Quantum Physics — CheriMathLab

Key Concepts & Laws

The strange, beautiful rules governing nature at the smallest scales.

Wave-Particle Duality

de Broglie wavelength: $$\lambda = \frac{h}{p} = \frac{h}{mv}$$ All matter has wave properties; all radiation has particle properties. The double-slit experiment demonstrates both.

Photoelectric Effect

Einstein's equation: $$KE_{\max} = hf - \phi$$ Threshold frequency: $f_0 = \phi/h$
Light is quantized into photons of energy $E = hf$.

Uncertainty Principle

Heisenberg: $$\Delta x \cdot \Delta p \ge \frac{\hbar}{2}$$ Energy-time: $\Delta E \cdot \Delta t \ge \frac{\hbar}{2}$
Fundamental limit, not a measurement limitation.

Schrödinger Equation

Time-independent: $$-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi$$ $|\psi|^2$ gives probability density.
Solutions must be normalized, continuous, and finite.

Hydrogen Atom

Energy levels: $$E_n = -\frac{13.6\;\text{eV}}{n^2}$$ Quantum numbers: $n, l, m_l, m_s$
Orbital angular momentum: $L = \hbar\sqrt{l(l+1)}$

Quantum Tunneling

Particles can penetrate classically forbidden barriers. Transmission probability depends exponentially on barrier width and height. Essential for nuclear fusion, tunnel diodes, and STMs.

Spin & Pauli Exclusion

Intrinsic angular momentum: $S = \hbar\sqrt{s(s+1)}$
Electrons: $s = \tfrac{1}{2}$, $m_s = \pm\tfrac{1}{2}$
No two identical fermions can share the same quantum state.

Quantum Entanglement

Entangled particles share a quantum state — measuring one instantly determines the other regardless of distance. Bell's theorem rules out local hidden variables. Basis for quantum computing and cryptography.

Problems

12 problems exploring the quantum realm — from photons to entanglement.

Easy

Ultraviolet light of wavelength 250 nm strikes a metal surface with work function $\phi = 3.5\;\text{eV}$. (a) Find the energy of each photon in eV. (b) Find the maximum kinetic energy of emitted electrons. (c) Find the stopping potential. (d) What is the threshold wavelength for this metal?

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$E = hc/\lambda$. Use $1\;\text{eV} = 1.6 \times 10^{-19}\;\text{J}$ or the shortcut $E(\text{eV}) = 1240/\lambda(\text{nm})$. Stopping potential: $eV_0 = KE_{\max}$. Threshold: $\lambda_0 = hc/\phi$.

Easy

Calculate the de Broglie wavelength of (a) an electron accelerated through 100 V, (b) a baseball (145 g) thrown at 40 m/s, and (c) a neutron with kinetic energy 0.025 eV (thermal neutron). Which of these would show observable wave behavior, and why?

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$\lambda = h/p = h/\sqrt{2mKE}$. For the electron: $KE = eV = 100\;\text{eV}$. Wave effects are observable when $\lambda$ is comparable to the size of structures the particle interacts with (crystal lattices ~0.1 nm).

Medium

An electron is confined to a one-dimensional box of length $L = 0.5\;\text{nm}$ (approximate atomic size). (a) Find the energies of the first three levels. (b) Find the wavelength of the photon emitted in a transition from $n = 3$ to $n = 1$. (c) In what region of the electromagnetic spectrum does this photon lie?

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Particle in a box: $E_n = \frac{n^2 h^2}{8mL^2} = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$. Photon energy: $\Delta E = E_3 - E_1$. Then $\lambda = hc/\Delta E$.

Medium

The position of an electron is measured to an accuracy of $\Delta x = 0.01\;\text{nm}$. (a) What is the minimum uncertainty in its momentum? (b) What is the corresponding minimum uncertainty in its velocity? (c) Compare this velocity uncertainty with the speed of light. What does this tell you about the electron?

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$\Delta p \ge \frac{\hbar}{2\Delta x}$. Then $\Delta v = \Delta p / m_e$. If $\Delta v$ approaches $c$, relativistic treatment is needed and the non-relativistic uncertainty principle alone is insufficient.

Hard

For the hydrogen atom, calculate: (a) the energy needed to ionize hydrogen from its ground state (in eV and J), (b) the wavelength of the photon emitted when an electron transitions from $n = 4$ to $n = 2$ (what color is this?), (c) the orbital radius and speed of the electron in the ground state using the Bohr model.

Show Hint

Ionization from ground state: $E = 13.6\;\text{eV}$. For transitions: $\frac{1}{\lambda} = R_H\!\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ where $R_H = 1.097 \times 10^7\;\text{m}^{-1}$. Bohr radius: $a_0 = 0.0529\;\text{nm}$. Speed: $v = \frac{e^2}{4\pi\varepsilon_0 \hbar} \cdot \frac{1}{n}$.

Hard

An electron with energy $E = 5\;\text{eV}$ encounters a rectangular potential barrier of height $V_0 = 8\;\text{eV}$ and width $a = 0.5\;\text{nm}$. (a) Calculate the decay constant $\kappa$ inside the barrier. (b) Estimate the transmission probability using the WKB approximation. (c) By what factor does the transmission probability change if the barrier width is doubled?

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Decay constant: $\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$. Transmission: $T \approx e^{-2\kappa a}$. If width doubles, the exponent doubles, so $T' = T^2$ — exponential sensitivity to barrier width.

Medium

A quantum harmonic oscillator has angular frequency $\omega = 5 \times 10^{14}\;\text{rad/s}$. (a) Find the zero-point energy. (b) Find the energy spacing between adjacent levels. (c) At what temperature is the thermal energy $k_B T$ equal to the energy spacing? (d) Below this temperature, why do quantum effects become important?

Show Hint

Energy levels: $E_n = (n + \tfrac{1}{2})\hbar\omega$. Zero-point energy $= \tfrac{1}{2}\hbar\omega$. Spacing $= \hbar\omega$. Set $k_B T = \hbar\omega$ to find $T$. Below this $T$, the system "freezes" into the ground state.

Hard

List all possible quantum states $(n, l, m_l, m_s)$ for $n = 3$ in hydrogen. (a) How many total states are there? (b) What are the possible orbital shapes ($l$ values) and their names? (c) If a magnetic field is applied, how many distinct energy levels does the $n = 3$ shell split into (ignoring spin-orbit coupling)?

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For $n = 3$: $l = 0, 1, 2$. For each $l$: $m_l = -l$ to $+l$. For each $(n, l, m_l)$: $m_s = \pm\tfrac{1}{2}$. Total states $= 2n^2 = 18$. In a magnetic field, each distinct $m_l$ gives a different energy (normal Zeeman effect).

Advanced

In a Stern-Gerlach experiment, silver atoms (one unpaired electron, spin-$\tfrac{1}{2}$) pass through an inhomogeneous magnetic field. (a) How many beams emerge and why? (b) If the field gradient is 100 T/m over a path length of 10 cm, and atoms move at 500 m/s, estimate the deflection of each beam. (c) What would happen with spin-1 particles?

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Two beams for spin-$\tfrac{1}{2}$ ($m_s = \pm\tfrac{1}{2}$). Force: $F = \mu_z \cdot \frac{dB}{dz}$ where $\mu_z = -g_s m_s \mu_B$. Time in field: $t = L/v$. Deflection: $d = \tfrac{1}{2}(F/m)t^2$. Spin-1 would give 3 beams ($m_s = -1, 0, +1$).

Advanced

Compton scattering: An X-ray photon of wavelength 0.05 nm scatters off a stationary electron at an angle of 90°. (a) Find the wavelength of the scattered photon. (b) Find the kinetic energy of the recoiling electron. (c) At what angle is the wavelength shift maximum, and what is that maximum shift?

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Compton formula: $\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$. Compton wavelength of electron: $\frac{h}{m_e c} = 0.00243\;\text{nm}$. At 90°: $\Delta\lambda = \frac{h}{m_e c}$. Maximum shift at 180° (backscatter): $\Delta\lambda = \frac{2h}{m_e c}$.

Advanced

Thought Experiment: In the EPR paradox, two entangled photons are produced with correlated polarizations. Alice and Bob measure polarization along different axes separated by angle $\theta$. (a) What does quantum mechanics predict for the correlation? (b) What would a local hidden variable theory predict? (c) Bell's inequality states $|S| \le 2$ for local hidden variables. Quantum mechanics predicts $|S| = 2\sqrt{2}$. What is the significance of this violation?

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QM predicts correlation $P = \cos^2\theta$. Bell's inequality is violated experimentally, ruling out local hidden variables. This means quantum entanglement involves genuine non-local correlations — not just shared classical information.

Advanced

Thought Experiment: Schrödinger's cat is in a superposition of alive and dead states until observed. (a) Write the superposition state mathematically. (b) Explain the measurement problem: why don't we see macroscopic superpositions? (c) Discuss decoherence — how does interaction with the environment effectively "collapse" the wave function? (d) How does this relate to the practical challenge of building quantum computers?

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State: $|\psi\rangle = \frac{1}{\sqrt{2}}\bigl(|\text{alive}\rangle + |\text{dead}\rangle\bigr)$. Decoherence: the cat interacts with $\sim 10^{26}$ air molecules, photons, etc., creating entanglement with the environment. This entanglement makes the off-diagonal terms of the density matrix vanish on absurdly short timescales ($\sim 10^{-39}\;\text{s}$ for a cat), yielding an effectively classical mixture.