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Ch. 1 Problems

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Basic

Medium

Advanced

Basic

B-1. Appreciating the Smallness of Planck's Constant

Let a typical frequency of visible light be \(\nu = 5.0 \times 10^{14}\;\mathrm{Hz}\). Calculate the energy of a single photon \(E = h\nu\) in SI units (J), and then convert it to electron volts (eV). Use \(h = 6.626 \times 10^{-34}\;\mathrm{J \cdot s}\) and \(1\;\mathrm{eV} = 1.602 \times 10^{-19}\;\mathrm{J}\).

Hint

Substitute the numerical values into \(E = h\nu\), and for the conversion to eV, use \(E\;[\mathrm{eV}] = E\;[\mathrm{J}] / (1.602 \times 10^{-19})\).

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B-2. Work Function and Threshold Frequency

The work function of sodium (Na) is \(W = 2.28\;\mathrm{eV}\). Find the threshold frequency \(\nu_0\) for the photoelectric effect to occur. Also, find the corresponding threshold wavelength \(\lambda_0\) in units of nm. Use \(c = 3.00 \times 10^8\;\mathrm{m/s}\).

Hint

The threshold condition is \(h\nu_0 = W\). The relationship with wavelength is \(c = \lambda_0 \nu_0\). Convert \(W\) to J before substituting.

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B-3. Kinetic Energy in the Photoelectric Effect

Ultraviolet light with wavelength \(\lambda = 200\;\mathrm{nm}\) is incident on a metal with work function \(W = 4.50\;\mathrm{eV}\). Find the maximum kinetic energy \(K\) of the emitted electrons in units of eV.

Hint

The photon energy is given by \(E = hc/\lambda\). Using \(hc \simeq 1240\;\mathrm{eV \cdot nm}\) simplifies the calculation. Then apply \(K = E - W\).

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B-4. Wavelength Calculation Using the Rydberg Formula

Using the Rydberg formula (1.6), find the wavelength \(\lambda\) in nm of the spectral line corresponding to \(m = 3\) in the Balmer series (\(n = 2\)) of the hydrogen atom. Use \(R_\infty = 1.097 \times 10^7\;\mathrm{m^{-1}}\).

Hint

Substitute numerical values into

\[\frac{1}{\lambda} = R_\infty \left(\frac{1}{2^2} - \frac{1}{3^2}\right)\]

and solve for \(\lambda\).

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B-5. Bohr's Quantization Condition and Orbital Radius

In the Bohr model of the hydrogen atom, the force balance for the electron's circular motion (Coulomb force = centripetal force) is given by

\[ \frac{e^2}{4\pi\varepsilon_0 r^2} = \frac{m_e v^2}{r} \]

By combining this with Bohr's quantization condition \(m_e v r = n\hbar\), express the \(n\)-th orbital radius \(r_n\) in terms of \(n\), \(\hbar\), \(m_e\), \(e\), and \(\varepsilon_0\).

Hint

Obtain \(v = n\hbar/(m_e r)\) from the quantization condition, substitute it into the force balance equation, and solve for \(r\).

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B-6. Numerical Calculation of the Bohr Radius

Setting \(n = 1\) in the result of D5 gives the value called the Bohr radius \(a_0\). Using the following constants, calculate the value of \(a_0\) to 3 significant figures.

  • \(\hbar = 1.055 \times 10^{-34}\;\mathrm{J \cdot s}\)
  • \(m_e = 9.109 \times 10^{-31}\;\mathrm{kg}\)
  • \(e = 1.602 \times 10^{-19}\;\mathrm{C}\)
  • \(4\pi\varepsilon_0 = 1.113 \times 10^{-10}\;\mathrm{C^2/(N \cdot m^2)}\)
Hint
\[a_0 = \frac{4\pi\varepsilon_0 \hbar^2}{m_e e^2}\]

Substitute the numerical values into this expression. The answer should be on the order of \(10^{-10}\;\mathrm{m}\).

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B-7. Limit of the Planck Distribution

Regarding Planck's formula for the average energy

\[ \langle E \rangle = \frac{h\nu}{e^{h\nu / k_B T} - 1} \]

show that in the limit \(h\nu \ll k_B T\), we obtain \(\langle E \rangle \simeq k_B T\), using the approximation \(e^x \simeq 1 + x\) (\(x \ll 1\)).

Hint

Setting \(x = h\nu/(k_B T) \ll 1\), the denominator becomes \(e^x - 1 \simeq x\).

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B-8. Comparison of Boltzmann Factors

At temperature \(T = 6000\;\mathrm{K}\) (approximately the temperature of the Sun's surface), calculate the Boltzmann factor \(e^{-h\nu / k_B T}\) for each of the following two frequencies.

(a) Visible light: \(\nu_1 = 5.0 \times 10^{14}\;\mathrm{Hz}\)

(b) Ultraviolet light: \(\nu_2 = 3.0 \times 10^{15}\;\mathrm{Hz}\)

Compare the results and confirm how high-frequency modes are suppressed.

Hint

First calculate \(k_B T\) (where \(k_B = 1.38 \times 10^{-23}\;\mathrm{J/K}\)), then determine \(h\nu/(k_B T)\), and finally evaluate the exponential function.

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Medium

M-1. Derivation of Hydrogen Atom Energy Levels Using the Bohr Model

In the Bohr model of the hydrogen atom, derive the energy levels \(E_n\) following the steps below.

(a) By simultaneously solving the force balance for the electron's circular motion (Coulomb force = centripetal force) and the Bohr quantization condition \(L = n\hbar\), find the radius \(r_n\) and velocity \(v_n\) of the \(n\)-th orbit.

(b) Express the total energy of the electron (kinetic energy + Coulomb potential energy) in terms of \(r_n\), and derive

\[ E_n = -\frac{m_e e^4}{2(4\pi\varepsilon_0)^2 \hbar^2} \cdot \frac{1}{n^2} \tag{*} \]

(c) Using the frequency condition \(h\nu = E_m - E_n\) and \(c = \lambda\nu\), reproduce the Rydberg formula (1.6) and express the Rydberg constant \(R_\infty\) in terms of fundamental constants.

Hint

(a) Find \(r_n\) using the same procedure as D5, then substitute back into the quantization condition to obtain \(v_n\). (b) The kinetic energy is \(\frac{1}{2}m_e v_n^2\), and the potential energy is \(-e^2/(4\pi\varepsilon_0 r_n)\). Using the force balance relation \(\frac{1}{2}m_e v_n^2 = e^2/(8\pi\varepsilon_0 r_n)\) allows for a concise expression. (c) Rearrange by writing \(1/\lambda = \nu/c\).

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M-2. Determination of Planck's Constant from Photoelectric Effect Experimental Data

In a photoelectric effect experiment, a metal is irradiated with light of various frequencies \(\nu\), and the maximum kinetic energy \(K\) of the ejected electrons is measured. Suppose the following data are obtained.

\(\nu\;[10^{14}\;\mathrm{Hz}]\) \(K\;[\mathrm{eV}]\)
6.0 0.21
7.5 0.83
9.0 1.45
10.5 2.07

(a) Sketch the general shape of the graph of \(K\) as a function of \(\nu\), and confirm that the linear relationship \(K = h\nu - W\) holds.

(b) Determine the value of Planck's constant \(h\) in units of eV·s from the data.

(c) Determine the work function \(W\) of this metal in units of eV.

(d) Convert the obtained value of \(h\) to SI units (J·s) and compare it with the literature value \(h = 6.626 \times 10^{-34}\;\mathrm{J \cdot s}\).

Hint

(b) The slope of the line corresponds to \(h\). Choose two points and calculate \(\Delta K / \Delta \nu\). (c) Either find \(W = h\nu_0\) from the \(\nu\)-intercept (the value of \(\nu_0\) where \(K = 0\)), or read it directly from the \(K\)-intercept of the line.

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M-3. Order-of-magnitude estimate of classical atomic collapse time

In classical electromagnetism, the energy radiated per unit time by a charge \(e\) moving with acceleration \(a\) (Larmor's formula) is given by

\[ P = \frac{e^2 a^2}{6\pi\varepsilon_0 c^3} \]

Assume that the electron in a hydrogen atom is in a circular orbit at the Bohr radius \(a_0 \simeq 5.3 \times 10^{-11}\;\mathrm{m}\), and answer the following.

(a) Express the centripetal acceleration \(a\) of the electron in terms of \(a_0\), \(m_e\), \(e\), and \(\varepsilon_0\) from the force balance, and calculate its numerical value.

(b) Use Larmor's formula to calculate the radiated power \(P\).

(c) Using the total energy of the electron \(E_1 = -13.6\;\mathrm{eV}\), estimate the classical collapse time \(\tau\) from the order of magnitude of \(|E_1|/P\), and confirm that it is consistent with \(\tau \sim 10^{-11}\;\mathrm{s}\) in Eq. (1.5).

Hint

(a) Obtain \(a\) from the Coulomb force \(F = e^2/(4\pi\varepsilon_0 a_0^2)\) and \(F = m_e a\). (c) Convert the energy to joules before dividing by \(P\). This is not a rigorous calculation of the collapse time, but it suffices if the order of magnitude agrees.

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M-4. High-Frequency Limit of the Planck Distribution and Wien's Law

The spectral radiance (per unit frequency) of Planck's blackbody radiation is given by

\[ B(\nu, T) = \frac{2h\nu^3}{c^2} \cdot \frac{1}{e^{h\nu/k_B T} - 1} \]

(a) Show that in the limit \(h\nu \gg k_B T\), \(B(\nu, T) \simeq \frac{2h\nu^3}{c^2} e^{-h\nu/k_B T}\) (Wien's radiation law).

(b) Show that the frequency \(\nu_{\max}\) at which \(B(\nu, T)\) is maximized is proportional to the temperature \(T\) (Wien's displacement law: \(\nu_{\max} \propto T\)) by rewriting \(\partial B / \partial \nu = 0\) in terms of \(x = h\nu/(k_B T)\). (It is not necessary to solve the transcendental equation; it suffices to show that \(x\) is a constant.)

Hint

(a) When \(h\nu \gg k_B T\), we have \(e^{h\nu/k_B T} \gg 1\), so the \(-1\) in the denominator can be neglected. (b) By substituting \(x = h\nu/(k_B T)\), the condition \(\partial B / \partial \nu = 0\) becomes a transcendental equation in \(x\) alone. Since \(x\) is a constant independent of \(T\), it follows that \(\nu_{\max} = (\text{constant}) \times k_B T / h\).

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Advanced

A-1. Derivation of the Rayleigh–Jeans Law from the Classical Equipartition Theorem and the Ultraviolet Catastrophe

Consider the number of standing wave modes of the electromagnetic field inside a cubic cavity of side length \(L\).

(a) Show that the number of modes (including the 2 degrees of freedom of polarization) in the wavenumber range from \(k\) to \(k + dk\) is

\[ g(k)\,dk = \frac{V}{\pi^2} k^2\,dk \]

where \(V = L^3\) is the volume of the cavity. (Use the boundary conditions \(k_x = n_x \pi / L\), \(k_y = n_y \pi / L\), \(k_z = n_z \pi / L\) (\(n_x, n_y, n_z = 1, 2, 3, \ldots\)) for each direction.)

(b) Using the relation \(k = 2\pi\nu/c\), rewrite the number of modes in the frequency range from \(\nu\) to \(\nu + d\nu\) as

\[ g(\nu)\,d\nu = \frac{8\pi V \nu^2}{c^3}\,d\nu \]

(c) By assigning an average energy \(k_B T\) to each mode according to the classical equipartition theorem, show that the spectral energy density per unit volume \(u(\nu, T)\) is

\[ u(\nu, T) = \frac{8\pi \nu^2}{c^3} k_B T \]

(the Rayleigh–Jeans law).

(d) Show that integrating \(u(\nu, T)\) from \(\nu = 0\) to \(\infty\) diverges, and explain that this is the ultraviolet catastrophe.

(e) Qualitatively explain why replacing the equipartition theorem with Planck's mean energy \(\langle E \rangle = h\nu/(e^{h\nu/k_B T} - 1)\) yields a finite total energy density.

Hint

(a) In the first octant of \(k\)-space (\(n_x, n_y, n_z > 0\)), take \(1/8\) of the spherical shell volume \(4\pi k^2 dk\), multiply by the density of lattice points \((\pi/L)^{-3}\), and multiply by 2 for polarization. (d) \(\int_0^\infty \nu^2 d\nu\) diverges. (e) Since \(\langle E \rangle\) decreases exponentially at high frequencies, the integral of \(\nu^2 \cdot \langle E \rangle\) converges.

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A-2. Generalization of the Bohr Model: Hydrogen-like Ions and the Correspondence Principle

Apply the Bohr model to a hydrogen-like ion (atomic number \(Z\), one electron).

(a) Taking into account that the nuclear charge is \(Ze\), derive the \(n\)-th orbital radius \(r_n\) and energy level \(E_n\) in a form that includes \(Z\).

(b) Calculate the energy and orbital radius of the ground state (\(n = 1\)) of \(\mathrm{He^+}\) (\(Z = 2\)), and compare them with those of the hydrogen atom.

(c) Bohr's correspondence principle: Show that when the quantum number \(n\) is sufficiently large, the frequency \(\nu_{n \to n-1}\) of light emitted in a transition between adjacent levels coincides with the classical orbital frequency \(f_n\) of the electron in the \(n\)-th orbit. That is, confirm that in the limit \(n \gg 1\),

\[ \nu_{n \to n-1} \simeq f_n \]

holds.

Hint

(a) The only change is that the Coulomb force becomes \(Ze^2/(4\pi\varepsilon_0 r^2)\). Follow the same procedure as S1, including \(Z\) in the calculation. (c) Calculate \(\nu_{n \to n-1} = (E_n - E_{n-1})/h\), and use \((1/(n-1)^2 - 1/n^2) \simeq 2/n^3\) for \(n \gg 1\). The classical orbital frequency is obtained from \(f_n = v_n/(2\pi r_n)\).

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