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Basic

Medium

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Basic

B-1. Photon Energy Calculation

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(a) Yellow light of sodium

Approach: Substitute directly into \(E = h\nu\).

\[ E = h\nu = (6.63 \times 10^{-34}\ \mathrm{J \cdot s}) \times (5.09 \times 10^{14}\ \mathrm{Hz}) \]
\[ E = 6.63 \times 5.09 \times 10^{-34+14}\ \mathrm{J} = 33.75 \times 10^{-20}\ \mathrm{J} \]
\[ \boxed{E \approx 3.38 \times 10^{-19}\ \mathrm{J}} \]

(b) Red laser (\(\lambda = 633\ \mathrm{nm}\))

Approach: Convert to frequency using \(\nu = c/\lambda\), then calculate \(E = h\nu = hc/\lambda\).

\[ \lambda = 633\ \mathrm{nm} = 633 \times 10^{-9}\ \mathrm{m} = 6.33 \times 10^{-7}\ \mathrm{m} \]
\[ E = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34})(3.00 \times 10^{8})}{6.33 \times 10^{-7}} \]
\[ E = \frac{19.89 \times 10^{-26}}{6.33 \times 10^{-7}} = 3.14 \times 10^{-19}\ \mathrm{J} \]
\[ \boxed{E \approx 3.14 \times 10^{-19}\ \mathrm{J}} \]

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

\[ E = h\nu = (6.63 \times 10^{-34})(2.0 \times 10^{9}) = 13.26 \times 10^{-25}\ \mathrm{J} \]
\[ \boxed{E \approx 1.33 \times 10^{-24}\ \mathrm{J}} \]

Verification: The order of frequency magnitudes is (a) > (b) > (c), and the energies follow the same order. The energy of visible light is on the order of \(10^{-19}\ \mathrm{J}\), while radio waves are on the order of \(10^{-24}\ \mathrm{J}\), which is physically reasonable.

B-2. Relationship Between Probability Amplitude and Probability

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(a) \(\phi^*\)

The complex conjugate of \(\phi = 3 + 4i\) is obtained by replacing \(i\) with \(-i\):

\[ \boxed{\phi^* = 3 - 4i} \]

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

\[ |\phi|^2 = (3 - 4i)(3 + 4i) = 9 + 12i - 12i - 16i^2 = 9 + 16 = 25 \]
\[ \boxed{|\phi|^2 = 25} \]

(c) Express \(|\phi|^2\) in terms of \(A\) and \(\theta\)

When \(\phi = Ae^{i\theta}\), we have \(\phi^* = Ae^{-i\theta}\) (where \(A > 0\) is real), so

\[ |\phi|^2 = \phi^*\phi = Ae^{-i\theta} \cdot Ae^{i\theta} = A^2 e^{0} = A^2 \]
\[ \boxed{|\phi|^2 = A^2} \]

Verification: For (b), \(|\phi| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\), so \(|\phi|^2 = 25\). ✓ For (c), we confirm that the result is independent of the phase \(\theta\), which is consistent with the fundamental property of quantum mechanics that probabilities do not depend on the absolute phase.

B-3. Calculation of Interference Terms

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(a) Expansion of \(|\phi_1 + \phi_2|^2\)

\[ |\phi_1 + \phi_2|^2 = (\phi_1^* + \phi_2^*)(\phi_1 + \phi_2) \]
\[ = |\phi_1|^2 + \phi_1^*\phi_2 + \phi_2^*\phi_1 + |\phi_2|^2 \]

Computing each term. Since \(\phi_1 = e^{i\alpha}\), \(\phi_2 = e^{i\beta}\):

  • \(|\phi_1|^2 = e^{-i\alpha}e^{i\alpha} = 1\)
  • \(|\phi_2|^2 = e^{-i\beta}e^{i\beta} = 1\)
  • \(\phi_1^*\phi_2 = e^{-i\alpha}e^{i\beta} = e^{i(\beta - \alpha)}\)
  • \(\phi_2^*\phi_1 = e^{-i\beta}e^{i\alpha} = e^{i(\alpha - \beta)}\)

Sum of the interference terms:

\[ \phi_1^*\phi_2 + \phi_2^*\phi_1 = e^{i(\beta-\alpha)} + e^{-i(\beta-\alpha)} = 2\cos(\alpha - \beta) \]

Therefore,

\[ \boxed{|\phi_1 + \phi_2|^2 = 2 + 2\cos(\alpha - \beta)} \]

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

\[ \boxed{|\phi_1|^2 + |\phi_2|^2 = 1 + 1 = 2} \]

(c) Expressing the interference term in terms of the phase difference \(\delta = \alpha - \beta\)

\[ |\phi_1 + \phi_2|^2 - (|\phi_1|^2 + |\phi_2|^2) = (2 + 2\cos\delta) - 2 \]
\[ \boxed{\text{Interference term} = 2\cos\delta} \]

(d) Special cases

  • \(\delta = 0\) (in phase): \(|\phi_1 + \phi_2|^2 = 2 + 2\cos 0 = 2 + 2 = \boxed{4}\)
  • \(\delta = \pi\) (out of phase): \(|\phi_1 + \phi_2|^2 = 2 + 2\cos\pi = 2 - 2 = \boxed{0}\)

Verification: When \(\delta = 0\), \(\phi_1 = \phi_2 = e^{i\alpha}\) so \(|\phi_1 + \phi_2|^2 = |2e^{i\alpha}|^2 = 4\). ✓ When \(\delta = \pi\), \(\phi_2 = e^{i(\alpha-\pi)} = -e^{i\alpha} = -\phi_1\) so \(|\phi_1 + \phi_2|^2 = 0\). ✓

B-4. Probability Distributions of Bullets and Electrons

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(a) Bullet case: \(x = 0\)

\[ P_1(0) = e^{-(0-a)^2} = e^{-a^2}, \quad P_2(0) = e^{-(0+a)^2} = e^{-a^2} \]
\[ \boxed{P_{12}^{\text{bullet}}(0) = P_1(0) + P_2(0) = 2e^{-a^2}} \]

(b) Electron case: \(x = 0\)

Calculate the probability amplitudes:

\[ \phi_1(0) = e^{-(0-a)^2/2} = e^{-a^2/2}, \quad \phi_2(0) = e^{-(0+a)^2/2} = e^{-a^2/2} \]
\[ P_{12}^{\text{electron}}(0) = |\phi_1(0) + \phi_2(0)|^2 = (e^{-a^2/2} + e^{-a^2/2})^2 = (2e^{-a^2/2})^2 \]
\[ \boxed{P_{12}^{\text{electron}}(0) = 4e^{-a^2}} \]

(c) Comparison

\[ P_{12}^{\text{electron}}(0) = 4e^{-a^2}, \quad P_{12}^{\text{bullet}}(0) = 2e^{-a^2} \]
\[ \frac{P_{12}^{\text{electron}}(0)}{P_{12}^{\text{bullet}}(0)} = \frac{4e^{-a^2}}{2e^{-a^2}} = 2 \]
\[ \boxed{P_{12}^{\text{electron}}(0) = 2 \times P_{12}^{\text{bullet}}(0) > P_{12}^{\text{bullet}}(0)} \]

The electron case is larger. This is because at \(x = 0\) (the midpoint between the two slits), the probability amplitudes overlap in phase, resulting in constructive interference.

Verification: \(|\phi_1|^2 = (e^{-a^2/2})^2 = e^{-a^2} = P_1(0)\), which is consistent with the problem setup. ✓

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

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

\[ e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0 = \boxed{-1} \]

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

\[ e^{i\pi/2} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = 0 + i = \boxed{i} \]

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

\[ e^{i\pi/4} = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \]
\[ \boxed{\text{Real part} = \frac{\sqrt{2}}{2}, \quad \text{Imaginary part} = \frac{\sqrt{2}}{2}} \]

(d) \(|e^{i\theta}|^2\)

\[ |e^{i\theta}|^2 = e^{-i\theta} \cdot e^{i\theta} = e^{0} = 1 \]

Alternatively, \(|e^{i\theta}|^2 = \cos^2\theta + \sin^2\theta = 1\).

\[ \boxed{|e^{i\theta}|^2 = 1 \quad (\text{independent of } \theta)} \]

Verification: (a) is the famous Euler identity \(e^{i\pi} + 1 = 0\). (d) corresponds to the fact that a point on the unit circle has absolute value 1. ✓

B-6. Wave Intensity and Interference

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(a) Expressing \(h_1 + h_2\) in product form

Using the sum-to-product formula \(\cos P + \cos Q = 2\cos\!\left(\frac{P+Q}{2}\right)\cos\!\left(\frac{P-Q}{2}\right)\).

Let \(P = \omega t\), \(Q = \omega t + \delta\):

\[ h_1 + h_2 = A(\cos\omega t + \cos(\omega t + \delta)) \]
\[ = 2A\cos\!\left(\frac{\omega t + \omega t + \delta}{2}\right)\cos\!\left(\frac{\omega t - (\omega t + \delta)}{2}\right) \]
\[ \boxed{h_1 + h_2 = 2A\cos\!\left(\frac{\delta}{2}\right)\cos\!\left(\omega t + \frac{\delta}{2}\right)} \]

(b) When \(\delta = 0\)

\[ h_1 + h_2 = 2A\cos(0)\cos(\omega t) = 2A\cos(\omega t) \]

The amplitude is \(\boxed{2A}\)

(c) When \(\delta = \pi\)

\[ h_1 + h_2 = 2A\cos\!\left(\frac{\pi}{2}\right)\cos\!\left(\omega t + \frac{\pi}{2}\right) = 2A \cdot 0 \cdot \cos\!\left(\omega t + \frac{\pi}{2}\right) \]
\[ \boxed{h_1 + h_2 = 0} \]

Verification: (b) In-phase waves interfere constructively, doubling the amplitude. (c) Out-of-phase waves cancel completely. Physically reasonable. ✓

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

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(a) Energy of the ground state and first excited state

\[ E_1 = -\frac{13.6\ \mathrm{eV}}{1^2} = \boxed{-13.6\ \mathrm{eV}} \]
\[ E_2 = -\frac{13.6\ \mathrm{eV}}{2^2} = -\frac{13.6}{4}\ \mathrm{eV} = \boxed{-3.4\ \mathrm{eV}} \]

(b) Energy of the emitted photon

\[ \Delta E = E_2 - E_1 = (-3.4) - (-13.6) = \boxed{10.2\ \mathrm{eV}} \]

(c) Frequency of the photon

First, convert eV to J:

\[ \Delta E = 10.2\ \mathrm{eV} \times 1.60 \times 10^{-19}\ \mathrm{J/eV} = 16.32 \times 10^{-19}\ \mathrm{J} = 1.632 \times 10^{-18}\ \mathrm{J} \]
\[ \nu = \frac{\Delta E}{h} = \frac{1.632 \times 10^{-18}}{6.63 \times 10^{-34}} = 2.46 \times 10^{15}\ \mathrm{Hz} \]
\[ \boxed{\nu \approx 2.46 \times 10^{15}\ \mathrm{Hz}} \]

(d) Does it fall within the visible light range?

The frequency range of visible light is \(4.3 \times 10^{14}\ \mathrm{Hz}\) to \(7.5 \times 10^{14}\ \mathrm{Hz}\).

\(\nu \approx 2.46 \times 10^{15}\ \mathrm{Hz}\) is greater than the upper limit of this range, \(7.5 \times 10^{14}\ \mathrm{Hz}\).

\[ \boxed{\text{It does not fall within the visible light range (it corresponds to ultraviolet radiation).}} \]

Verification: It is known that the hydrogen Lyman series (\(n \geq 2 \to n = 1\)) lies in the ultraviolet region, so the result is reasonable. Checking the wavelength: \(\lambda = c/\nu = 3.00 \times 10^8 / 2.46 \times 10^{15} \approx 122\ \mathrm{nm}\), which agrees with the Lyman-\(\alpha\) line (121.6 nm). ✓

B-8. Superposition of Probability Amplitudes — Numerical Example

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(a) \(P_1 = |\phi_1|^2\)

\[ P_1 = \left|\frac{1}{\sqrt{3}}\right|^2 = \boxed{\frac{1}{3}} \]

(b) \(P_2 = |\phi_2|^2\)

\[ P_2 = \left|\sqrt{\frac{2}{3}}\, e^{i\pi/3}\right|^2 = \frac{2}{3} \cdot |e^{i\pi/3}|^2 = \frac{2}{3} \cdot 1 = \boxed{\frac{2}{3}} \]

(c) Verification of the normalization condition

\[ P_1 + P_2 = \frac{1}{3} + \frac{2}{3} = 1 \quad \checkmark \]
\[ \boxed{P_1 + P_2 = 1 \text{ is satisfied.}} \]

(d) Real and imaginary parts of \(\phi_2\)

\[ \phi_2 = \sqrt{\frac{2}{3}}\, e^{i\pi/3} = \sqrt{\frac{2}{3}}\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) = \sqrt{\frac{2}{3}}\left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) \]
\[ \boxed{\text{Real part} = \frac{1}{2}\sqrt{\frac{2}{3}} = \frac{1}{\sqrt{6}}, \quad \text{Imaginary part} = \frac{\sqrt{3}}{2}\sqrt{\frac{2}{3}} = \frac{1}{\sqrt{2}}} \]

Verification: \(|\phi_2|^2 = \left(\frac{1}{\sqrt{6}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{6} + \frac{1}{2} = \frac{1}{6} + \frac{3}{6} = \frac{4}{6} = \frac{2}{3}\). ✓

Medium

M-1. Comparison of Classical and Quantum Probability

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(a) Why \(P_{12} = P_1 + P_2\) holds for bullets

Bullets are macroscopic particles that always pass through one slit or the other when traversing a double slit. The event of passing through slit 1 and the event of passing through slit 2 are mutually exclusive events (they cannot occur simultaneously). Therefore, the classical addition rule of probability applies directly, giving

\[ P_{12}(x) = P_1(x) + P_2(x) \]

When a bullet passes through one slit, whether the other slit is open or not does not affect the bullet's trajectory, so the probability distributions from each slit can be independently added together.

(b) Why \(P_{12} \neq P_1 + P_2\) for electrons

In the case of electrons, probability amplitudes (complex numbers) \(\phi_1(x)\), \(\phi_2(x)\) are assigned to the process of passing through each slit. According to the rules of quantum mechanics, one must first add the probability amplitudes for indistinguishable paths and then take the squared modulus:

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

Expanding this gives:

\[ P_{12} = |\phi_1|^2 + |\phi_2|^2 + \underbrace{\phi_1^*\phi_2 + \phi_2^*\phi_1}_{\text{interference term}} \]
\[ = P_1 + P_2 + 2\,\mathrm{Re}(\phi_1^*\phi_2) \]

Since the interference term \(2\,\mathrm{Re}(\phi_1^*\phi_2)\) is generally not zero, \(P_{12} \neq P_1 + P_2\). This interference term can be positive or negative depending on the position \(x\), producing a pattern of bright and dark fringes (interference pattern) on the screen. Although each electron is detected as a single particle, when many electrons are accumulated, this interference pattern emerges statistically.

(c) If probability amplitudes were restricted to real numbers

Interference terms still appear even when amplitudes are real. When \(\phi_1, \phi_2\) are real:

\[ \phi_1^*\phi_2 + \phi_2^*\phi_1 = 2\phi_1\phi_2 \]

This is not necessarily zero, so interference itself still occurs.

However, a serious limitation arises. With real probability amplitudes, the "phase difference" between two amplitudes can only take 2 values: \(0\) (same sign: \(\phi_1\phi_2 > 0\)) or \(\pi\) (opposite sign: \(\phi_1\phi_2 < 0\)). Consequently, the interference term can only be \(+2|\phi_1||\phi_2|\) (constructive interference) or \(-2|\phi_1||\phi_2|\) (destructive interference), making it impossible to reproduce a smooth interference pattern with continuously varying phase differences.

In the complex case, the phase difference \(\delta\) can vary continuously from \(0\) to \(2\pi\), and the interference term \(2|\phi_1||\phi_2|\cos\delta\) varies smoothly from \(-2|\phi_1||\phi_2|\) to \(+2|\phi_1||\phi_2|\). This allows for a correct description of the continuous interference fringes observed in experiments.

M-2. Light Quantum Hypothesis and the Photoelectric Effect

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(a) Maximum Kinetic Energy \(K_{\max}\)

The energy of a single photon \(h\nu\) is transferred to an electron in the metal. The minimum energy required for an electron to escape from the metal surface is the work function \(W\). From conservation of energy:

\[ h\nu = W + K_{\max} \]
\[ \boxed{K_{\max} = h\nu - W} \]

(b) Threshold Frequency \(\nu_0\)

The condition for an electron to barely escape is \(K_{\max} = 0\), so:

\[ h\nu_0 - W = 0 \]
\[ \boxed{\nu_0 = \frac{W}{h}} \]

(c) Contradiction with Classical Theory

In classical wave theory, the energy of light is proportional to the square of the amplitude (i.e., the intensity) and does not depend on frequency. Therefore, the classical assumption is:

"By increasing the intensity of light, one can impart an arbitrarily large amount of energy to an electron, regardless of frequency."

However, experiments show that light with \(\nu < \nu_0\) cannot eject electrons no matter how much the intensity is increased. This is because the energy of light is transferred to electrons in discrete units (photons) of \(h\nu\), and if the energy of a single photon \(h\nu\) is less than \(W\), no number of photons can cause a single electron to escape (since each photon interacts with an electron independently). This contradicts the classical assumption that "energy can be accumulated continuously."

(d) Numerical Calculation

Photon energy:

\[ E = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34})(3.00 \times 10^8)}{400 \times 10^{-9}} = \frac{19.89 \times 10^{-26}}{4.00 \times 10^{-7}} = 4.97 \times 10^{-19}\ \mathrm{J} \]

Converting to eV:

\[ E = \frac{4.97 \times 10^{-19}}{1.60 \times 10^{-19}}\ \mathrm{eV} \approx 3.11\ \mathrm{eV} \]

Maximum kinetic energy:

\[ K_{\max} = E - W = 3.11 - 2.3 = \boxed{0.8\ \mathrm{eV}} \]

Verification: Light with wavelength 400 nm is violet light with energy of approximately 3.1 eV. A work function of 2.3 eV is close to values for metals such as cesium. \(K_{\max} \approx 0.8\ \mathrm{eV}\) is a reasonable value. ✓

M-3. Models in Physics and Falsifiability

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(a) What it means for Newtonian mechanics to be a "hypothesis"

That Newtonian mechanics is a "hypothesis" means that it is not the "truth" of nature, but merely "the best model that does not currently contradict experiment." If refuted by experiment, it is replaced by a better model.

  • Successful domain: Celestial motion (planetary orbits, lunar motion, etc.). Newton's equations of motion and the law of universal gravitation can predict planetary positions with high precision. The discovery of Neptune was based on predictions from Newtonian mechanics.

  • Domain of failure: Phenomena at the atomic scale. For example, the discreteness of spectral lines of the hydrogen atom and the interference pattern in the double-slit experiment with electrons cannot be explained by Newtonian mechanics. Furthermore, the precise value of Mercury's perihelion precession cannot be reproduced by Newtonian mechanics, requiring general relativity.

(b) Why we call quantum mechanics a "hypothesis"

For over 100 years, the predictions of quantum mechanics have been confirmed in every experiment across atomic physics, solid-state physics, particle physics, and more. Nevertheless, the reasons for still calling it a "hypothesis" are as follows:

Quantum mechanics and general relativity (the model of gravity) are each extremely accurate within their respective domains of applicability, but in situations where both must be applied simultaneously (the center of black holes, the beginning of the universe, and other regimes where gravity is strong at extremely small scales), they contradict each other. A "quantum gravity" model that unifies both has not yet been found. Therefore, quantum mechanics in its current form may be incomplete, and it may eventually be situated as an approximation to a more comprehensive model. No matter how successful it has been, as long as the possibility of being refuted in principle cannot be excluded, it remains a "hypothesis."

(c) What falsifiability is

Falsifiability is the property that, for a given claim, one can clearly define "what observational result would allow us to judge that claim to be wrong." A scientific hypothesis must be falsifiable.

The statement "tomorrow's weather will be sunny, rainy, or cloudy" covers all conceivable weather conditions, so no observational result can contradict it. Whether it is sunny, rainy, or cloudy, this statement turns out to be "correct." Since there is no way to refute it, this statement is not falsifiable and has no value as a scientific prediction.

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

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(a) Grounds as a Founder

The Light Quantum Hypothesis of 1905:

At the time, it was established through Maxwell's electromagnetism that light is a wave. However, in the photoelectric effect (the phenomenon where electrons are ejected when light strikes a metal), the experimental fact that the energy of emitted electrons depends not on the intensity of light but on its frequency could not be explained by wave theory. Einstein proposed that light consists of particles (light quanta, later called photons) possessing energy \(E = h\nu\) proportional to the frequency \(\nu\). This quantitatively explained the photoelectric effect and established the particle nature of light.

The Prediction of Stimulated Emission in 1917:

Through statistical mechanical considerations of atoms and radiation fields in thermal equilibrium, Einstein showed that in addition to absorption and spontaneous emission of light, stimulated emission (a process in which an external photon induces the emission of a photon with the same frequency, same direction, and same phase) must exist for consistency with Planck's blackbody radiation formula. This stimulated emission is the mechanism that creates a state in which all photons are aligned, and it is precisely the operating principle of the laser (LASER: Light Amplification by Stimulated Emission of Radiation), realized in 1960.

(b) Reasons for Becoming a Critic

The core of quantum mechanics, completed in the 1920s, is a probabilistic, non-deterministic worldview in which "physical quantities do not have definite values until measured" and "only probabilities can be predicted." Einstein fundamentally refused to accept this worldview.

The phrase "God does not play dice (Gott würfelt nicht)" expressed Einstein's conviction that deterministic laws must underlie nature. While Einstein acknowledged that quantum mechanics yields correct experimental predictions, he believed it to be an "incomplete" description, and that a deeper (deterministic) model must exist that determines the values of physical quantities even before measurement. Einstein's position was that probabilities appear only because we are ignoring variables we do not yet know about (hidden variables).

(c) The EPR Paradox and Bell's Theorem

In the 1935 EPR paper, Einstein argued for the incompleteness of quantum mechanics. In the correlations between two distant particles (quantum entanglement), the fact that a measurement on one particle appears to instantaneously determine the result for the other led him to argue that hidden variables must exist that fix the values before measurement. In 1964, Bell derived Bell's inequality as a statistical constraint following from the assumption of hidden variables. If hidden variables exist, the correlations between measurement results must satisfy this inequality. Subsequent experiments (the Aspect experiment (1982), among others) confirmed that Bell's inequality is violated, thereby ruling out the local hidden variable models that Einstein had envisioned.

M-5. Interference of Probability Amplitudes — Quantitative Analysis

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(a) Combined probability \(P_{12}(x)\)

\[ \phi_1 + \phi_2 = Ae^{ikr_1} + Ae^{ikr_2} = Ae^{ikr_1}\left(1 + e^{ik(r_2 - r_1)}\right) \]

Taking the absolute value squared:

\[ P_{12} = |\phi_1 + \phi_2|^2 = A^2 \left|1 + e^{-ik\Delta r}\right|^2 \]

Here we defined \(\Delta r = r_1 - r_2\), so \(r_2 - r_1 = -\Delta r\).

\[ \left|1 + e^{-ik\Delta r}\right|^2 = (1 + e^{ik\Delta r})(1 + e^{-ik\Delta r}) = 1 + e^{ik\Delta r} + e^{-ik\Delta r} + 1 \]
\[ = 2 + 2\cos(k\Delta r) \]

Therefore:

\[ \boxed{P_{12}(x) = 2A^2\left[1 + \cos(k\Delta r(x))\right] = 4A^2\cos^2\!\left(\frac{k\Delta r(x)}{2}\right)} \]

Alternative derivation (direct expansion):

\[ P_{12} = |\phi_1|^2 + |\phi_2|^2 + \phi_1^*\phi_2 + \phi_2^*\phi_1 \]
\[ = A^2 + A^2 + A^2 e^{-ikr_1}e^{ikr_2} + A^2 e^{-ikr_2}e^{ikr_1} \]
\[ = 2A^2 + A^2\left(e^{ik(r_2-r_1)} + e^{-ik(r_2-r_1)}\right) = 2A^2 + 2A^2\cos(k(r_2-r_1)) \]
\[ = 2A^2(1 + \cos(k\Delta r)) \quad (\text{since } r_2 - r_1 = -\Delta r \text{, we have } \cos(k(r_2-r_1)) = \cos(k\Delta r)) \]

The result is the same.

(b) Condition for maxima

\(P_{12}\) is maximized when \(\cos(k\Delta r) = 1\), i.e.:

\[ k\Delta r = 2n\pi \quad (n = 0, \pm 1, \pm 2, \ldots) \]
\[ \boxed{\Delta r = \frac{2n\pi}{k} \quad (n = 0, \pm 1, \pm 2, \ldots)} \]

In this case:

\[ \boxed{P_{12}^{\max} = 2A^2(1 + 1) = 4A^2} \]

(c) Condition for minima (zero)

\(P_{12} = 0\) when \(\cos(k\Delta r) = -1\), i.e.:

\[ k\Delta r = (2n+1)\pi \quad (n = 0, \pm 1, \pm 2, \ldots) \]
\[ \boxed{\Delta r = \frac{(2n+1)\pi}{k} \quad (n = 0, \pm 1, \pm 2, \ldots)} \]

(d) Rewriting in terms of \(\lambda = 2\pi/k\)

From \(k = 2\pi/\lambda\), we get \(2\pi/k = \lambda\).

Constructive interference (maximum) condition:

\[ \Delta r = \frac{2n\pi}{k} = n\lambda \]
\[ \boxed{\Delta r = n\lambda \quad (n = 0, \pm 1, \pm 2, \ldots)} \]

Destructive interference (minimum) condition:

\[ \Delta r = \frac{(2n+1)\pi}{k} = \left(n + \frac{1}{2}\right)\lambda \]
\[ \boxed{\Delta r = \left(n + \frac{1}{2}\right)\lambda \quad (n = 0, \pm 1, \pm 2, \ldots)} \]

Verification: Constructive interference occurs when the path difference is an integer multiple of the wavelength, and destructive interference occurs when it is a half-integer multiple. This is consistent with the classical wave interference conditions. Also, \(P_{12}^{\max} = 4A^2 = 4 \times |\phi_1|^2\), which is 4 times the probability \(A^2\) from each individual slit. This corresponds to the fact that when the amplitude doubles, the probability quadruples. ✓

Advanced

A-1. Indispensability of Complex Probability Amplitudes

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(a) Geometric Condition for Real Numbers

When both \(\phi_1\) and \(\phi_2\) are real, the normalization condition is:

\[ |\phi_1|^2 + |\phi_2|^2 = \phi_1^2 + \phi_2^2 = 1 \]

This represents a point on the unit circle in the \((\phi_1, \phi_2)\) plane. That is, \((|\phi_1|, |\phi_2|)\) satisfies the condition of being a point on the unit circle in the first quadrant (\(|\phi_1|^2 + |\phi_2|^2 = 1\), \(|\phi_1| \geq 0\), \(|\phi_2| \geq 0\)).

However, since \(\phi_1, \phi_2\) themselves can have signs, the degrees of freedom consist of only one angular parameter (which can be written as \(\phi_1 = \cos\theta\), \(\phi_2 = \pm\sin\theta\)) plus the choice of sign (\(+\) or \(-\)) for each amplitude.

(b) Increase of Degrees of Freedom for Complex Numbers

In the complex case, we can write \(\phi_1 = |\phi_1|e^{i\alpha}\), \(\phi_2 = |\phi_2|e^{i\beta}\). Counting the independent real parameters:

  • \(|\phi_1|\): Due to the normalization condition \(|\phi_1|^2 + |\phi_2|^2 = 1\), once \(|\phi_1|\) is determined, \(|\phi_2|\) is also determined → 1 parameter
  • \(\alpha\): 1 parameter
  • \(\beta\): 1 parameter

Total: 3 parameters. However, since the overall phase \(e^{i\gamma}\) (\(\phi_1 \to \phi_1 e^{i\gamma}\), \(\phi_2 \to \phi_2 e^{i\gamma}\)) is physically unobservable, we subtract 1 parameter.

Therefore, the number of physically independent parameters is \(3 - 1 = 2\).

For real numbers: \(|\phi_1|\) (1 parameter, constrained by normalization) and the choice of signs. Effectively, writing \(\phi_1 = \cos\theta\), \(\phi_2 = \sin\theta\) gives 1 parameter (\(\theta\)). Even including the sign freedom, the phase difference can only take discrete values of \(0\) or \(\pi\), so there is only 1 continuous parameter.

For complex numbers, the physical degrees of freedom are 2 (for example, \(|\phi_1|\) and the relative phase \(\alpha - \beta\)).

\[ \boxed{\text{The degrees of freedom increase by one (the relative phase } \alpha - \beta \text{ is added as a new continuous degree of freedom).}} \]

(c) Restrictions on Interference Patterns for Real Numbers

For real probability amplitudes, the interference term is:

\[ \phi_1^*\phi_2 + \phi_2^*\phi_1 = 2\phi_1\phi_2 \]

Since \(\phi_1(x)\) and \(\phi_2(x)\) take real values at each position \(x\), the sign of their product \(\phi_1(x)\phi_2(x)\) is positive when both have the same sign and negative when they have opposite signs. Thus the interference term:

\[ 2\phi_1\phi_2 = \pm 2|\phi_1||\phi_2| \]

can only take two values (at each point, it is either constructive or destructive interference—a binary choice).

On the other hand, the interference fringes observed experimentally exhibit a continuous pattern where the probability varies smoothly as a \(\cos\) function depending on position \(x\). With complex probability amplitudes, the phase difference \(\delta(x) = k(r_1(x) - r_2(x))\) varies continuously with position \(x\), and the interference term becomes \(2|\phi_1||\phi_2|\cos\delta(x)\), which varies smoothly from \(-2|\phi_1||\phi_2|\) to \(+2|\phi_1||\phi_2|\).

In the real case, the interference term can only produce discontinuous patterns (or patterns that switch discretely across zero), making it impossible to reproduce the smooth \(\cos\)-type interference fringes observed experimentally.

(d) Summary of Physical Significance

The physical significance of probability amplitudes being complex numbers in quantum mechanics lies in possessing the degree of freedom of a continuous phase. The phase of a complex number can vary continuously from \(0\) to \(2\pi\), which allows the interference between probability amplitudes of two paths to vary smoothly from constructive to destructive through \(\cos\delta\). This continuous phase degree of freedom enables the smooth interference fringes observed experimentally, phase rotation in time evolution (\(e^{-iEt/\hbar}\)), and non-classical correlations in quantum entanglement. With real numbers, the phase difference is restricted to discrete values of \(0\) or \(\pi\), making it impossible to reproduce the rich predictions of quantum mechanics.

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

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(a) Limits of the Boltzmann Distribution

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

When \(T \to \infty\):

\[ \frac{E_2 - E_1}{k_B T} \to 0 \quad \Longrightarrow \quad \frac{N_2}{N_1} \to e^0 = 1 \]
\[ \boxed{T \to \infty: \quad N_2/N_1 \to 1} \]

Physical meaning: In the high-temperature limit, the thermal energy \(k_B T\) is much larger than the energy difference \(E_2 - E_1\), so the ground state and excited state are occupied nearly equally (equipartition).

When \(T \to 0\):

\[ \frac{E_2 - E_1}{k_B T} \to +\infty \quad \Longrightarrow \quad \frac{N_2}{N_1} \to e^{-\infty} = 0 \]
\[ \boxed{T \to 0: \quad N_2/N_1 \to 0} \]

Physical meaning: At absolute zero, all atoms fall into the lowest energy state (ground state), and the population of the excited state becomes zero.

(b) Deriving \(\rho(\nu)\) from the Equilibrium Condition

Thermal equilibrium condition:

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

Solving for \(\rho(\nu)\):

\[ B'\rho(\nu) N_1 - B\rho(\nu) N_2 = A N_2 \]
\[ \rho(\nu)(B' N_1 - B N_2) = A N_2 \]
\[ \rho(\nu) = \frac{A N_2}{B' N_1 - B N_2} = \frac{A}{B'(N_1/N_2) - B} \]

Substituting the Boltzmann distribution \(N_1/N_2 = \exp\!\left(\frac{E_2 - E_1}{k_B T}\right)\). Since \(E_2 - E_1 = h\nu\):

\[ \boxed{\rho(\nu) = \frac{A}{B'\, e^{h\nu/(k_B T)} - B}} \]

(c) Comparison with Planck's Formula

Planck's blackbody radiation formula:

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

Comparing with the result from (b). For both to agree at all temperatures \(T\), the structure of the denominators must match.

The denominator of the result from (b) is \(B' e^{h\nu/(k_BT)} - B\), while the denominator of Planck's formula (rewritten in a form divided by \(A\)) corresponds to \((A \cdot c^3/(8\pi h\nu^3))(e^{h\nu/(k_BT)} - 1)\).

For direct comparison, rewrite (b) as:

\[ \rho(\nu) = \frac{A/B}{(B'/B)\, e^{h\nu/(k_B T)} - 1} \]

Planck's formula:

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

For both to agree at all \(T\):

  1. The coefficients of \(e^{h\nu/(k_BT)}\) in the denominators must match: \(B'/B = 1\), i.e.,
\[ \boxed{B' = B} \]
  1. The numerators must match:
\[ \frac{A}{B} = \frac{8\pi h\nu^3}{c^3} \]
\[ \boxed{\frac{A}{B} = \frac{8\pi h\nu^3}{c^3}} \]

Verification (the \(T \to \infty\) limit): When \(T \to \infty\), \(e^{h\nu/(k_BT)} \approx 1 + h\nu/(k_BT)\), so:

Result from (b): \(\rho(\nu) \approx \frac{A}{B'(1 + h\nu/(k_BT)) - B} = \frac{A}{(B'-B) + B' h\nu/(k_BT)}\)

With \(B' = B\): \(\rho(\nu) \approx \frac{A}{B \cdot h\nu/(k_BT)} = \frac{A k_B T}{B h\nu}\)

Planck's formula: \(\rho(\nu) \approx \frac{8\pi h\nu^3}{c^3} \cdot \frac{k_BT}{h\nu} = \frac{8\pi\nu^2 k_BT}{c^3}\)

This is the Rayleigh-Jeans law, consistent with \(A/B = 8\pi h\nu^3/c^3\). ✓

(d) Case Without Stimulated Emission (\(B = 0\))

Setting \(B = 0\), the equilibrium condition becomes:

\[ B'\rho(\nu) N_1 = A N_2 \]
\[ \rho(\nu) = \frac{A N_2}{B' N_1} = \frac{A}{B'} \cdot \frac{N_2}{N_1} = \frac{A}{B'}\, e^{-h\nu/(k_BT)} \]
\[ \rho(\nu) = \frac{A}{B'}\, e^{-h\nu/(k_B T)} \]

This has the form of Wien's radiation law and contradicts Planck's formula:

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

Specifically:

  1. The \(\nu^3\) factor is missing. In the \(B = 0\) result, the frequency dependence of \(\rho(\nu)\) is only \(e^{-h\nu/(k_BT)}\), lacking the \(\nu^3\) prefactor contained in Planck's formula (assuming \(A/B'\) is independent of \(\nu\)).

  2. Disagreement in the low-frequency (high-temperature) limit. When \(h\nu \ll k_BT\), the \(B = 0\) result gives \(\rho(\nu) \approx A/B'\) (a constant), whereas Planck's formula (and the experimentally confirmed Rayleigh-Jeans law) gives \(\rho(\nu) \propto \nu^2 T\).

  3. The structure of the denominator differs. The denominator of Planck's formula is \(e^{h\nu/(k_BT)} - 1\), whereas the \(B = 0\) case yields a simple exponential \(e^{-h\nu/(k_BT)}\), with the \(-1\) term absent.

Therefore, stimulated emission (\(B \neq 0\)) is logically necessary from the requirement of thermal equilibrium. Since Planck's blackbody radiation formula is experimentally correct, thermal equilibrium cannot be achieved with absorption and spontaneous emission alone; without stimulated emission, detailed balance between atoms and the radiation field cannot hold. Einstein deductively derived the existence of stimulated emission as a new physical process solely from the requirement of thermodynamic consistency.

Verification: The \(B = 0\) result \(\rho \propto e^{-h\nu/(k_BT)}\) corresponds to Wien's radiation law, which is known to be a good approximation to Planck's formula only at high frequencies (\(h\nu \gg k_BT\)). It breaks down at low frequencies, so \(B = 0\) cannot hold in general. ✓