Quantum Mechanics: Unit Review

T = 2.90 × 10⁻³ ÷ 4.14 × 10⁻⁷ = 7.0 × 10³ K.

Classical theory predicted unlimited intensity at short wavelengths. Planck assumed oscillator energies are quantized (E = nhf), so high-frequency quanta are too large to be excited often, and the curve falls at short wavelengths.

E = (6.63 × 10⁻³⁴)(3.0 × 10¹³) = 2.0 × 10⁻²⁰ J.

E = 1240 ÷ 310 = 4.0 eV = 6.4 × 10⁻¹⁹ J.

E = 1240 ÷ 532 = 2.33 eV = 3.74 × 10⁻¹⁹ J; N = 3.0 × 10⁻³ ÷ 3.74 × 10⁻¹⁹ = 8.0 × 10¹⁵ /s.

The number of photons per second; each photon's energy stays the same.

Below a threshold frequency no electrons are emitted at any intensity. In the wave model, enough intensity (or time) should always supply enough energy, regardless of frequency.

1.6 eV = 1.6 × 1.60 × 10⁻¹⁹ = 2.6 × 10⁻¹⁹ J.

The photocurrent (it roughly doubles); the stopping voltage is unchanged.

hf = 1240 ÷ 250 = 4.96 eV; E(k,max) = 0.86 eV; V₀ = 0.86 V.

λ₀ = 1240 ÷ 3.1 = 400 nm.

Slope = h; horizontal intercept = threshold frequency f₀.

h = 4.0 × 10⁻¹⁵ × 1.60 × 10⁻¹⁹ = 6.4 × 10⁻³⁴ J·s; error = 0.23 ÷ 6.63 × 100% ≈ 3.5%.

The slope removes the effect of the unknown work function (and any constant offset), and using many points reduces random error.

h ≈ eVλ/c = (1.60 × 10⁻¹⁹)(2.6)(4.70 × 10⁻⁷) ÷ (3.00 × 10⁸) = 6.5 × 10⁻³⁴ J·s.

p = 6.63 × 10⁻³⁴ ÷ 2.0 × 10⁻¹¹ = 3.3 × 10⁻²³ kg·m/s.

Δλ = 2 × 2.43 pm = 4.86 pm; λ′ = 0.0849 nm.

The wavelength shift and its angle dependence match a collision in which a photon with momentum h/λ hits an electron, conserving energy and momentum. A classical wave would scatter with no change in wavelength.

Each photon is detected at one spot (particle); many spots build an interference pattern (wave).

1.022 MeV (two electron rest energies).

Positrons from the tracer annihilate with electrons, producing two 0.511 MeV photons travelling in opposite directions; detectors on opposite sides of a ring record them at the same moment, defining a line through the source.

λ = 6.63 × 10⁻³⁴ ÷ (9.11 × 10⁻³¹ × 3.0 × 10⁶) = 2.4 × 10⁻¹⁰ m (0.24 nm).

λ ≈ 1.226 ÷ √400 = 0.0613 nm.

The proton. λ = h/√(2mE(k)); with the same E(k), the larger mass gives larger momentum and a shorter wavelength.

λ = 1.226 ÷ √81 = 0.136 nm. sin θ = 0.136 ÷ 0.250 = 0.545; θ ≈ 33°.

They get larger (longer wavelength → larger angles).

The wavelength from the diffraction geometry (d sin θ) matched the wavelength predicted by λ = h/p to about 1%.

ΔE = −0.544 − (−3.40) = 2.86 eV; λ = 1240 ÷ 2.86 = 434 nm.

E₄ − E₁ = −0.85 − (−13.6) = 12.75 eV (about 12.8 eV).

Only orbits where a whole number of electron wavelengths fits around the circumference (2πr = nλ) form stable standing waves.

Δp ≥ 5.28 × 10⁻³⁵ ÷ 2.0 × 10⁻¹⁰ = 2.64 × 10⁻²⁵; Δv = 2.64 × 10⁻²⁵ ÷ 9.11 × 10⁻³¹ = 2.9 × 10⁵ m/s.

The probability density of finding the particle at a location.

For example the scanning tunnelling microscope or flash memory.

λ = 1240 ÷ 1.9 = 650 nm (about 653 nm).

A photon of the right energy causes an excited atom to drop to a lower level and emit a second photon identical to the first (same energy, direction and phase).

Benefit: e.g., no emissions while generating, power for remote areas. Cost: e.g., mining and manufacturing impacts, recycling at end of life, lower output in winter.

Electrons accelerated through kilovolts have much shorter wavelengths (picometres) than visible light, and resolution is limited by wavelength.

2⁴⁰ ≈ 1.1 × 10¹².

Future quantum computers could break current encryption, and data stolen now could be decrypted later.

+2/3 + 2/3 − 1/3 = +1 (the Σ⁺).

Charge: +1 → 0 + 0 = 0. Not conserved, so not allowed.

That neutrinos change type as they travel, so they have mass (Arthur McDonald, 2015 Nobel Prize, shared with Takaaki Kajita).