The dimensional formula of Planck's constant (h) is [M¹L²T⁻¹]. It is derived from Planck's quantum relation E = hν, where E is energy (dimensions [ML²T⁻²]) and ν (nu) is frequency (dimensions [T⁻¹]). The value of Planck's constant is h = 6.626 × 10⁻³⁴ J·s, and its SI unit is Joule-second (J·s).
Dimensional formula of Planck's constant h = [M¹L²T⁻¹].
Derived from h = E/ν: [h] = [ML²T⁻²] / [T⁻¹] = [ML²T⁻¹].
Value of h = 6.626 × 10⁻³⁴ J·s (Joule-second).
Planck's constant has the same dimensional formula as angular momentum [ML²T⁻¹].
Reduced Planck's constant ℏ = h/2π = 1.055 × 10⁻³⁴ J·s.
Introduced by Max Planck in 1900; foundational to quantum mechanics.
de Broglie wavelength λ = h/p uses Planck's constant to link wave and particle properties.
Planck's quantum relation: E = hν
Where: • E = energy of a photon • h = Planck's constant • ν (nu) = frequency of radiation
Solving for h: h = E / ν
Dimensions of each quantity: • [E] = [M¹L²T⁻²] (dimensional formula of energy) • [ν] = [T⁻¹] (frequency = cycles per second = s⁻¹)
Substituting: [h] = [M¹L²T⁻²] / [T⁻¹] [h] = [M¹L²T⁻²] × [T¹] [h] = [M¹L²T⁻²⁺¹] [h] = [M¹L²T⁻¹]
Dimensional Formula of Planck's Constant = [M¹L²T⁻¹]
SI unit: h has unit J·s (Joule-second) = kg·m²·s⁻² × s = kg·m²·s⁻¹ This confirms [M¹L²T⁻¹] ✓
Value of Planck's Constant: h = 6.626 × 10⁻³⁴ J·s h = 6.626 × 10⁻³⁴ kg·m²·s⁻¹
Reduced Planck's constant (h-bar): ℏ = h / (2π) = 1.055 × 10⁻³⁴ J·s (Used in quantum mechanics, Schrödinger's equation, etc.)
Historical significance: • Introduced by Max Planck in 1900 to solve the ultraviolet catastrophe • Marks the boundary between classical and quantum physics • Extremely small value means quantum effects only matter at atomic/subatomic scales
Planck's constant appears in: • Photoelectric effect: E = hν (Einstein, 1905) • de Broglie wavelength: λ = h/p • Heisenberg uncertainty principle: Δx·Δp ≥ ℏ/2 • Bohr's model: angular momentum = nℏ • Schrödinger equation (via ℏ)
Comparison of dimensional formulas:
Planck's constant h: [ML²T⁻¹] Angular momentum L: [ML²T⁻¹]
h and angular momentum have identical dimensional formulas! This is not coincidental — Planck's constant is fundamentally a quantum of angular momentum.
de Broglie wavelength: λ = h/p [h/p] = [ML²T⁻¹] / [MLT⁻¹] = [L] ✓ (wavelength has dimensions of length)
Heisenberg Uncertainty Principle: Δx · Δp ≥ ℏ/2 [Δx·Δp] = [L][MLT⁻¹] = [ML²T⁻¹] = [ℏ] ✓
Bohr's angular momentum quantization: L = nℏ → [nℏ] = (dimensionless)[ML²T⁻¹] = [ML²T⁻¹] ✓
Photon momentum: p = h/λ = E/c [h/λ] = [ML²T⁻¹]/[L] = [MLT⁻¹] ✓ (momentum)
Einstein used Planck's constant to explain the photoelectric effect (1905), for which he won the Nobel Prize in 1921.
Photoelectric equation: KE_max = hν - φ
Where: • KE_max = maximum kinetic energy of emitted electron • h = Planck's constant = 6.626 × 10⁻³⁴ J·s • ν = frequency of incident light • φ = work function of the metal
Threshold frequency (ν₀): ν₀ = φ/h (minimum frequency needed to eject an electron)
Numerical example: For sodium metal: φ = 2.3 eV = 3.68 × 10⁻¹⁹ J Threshold frequency: ν₀ = 3.68×10⁻¹⁹ / 6.626×10⁻³⁴ = 5.55 × 10¹⁴ Hz (visible light range)
Energy of a photon of green light (λ = 550 nm): E = hc/λ = (6.626×10⁻³⁴ × 3×10⁸) / (550×10⁻⁹) = 3.62 × 10⁻¹⁹ J = 2.26 eV
Quick Summary Table:
Quantity | Formula | Dimensions Planck's constant | h = E/ν | [ML²T⁻¹] Angular momentum | L = Iω | [ML²T⁻¹] Energy | E = hν | [ML²T⁻²] Frequency | ν = 1/T | [T⁻¹] Momentum | p = mv | [MLT⁻¹]
Memory tip: Planck's constant [ML²T⁻¹] is energy [ML²T⁻²] divided by frequency [T⁻¹]. The T exponent: -2 - (-1) = -1 → [ML²T⁻¹].
Important note: h ≠ ℏ • h = 6.626 × 10⁻³⁴ J·s (Planck's constant) • ℏ = h/2π = 1.055 × 10⁻³⁴ J·s (reduced Planck's constant / Dirac constant) Both have the same dimensional formula [ML²T⁻¹].
[M¹L²T⁻¹]. Derived from h = E/ν: [ML²T⁻²] / [T⁻¹] = [ML²T⁻¹].
Joule-second (J·s) or equivalently kg·m²·s⁻¹.
h = 6.626 × 10⁻³⁴ J·s. The reduced form ℏ = h/2π = 1.055 × 10⁻³⁴ J·s.
Angular momentum [ML²T⁻¹] has the same dimensional formula as Planck's constant.
From E = hν (energy = Planck's constant × frequency), rearranging: h = E/ν gives [h] = [ML²T⁻²]/[T⁻¹] = [ML²T⁻¹].
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