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Prologue Problems

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Basic

Medium

Advanced

Basic

B-1. Photon Energy Calculation

According to Einstein's light quantum hypothesis, the energy of a photon with frequency \(\nu\) is given by \(E = h\nu\). Using Planck's constant \(h = 6.63 \times 10^{-34}\ \mathrm{J \cdot s}\), find the energy of a single photon for each of the following types of light.

(a) Yellow light from sodium (frequency \(\nu = 5.09 \times 10^{14}\ \mathrm{Hz}\))

(b) Red laser (wavelength \(\lambda = 633\ \mathrm{nm}\), use the speed of light \(c = 3.00 \times 10^{8}\ \mathrm{m/s}\))

(c) Cell phone radio waves (frequency \(\nu = 2.0 \times 10^{9}\ \mathrm{Hz}\))

Hint

Directly substitute into \(E = h\nu\). For (b), convert to frequency using the relation \(\nu = c/\lambda\) before calculating.

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B-2. Relationship Between Probability Amplitude and Probability

Given a complex probability amplitude \(\phi = 3 + 4i\), calculate the following.

(a) The complex conjugate \(\phi^*\) of \(\phi\)

(b) \(|\phi|^2 = \phi^* \phi\)

(c) When the probability amplitude is written as \(\phi = A e^{i\theta}\) (\(A > 0\), \(\theta\) is real), express \(|\phi|^2\) in terms of \(A\) and \(\theta\).

Hint

(a) Replace \(i\) with \(-i\). (b) Expand \((3-4i)(3+4i)\). (c) Use the fact that the complex conjugate of \(e^{i\theta}\) is \(e^{-i\theta}\).

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B-3. Calculation of Interference Terms

Two probability amplitudes are given by \(\phi_1 = e^{i\alpha}\) and \(\phi_2 = e^{i\beta}\) (where \(\alpha, \beta\) are real numbers).

(a) Expand \(|\phi_1 + \phi_2|^2\) and express it in terms of \(\alpha\) and \(\beta\).

(b) Calculate \(|\phi_1|^2 + |\phi_2|^2\).

(c) Express the interference term \(|\phi_1 + \phi_2|^2 - (|\phi_1|^2 + |\phi_2|^2)\) using the phase difference \(\delta = \alpha - \beta\).

(d) Find \(|\phi_1 + \phi_2|^2\) for \(\delta = 0\) (in phase) and \(\delta = \pi\) (out of phase), respectively.

Hint

Expand \(|\phi_1 + \phi_2|^2 = (\phi_1^* + \phi_2^*)(\phi_1 + \phi_2)\) and show that \(\phi_1^*\phi_2 + \phi_2^*\phi_1 = 2\cos(\alpha - \beta)\).

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B-4. Probability Distributions of Bullets and Electrons

In a double-slit experiment, the probability (for bullets) or the absolute square of the probability amplitude (for electrons) when only slit 1 is open is given at position \(x\) on the screen as follows:

\[ P_1(x) = |\phi_1(x)|^2 = e^{-(x-a)^2}, \quad P_2(x) = |\phi_2(x)|^2 = e^{-(x+a)^2} \]

where \(a > 0\).

(a) Evaluate the combined probability for bullets \(P_{12}^{\text{bullet}}(x) = P_1(x) + P_2(x)\) at \(x = 0\).

(b) For electrons, given that the probability amplitudes are \(\phi_1(x) = e^{-(x-a)^2/2}\) and \(\phi_2(x) = e^{-(x+a)^2/2}\) (here we consider the real-valued case), evaluate \(P_{12}^{\text{electron}}(x) = |\phi_1(x) + \phi_2(x)|^2\) at \(x = 0\).

(c) Compare the results of (a) and (b), and determine which is larger: \(P_{12}^{\text{electron}}(0)\) or \(P_{12}^{\text{bullet}}(0)\).

Hint

Substituting \(x = 0\) gives \(P_1(0) = P_2(0) = e^{-a^2}\). In (b), compute \((\phi_1(0) + \phi_2(0))^2\) and compare it with \(P_1(0) + P_2(0)\) from (a).

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B-5. Polar Form of Complex Numbers and Euler's Formula

Using Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\), calculate the following.

(a) \(e^{i\pi}\)

(b) \(e^{i\pi/2}\)

(c) The real and imaginary parts of \(e^{i\pi/4}\)

(d) Find \(|e^{i\theta}|^2\) independent of \(\theta\).

Hint

Substitute the values of \(\theta\) into Euler's formula. For (d), calculate \(|e^{i\theta}|^2 = e^{-i\theta} \cdot e^{i\theta}\).

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B-6. Wave Intensity and Interference

Following the notation in the text, suppose the heights of two waves are given by

\[ h_1 = A\cos(\omega t), \quad h_2 = A\cos(\omega t + \delta) \]

where \(A > 0\) and \(\delta\) is the phase difference.

(a) Using the sum-to-product formula, rewrite \(h_1 + h_2\) in product form.

(b) Find the amplitude of \(h_1 + h_2\) when \(\delta = 0\).

(c) Find \(h_1 + h_2\) when \(\delta = \pi\).

Hint

Use \(\cos P + \cos Q = 2\cos\!\left(\frac{P+Q}{2}\right)\cos\!\left(\frac{P-Q}{2}\right)\).

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B-7. Discreteness of Energy — The Staircase Analogy with Numbers

As stated in the main text, "the energy of an atom is discrete, like steps on a staircase." The energy levels of the hydrogen atom are given by

\[ E_n = -\frac{13.6\ \mathrm{eV}}{n^2} \quad (n = 1, 2, 3, \ldots) \]

(This will be derived in Ch. 16.) Calculate the following.

(a) Find the energies of the ground state (\(n = 1\)) and the first excited state (\(n = 2\)), respectively.

(b) Find the energy of the photon emitted when a transition occurs from \(n = 2\) to \(n = 1\): \(\Delta E = E_2 - E_1\).

(c) Using \(E = h\nu\), find the frequency \(\nu\) of this photon (use \(1\ \mathrm{eV} = 1.60 \times 10^{-19}\ \mathrm{J}\)).

(d) Does this light fall within the visible light range (frequency \(4.3 \times 10^{14}\ \mathrm{Hz}\) to \(7.5 \times 10^{14}\ \mathrm{Hz}\))?

Hint

(b) Computing \(\Delta E = E_2 - E_1\) yields a positive value (the energy of the emitted photon). (c) Use \(\nu = \Delta E / h\), converting \(\Delta E\) to J (joules) before calculating.

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B-8. Superposition of Probability Amplitudes — Numerical Example

Consider a 2-state system where the probability amplitude for being in state \(|1\rangle\) is \(\phi_1 = \dfrac{1}{\sqrt{3}}\), and the probability amplitude for being in state \(|2\rangle\) is \(\phi_2 = \sqrt{\dfrac{2}{3}}\, e^{i\pi/3}\).

(a) Find the probability \(P_1 = |\phi_1|^2\) of observing state \(|1\rangle\).

(b) Find the probability \(P_2 = |\phi_2|^2\) of observing state \(|2\rangle\).

(c) Verify that the normalization condition \(P_1 + P_2 = 1\) holds.

(d) Find the real and imaginary parts of \(\phi_2\).

Hint

Use the fact that \(|c\, e^{i\theta}|^2 = |c|^2\). For (d), substitute \(e^{i\pi/3} = \cos(\pi/3) + i\sin(\pi/3)\).

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Medium

M-1. Comparison of Classical and Quantum Probability

Based on the Feynman-style comparison of bullets, waves, and electrons described in the main text, answer the following questions.

(a) Explain the physical reason why \(P_{12} = P_1 + P_2\) holds in the double-slit experiment with bullets, based on the fact that "a bullet passes through one slit or the other."

(b) In the double-slit experiment with electrons, each electron is detected as a single particle, yet when many electrons are accumulated, an interference pattern appears. Explain this fact using the mathematical structure of probability amplitudes

\[ P_{12} = |\phi_1 + \phi_2|^2 \]

In particular, clearly identify the reason why \(P_{12} \neq P_1 + P_2\).

(c) Suppose probability amplitudes were restricted to real numbers rather than complex numbers. Would interference terms still appear in that case? Discuss what restrictions arise regarding the sign of the interference term compared to the complex number case.

Hint

(b) Expand \(|\phi_1 + \phi_2|^2 = |\phi_1|^2 + |\phi_2|^2 + \phi_1^*\phi_2 + \phi_2^*\phi_1\) and show that the last two terms are the interference terms. (c) In the real case, \(\phi_1^*\phi_2 + \phi_2^*\phi_1 = 2\phi_1\phi_2\), and consider that the phase difference \(\delta\) can only take the values \(0\) or \(\pi\).

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M-2. Light Quantum Hypothesis and the Photoelectric Effect

Based on Einstein's light quantum hypothesis \(E = h\nu\), answer the following questions.

(a) Let the work function of a metal be \(W\). When light of frequency \(\nu\) is incident on the metal, express the maximum kinetic energy \(K_{\max}\) of the ejected electrons in terms of \(h\), \(\nu\), and \(W\).

(b) In the photoelectric effect, if the frequency of light is below a certain value \(\nu_0\), no electrons are ejected regardless of how much the light intensity is increased. Express this threshold frequency \(\nu_0\) in terms of \(W\) and \(h\).

(c) According to classical wave theory (Maxwell's electromagnetism), increasing the intensity of light should be able to provide sufficient energy to the electrons. Nevertheless, the experimental fact that no electrons are ejected below the threshold frequency contradicts a certain assumption of classical theory. Explain what assumption this is.

(d) Light of wavelength \(\lambda = 400\ \mathrm{nm}\) is incident on a metal with work function \(W = 2.3\ \mathrm{eV}\). Find the maximum kinetic energy \(K_{\max}\) of the ejected electrons in units of eV.

Hint

(a) Derive \(h\nu = W + K_{\max}\) from energy conservation. (b) Consider the condition \(K_{\max} = 0\). (d) First calculate the photon energy using \(E = hc/\lambda\).

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M-3. Models in Physics and Falsifiability

Based on the philosophy of science standpoint described in Introduction — Before the Four Journeys at the beginning of this site (models are hypotheses; equations are tools for falsifiability), answer the following questions.

(a) What does it mean to say that Newtonian mechanics is a "hypothesis"? Give one example each of a domain where Newtonian mechanics succeeded and a domain where it broke down, and explain.

(b) The text states that quantum mechanics "has not had a single incorrect prediction in over 100 years." Explain why quantum mechanics is nevertheless called a "hypothesis," referring to its relationship with general relativity.

(c) What is "falsifiability"? Explain why the claim "tomorrow's weather will be sunny, rainy, or cloudy" is not falsifiable.

Hint

(a) Contrast celestial motion (success) with the atomic scale (breakdown). (b) Refer to the quantum gravity problem. (c) A claim that is consistent with any possible observational outcome is not falsifiable.

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M-4. Einstein's Duality — Founder and Critic

Based on the section "Einstein and Quantum Mechanics" in the main text, answer the following questions.

(a) Explain the grounds for calling Einstein "one of the founders" of quantum theory by separately describing the physical content of his 1905 light quantum hypothesis and his 1917 prediction of stimulated emission.

(b) Explain why Einstein became a "critic" of quantum mechanics, taking into account the meaning of his statement "God does not play dice."

(c) Explain the relationship between the EPR paradox (1935) and Bell's theorem (1964) in approximately 200 characters, using ALL of the following keywords: incompleteness, hidden variables, inequality, experiment.

Hint

(c) Einstein argued for the "incompleteness" of quantum mechanics and assumed the existence of "hidden variables" whose values are determined before measurement. Bell derived an "inequality" from that assumption, and "experiments" violated it, thereby refuting Einstein's position—summarize this logical flow.

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M-5. Interference of Probability Amplitudes — Quantitative Analysis

In the double-slit experiment, the probability amplitudes from the two apertures at position \(x\) on the screen are given by

\[ \phi_1(x) = A\, e^{i k r_1(x)}, \quad \phi_2(x) = A\, e^{i k r_2(x)} \]

where \(A > 0\) is a real constant, \(k\) is the wavenumber, and \(r_1(x)\) and \(r_2(x)\) are the distances from aperture 1 and aperture 2 to position \(x\) on the screen, respectively.

(a) Calculate the combined probability \(P_{12}(x) = |\phi_1(x) + \phi_2(x)|^2\) and express it in terms of \(\Delta r(x) = r_1(x) - r_2(x)\) (the path difference).

(b) Express the condition for \(P_{12}(x)\) to take its maximum value in terms of \(\Delta r\) and \(k\). Also, find the value of \(P_{12}\) at that point.

(c) Find the condition for \(P_{12}(x)\) to take its minimum value of \(0\).

(d) Using the wavelength \(\lambda = 2\pi/k\), rewrite the conditions from (b) and (c) in terms of \(\Delta r\) and \(\lambda\).

Hint

(a) Factor as \(\phi_1 + \phi_2 = A e^{ikr_1}(1 + e^{ik(r_2 - r_1)})\) and compute \(|\cdot|^2\). Simplify into a form containing \(\cos\).

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Advanced

A-1. Indispensability of Complex Probability Amplitudes

In the main text, it is stated that "probability amplitudes must be complex numbers. Real numbers alone cannot reproduce the predictions of quantum mechanics." Examine this claim by following the steps below.

(a) Consider a two-state system. Let \(\phi_1\), \(\phi_2\) be the probability amplitudes for states \(|1\rangle\) and \(|2\rangle\), satisfying the normalization condition \(|\phi_1|^2 + |\phi_2|^2 = 1\). If \(\phi_1\) and \(\phi_2\) are both real, state the geometric condition satisfied by \((|\phi_1|, |\phi_2|)\).

(b) If \(\phi_1\) and \(\phi_2\) are complex, they can be written as \(\phi_1 = |\phi_1| e^{i\alpha}\), \(\phi_2 = |\phi_2| e^{i\beta}\). How many degrees of freedom (number of independent real parameters) are gained compared to the real case? Take into account that the overall phase (a common phase factor \(e^{i\gamma}\) applied to both \(\phi_1\) and \(\phi_2\)) is physically unobservable.

(c) In the interference pattern of a double-slit experiment, the phase difference \(\delta(x)\) varies continuously with position \(x\) on the screen, and \(P_{12}(x)\) is a smooth function containing \(\cos\delta(x)\). If probability amplitudes were restricted to real numbers, the phase difference could only take the two values \(0\) or \(\pi\). What restrictions would this impose on the interference pattern? Discuss in contrast with the smooth interference fringes observed in experiments.

(d) Based on the above considerations, describe "the physical significance of probability amplitudes being complex numbers in quantum mechanics" in approximately 200 characters (about 3–4 sentences).

Hint

(a) \(|\phi_1|^2 + |\phi_2|^2 = 1\) represents a point on the unit circle. In the real case, the only freedom is the sign of \(\phi\) itself. (b) The relative phase \(\alpha - \beta\), after removing the overall phase, constitutes a new degree of freedom. (c) The real-valued interference term can only take the two values \(\pm 2|\phi_1||\phi_2|\) and cannot produce smooth fringe patterns.

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A-2. From the Light Quantum Hypothesis to Stimulated Emission — The Chain of Einstein's Logic

Einstein proposed the light quantum hypothesis \(E = h\nu\) in 1905 and predicted stimulated emission in 1917. These two achievements are not independent but are logically connected. Follow the questions below to trace this chain of reasoning.

(a) Consider a two-level atomic system in thermal equilibrium at temperature \(T\). The ratio of populations of energy levels \(E_1\) (ground state) and \(E_2\) (excited state, \(E_2 > E_1\)) follows the Boltzmann distribution:

\[ \frac{N_2}{N_1} = \exp\!\left(-\frac{E_2 - E_1}{k_B T}\right) \]

where \(k_B\) is the Boltzmann constant. What does \(N_2/N_1\) approach as \(T \to \infty\)? And as \(T \to 0\)? Explain the physical meaning.

(b) When atoms and light (with frequency \(\nu = (E_2 - E_1)/h\)) coexist in thermal equilibrium, atoms repeatedly undergo transitions: absorbing light (\(E_1 \to E_2\)) and emitting light (\(E_2 \to E_1\)). There are two types of emission:

  • Spontaneous emission: An excited atom spontaneously emits a photon. The transition rate per unit time is \(A \cdot N_2\) (\(A\) is a constant).
  • Stimulated emission: Emission of a photon induced by an external photon. The transition rate per unit time is \(B \cdot \rho(\nu) \cdot N_2\) (\(B\) is a constant, \(\rho(\nu)\) is the energy density of light).

The absorption rate is \(B' \cdot \rho(\nu) \cdot N_1\). From the condition that the absorption rate equals the emission rate in thermal equilibrium,

\[ B'\, \rho(\nu)\, N_1 = A\, N_2 + B\, \rho(\nu)\, N_2 \]

express \(\rho(\nu)\) in terms of \(A\), \(B\), \(B'\), \(E_2 - E_1\), and \(k_B T\).

(c) Planck's blackbody radiation formula (studied in detail in Ch. 1) is given by

\[ \rho(\nu) = \frac{8\pi h \nu^3}{c^3} \cdot \frac{1}{e^{h\nu/(k_B T)} - 1} \]

By comparing the result of (b) with this formula, show that \(B' = B\) and express the value of \(A/B\) in terms of \(h\), \(\nu\), and \(c\).

(d) If stimulated emission did not exist (\(B = 0\)), what form would \(\rho(\nu)\) take from the equilibrium condition in (b)? Show that this contradicts Planck's formula, and explain that "stimulated emission is a logical necessity required by thermal equilibrium."

Hint

(b) Solve the equilibrium condition equation for \(\rho(\nu)\). Substitute the Boltzmann distribution from (a) for \(N_2/N_1\). (c) Comparing the result of (b) with Planck's formula in the limit \(T \to \infty\) yields \(B' = B\). Then compare at finite temperature to determine \(A/B\). (d) Setting \(B = 0\) gives \(\rho(\nu) \propto e^{-h\nu/(k_BT)}\), and the \(\nu^3\) factor is missing.

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