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Wave Properties of particles. Schrodinger Equation.

Problem 6.54

Calculate the most probable de Broglie wavelength of hydrogen molecules being in thermodynamic equilibrium at room temperature.

Reveal Answer
6.54.6.54 . First, let us find the distribution of molecules over de (\lambda) dλd \lambda where Broglie wavelengths. From the relation f(v)dv=−φf(v) d v=-\varphi f(v)f(v) is Maxwell's distribution of velocities, we obtain φ(λ)=Aλ−4e−a/λ2,a=2π2ℏ2/nkT \varphi(\lambda)=A \lambda^{-4} \mathrm{e}^{-a / \lambda 2}, \quad a=2 \pi^{2} \hbar^{2} / n k T The condition dφ/dλ=0d \varphi / d \lambda=0 provides λpr=πℏ/mkT=0.09nm\lambda_{p r}=\pi \hbar / \sqrt{m k T}=0.09 \mathrm{nm}
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