* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Thermal de Broglie Wavelength
X-ray fluorescence wikipedia , lookup
Scalar field theory wikipedia , lookup
Hydrogen atom wikipedia , lookup
Elementary particle wikipedia , lookup
Density matrix wikipedia , lookup
Wave function wikipedia , lookup
Quantum dot wikipedia , lookup
Quantum field theory wikipedia , lookup
Probability amplitude wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum fiction wikipedia , lookup
Renormalization group wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Coherent states wikipedia , lookup
Renormalization wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Quantum computing wikipedia , lookup
Quantum entanglement wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
De Broglie–Bohm theory wikipedia , lookup
Quantum machine learning wikipedia , lookup
Bell's theorem wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Identical particles wikipedia , lookup
History of quantum field theory wikipedia , lookup
Double-slit experiment wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum group wikipedia , lookup
Path integral formulation wikipedia , lookup
Particle in a box wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum state wikipedia , lookup
Canonical quantization wikipedia , lookup
Wave–particle duality wikipedia , lookup
Thermal de Broglie Wavelength—C.E. Mungan, Spring 2009 In this brief note I show that the quantum length, equal to the cube root of the quantum volume (which in turn is the reciprocal of the quantum concentration), given by ≡ h2 2π mkT (1) and obtained by calculating the partition function Z = L / of a 1D particle in a box (i.e., infinite square well of width L) is equal to half of the thermally averaged value of the de Broglie wavelength, ∞ 1 1 h h 1 h 1 λ = = = D(υ )dυ 2 2 p 2m υ 2m ∫0 υ (2) usually called the “thermal de Broglie wavelength” Λ. Here the Maxwell distribution of molecular speeds in an ideal gas is ⎛ m ⎞ D(υ ) = ⎜ ⎝ 2π kT ⎟⎠ 3/2 4πυ 2 e−mυ 2 /2kT . (3) Substituting Eq. (3) into (2) and changing variables to x = υ m / 2kT , we get 1 4π h ⎛ m ⎞ λ = ⎜ ⎟ 2 2m ⎝ 2π kT ⎠ 3/2 ∞ −mυ ∫ υe 0 2 /2kT dυ = 2 h2 2π mkT ∞ ∫ xe − x2 dx . (4) 0 But the indefinite integral is − 12 exp(−x 2 ) , and so substituting the limits we obtain = λ / 2 = Λ as desired. One could try to rationalize away the annoying factor of 1/2 that appears in this result by arguing that we should restrict attention to only the half of the particles that are traveling in the positive (rather than the negative) direction, but that is a rather handwaving argument.