfine structure** arise from relativistic corrections, spin-orbit coupling, and the spin-spin coupling for the electron and the proton. One needs to apply perturbation theory to the solutions obtained from the Schrodinger equation (or one can use the Dirac equation). The Bohr model is too simplistic to take into account these effects.

Question 32

What is the difference between **LS coupling** and **jj coupling** in atomic spectra?

Answer 32

* In **light atoms**, electron orbitals combine to form total L, and spins combine to form total S. Then, L and S couple to give total J. This is **LS coupling.** * In **heavy atoms**, each electon's l and s combine to form j, and all the j's add up to a total J. This is **jj coupling**.

Question 33

# Fill in the blank: The **hyperfine structure** in atomic spectra results from interactions between the **nuclear magnetic moment** and \_\_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_\_.

Answer 33

electron magnetic moment ## Footnote One can think that the magnetic moment of the nucleus produces a magnetic field that then interacts with the magnetic moment of the electrons. Another cause of the hyperfine structure is the electric quadrupole interaction, where the nuclear quadrupole moment (if this is non-zero) interacts with the electric field gradient from the electrons. Note that the hyperfine structure provides insights into **nuclear properties** and is essential for applications like atomic clocks.

Question 34

Explain the concept of **quantum numbers** in describing atomic orbitals and the physical significance of each.

Answer 34

* **Principal quantum number (n):** Determines the energy level and size of the orbital. * **Azimuthal quantum number (l):** Defines the shape of the orbital. * **Magnetic quantum number (mℓ):** Describes the orientation of the orbital. * **Spin quantum number (m_s):** Represents the intrinsic angular momentum of the electron. ## Footnote These quantum numbers arise from the solution of the **Schrödinger equation in spherical coordinates** and are fundamental to atomic structure. Note that these numbers specify the gross structure. If we include the fine structure, we need to talk about the total angular momentum (spin plus orbital) as well. If we include hyperfine, then the angular momentum due to the nuclear spin also comes into play.

Question 35

What is the **Lamb shift**, and why is it significant in the context of **atomic physics**?

Answer 35

It is a small **difference in energy levels** of hydrogen (specifically 2s^1/2 and 2p^1/2) not predicted by **Dirac's theory** since both these states have j = 1/2. ## Footnote It arises from **quantum electrodynamics (QED) effects**, such as vacuum polarization. The Lamb shift provided one of the first confirmations of QED and illustrates the importance of quantum field theory in atomic physics.

Question 36

Describe **isotope shifts** and their impact on atomic spectra.

Answer 36

The shifts in energy levels (and hence the atomic spectral lines) that result from **differences in nuclear mass and the nuclear charge distribution** in the isotopes. | Two isotopes differ in the number of neutrons in the nucleus. ## Footnote **Mass shift:** Changes in the mass of the nucleus from one isotope to another mean that we have a different reduced mass. This affects the energy levels. **Field shift:** Differences in nuclear charge distribution (different nuclei will have a different charge distribution in general) alters the interaction between the electrons and the nucleus. Isotope shifts are critical for precision measurements in spectroscopy and help study nuclear properties.

Question 37

Compare and contrast the **Lyman, Balmer, and Paschen** series in hydrogen spectra.

Answer 37

* **Lyman series:** Transitions to n=1, correspond to the ultraviolet region of the electromagnetic spectrum * **Balmer series:** Transitions to n=2, correspond to the visible region * **Paschen series:** Transitions to n=3, correspond to the infrared region ## Footnote These series are key to understanding the electromagnetic spectrum of hydrogen and illustrate the quantized nature of electronic transitions.

Question 38

What is the shortest wavelength in the **Lyman series** of hydrogen?

Answer 38

## Footnote The shortest wavelength corresponds to the maximum energy difference. In the Lyman series, the final value of n is 1. Consider then the initial value of n to be very large (approaching infinity). The Lyman series corresponds to transitions ending at n=1; shortest λ comes from infinite n.

Question 39

What are the **Rydberg states**?

Answer 39

The **highly excited states** of an atom with large principal quantum numbers. ## Footnote They exhibit **exaggerated atomic properties** and are highly sensitive to external fields. For example, their electrical polarizibility is very high due to their large quantum number. Rydberg atoms are instrumental in studying fundamental interactions and quantum phenomena at macroscopic scales.

Question 40

# True or False: The spin state of the electrons in the ground state of helium is a triplet state.

Answer 40

False ## Footnote In the ground state, the spatial state will be symmetric. The spin state then has to be antisymmetric (that is, the singlet state), since the overall state has to be antisymmetric (electrons are fermions).

Question 41

# True or False: In helium, states with antisymmetric spin states (parahelium) have a higher energy than the corresponding states with symmetric spin states (orthohelium).

Answer 41

True ## Footnote In parahelium, the spatial state will be symmetric, so the electrons will be closer to each other, leading to greater repulsive energy between them.

Question 42

Discuss the principle of the **Franck-Condon factor** and its role in **molecular spectroscopy.**

Answer 42

* The Franck-Condon factor allows us to explain the intensity distribution of vibronic transitions (transitions involving both electronic and vibrational energy levels) * During such a transition, the nuclei are assumed to be stationary. We then look at the overlap of the initial and final vibrational wavefunctions. The Frank-Condon factor quantifies this overlap. * **The most probable transitions are those where this overlap is the greatest.** ## Footnote The principle is essential for interpreting molecular **spectra and understanding** electronic-vibrational coupling.

Question 43

# Fill in the blank: For electric-dipole allowed **electronic transitions** in a diatomic molecule, the selection rule requires that the change in rotational quantum number J is \_\_\_\_\_\_\_.

Answer 43

ΔJ = 0, ±1 ## Footnote This rule arises from the conservation of angular momentum and that the photon is a spin-1 particle. Use addition or angular momenta.

Question 44

Describe the **selection rules for vibrational transitions** in diatomic molecules and the physical reasoning behind them.

Answer 44

Δv = ±1 | (v is the vibrational quantum number) ## Footnote Vibrational transitions typically follow this rule due to the diatomic molecule behaving like a quantum harmonic oscillator. The **harmonic oscillator model approximates vibrational energy levels**, and the linear term in the Taylor expansion of the dipole moment function leads to these rules - the matrix element of the dipole moment is only non-zero for Δv = ±1.

Question 45

What are the selection rules for **hyperfine transitions** in an atom?

Answer 45

* ΔF = 0, ±1 (where F is the total angular momentum quantum number) * Δm_F = 0, ±1 (where m_F is the projection of F) ## Footnote Hyperfine transitions are essential in atomic clocks. The 21 cm line arises from the hyperfine splitting of the ground electronic state of hydrogen

Question 46

What assumptions are made when analyzing the **hyperfine structure** in an atomic spectrum?

Answer 46

* **Nuclear spin is non-zero** (otherwise there is hyperfine splitting). * The electron-nucleus magnetic interaction is treated as perturbation. ## Footnote We typically assume a point-like nucleus. For s-electrons, we usually use the Fermi contact term.

Question 47

Discuss the role of **symmetry** in determining the selection rules in molecular spectroscopy.

Answer 47

* The selection rules are determined by the symmetry properties of the **initial and final states** and the transition operator. * Transitions are allowed if the product of the representations of the initial state, transition operator, and final state contains the **totally symmetric representation**. ## Footnote **Group theory** provides a powerful tool for determining selection rules by analyzing the symmetry properties of molecular states and the operators involved in transitions.

Question 48

# True or False: The **intensity of X-rays** is independent of the tube current in an X-ray tube.

Answer 48

False ## Footnote The intensity of X-rays is **directly proportional** to the tube current, as more electrons striking the target result in more X-ray photons being produced.

Question 49

What **atomic process** produces the **sharp lines** observed in the X-ray spectrum of a metal target?

Answer 49

Electronic transitions to **lower inner-shell** energy levels by an electron from a higher energy level (characteristic X-ray emission). ## Footnote High-energy electrons eject core electrons; outer electrons fall in, releasing photons with discrete energies. We have sharp peaks in the X-ray spectrum.

Question 50

Describe the distinction between **Bremsstrahlung** and characteristic **X-ray emissions.**

Answer 50

* **Bremsstrahlung Radiation**: Continuous spectrum. It is produced by the deceleration of electrons in the electric field of atomic nuclei. * **X-Rays**: Discrete spectral lines. They result from electronic transitions to inner atomic shells after electron ejection. ## Footnote Both occur in X-ray tubes, but mechanisms and spectra differ.

Question 51

In X-ray diffraction, what is the condition for **constructive interference** in terms of Bragg's Law?

Answer 51

nλ=2dsinθ ## Footnote This describes constructive interference from crystal planes separated by distance d.

Question 52

Explain how the **attenuation of X-rays** in a material is described mathematically.

Answer 52

## Footnote Attenuation is described as an **exponential law**. * ( I₀ ): Initial intensity * ( I ): Intensity after passing distance x through the material * ( μ ): Linear attenuation coefficient * ( x ): Thickness of the material The **intensity of X-rays decreases exponentially** with the thickness of the material. We have assumed a homogeneous material and X-rays with a single energy.

Question 53

# Fill in the blank: The **minimum wavelength of X-rays** produced in an X-ray tube is determined by the \_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ of the electrons striking the target.

Answer 53

maximum kinetic energy ## Footnote The **minimum wavelength**, λ_min, can be found using the equation λ_min = (hc)/(eV), where V is the accelerating voltage, illustrating the inverse relation between wavelength and energy. The higher the energy of the electrons, the higher the energy of the X-rays and so the smaller their wavelength.

Question 54

Derive the expression for the **Zeeman splitting** of an atomic energy level in a weak magnetic field.

Answer 54

## Footnote This is the Linear Zeeman effect. We need to consider for the orbital angular momentum and spin angular momentum. * g_J is Landé g-factor * 𝜇_𝐵 is Bohr magneton * m_J is the magnetic quantum number associated with total angular momentum J.

Question 55

# True or False: The **Paschen-Back effect** is the splitting of atomic energy levels in the presence of a strong magnetic field.

Answer 55

True ## Footnote In the Paschen-Back effect, the **interaction of the magnetic field with the atomic levels becomes so strong** that it decouples the spin and orbital angular momenta, leading to a different splitting pattern than the linear Zeeman effect.

Question 56

# Fill in the blank: The **interaction energy** of an atom in a **uniform magnetic field** is given by the dot product of the **magnetic moment** and the \_\_\_\_\_\_\_ \_\_\_\_\_\_\_.

Answer 56

magnetic field ## Footnote This interaction energy is central to understanding the **Zeeman effect** and is mathematically expressed as E = -**μ**⋅**B**.

Question 57

Describe the **level crossing phenomenon**.

Answer 57

Level crossing occurs when two **energy levels become degenerate** as a system parameter, like an external field, is varied. ## Footnote Level crossings can significantly affect the populations of atomic states in dynamical systems, especially in driven systems. For example, suppose we have a time-dependent Hamiltonian that changes very slowly. If we start off from an eigenstate of this Hamiltonian (at a particular time), we remain in the same instantaneous eigenstate (via the adiabatic theorem). If, however, we are near a level crossing and the Hamiltonian changes rapidly, this rapid transition induces transitions between the states.

Question 58

Explain how the quantum mechanical treatment of the **Stark effect** differs from the classical approach.

Answer 58

* **Quantum mechanically**, the Stark effect is treated **using perturbation theory** where the external electric field is a perturbation to the atomic Hamiltonian. * **Classically**, the Stark effect would be considered as the **interaction of the dipole moment** with the electric field, without quantization of energy levels. ## Footnote The quantum mechanical results make intuitive sense once we understand the classical approach. For example, the electric dipole moment in hydrogen for the 1s state is zero due to spherical symmetry. Corresponding, the linear Stark effect calculated for this state quantum mechanically is also zero. The quantum approach captures the discrete nature of atomic energy levels and their shifts.

Question 59

In the **Zeeman effect**, how many spectral components are expected from the 2p→1s transition in hydrogen under a weak magnetic field?

Answer 59

Three components are expected: Δm = −1, 0, +1 ## Footnote Zeeman splitting lifts degeneracy of ml; electric dipole selection rule still applies.

Question 60

How does the **Stark effect modify the energy levels** of hydrogen in an external electric field?

Answer 60

It describes the interaction of an atom with an external electric field via its electric dipole moment. As a result, the **energy levels split and shift**. ## Footnote We have both DC Stark shift (due to constant electric field) and AC Stark shift (due to an oscillating electric field). The calculation of the DC Stark shift in hydrogen is an excellent exercise in perturbation theory.

Question 61

What are the differences between **absorption, stimulated emission**, and **spontaneous emission**?

Answer 61

* **Absorption**: The absorption of a photon by an atom, causing it to jump to a higher energy level. * **Stimulated Emission:** An incoming photon can cause an excited atom to emit a second photon * **Spontaneous Emission:** An excited atom can also randomly emit a photon and drop to a lower energy level. ## Footnote Spontaneous emission takes place by itself. The photon emitted is random in direction (and has a random phase). On the other hand, in stimulated emission, the emitted photon is coherent with the incoming one. Stimulated emission forms the basis of lasers.

Question 62

How are atoms cooled using **Doppler shifts**?

Answer 62

* Atoms moving toward a laser absorb more photons due to the Doppler effect. * This results in momentum kicks that slow them down. * Spontaneous emission averages out, so the atoms cool overall. ## Footnote Doppler cooling is often used in laser cooling experiments to bring atoms close to absolute zero. The technique works best when the laser is slightly red-detuned from an atomic transition.