Show that at limitħookgT, the following expression, h w³ e(w, T)= e exp(Bhw)-1 reduces to the classical form given by: e(w 7) = Ac^{(k₂7)u^² = ³ [ Ac²³ (²7)]. T)
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![Show that at limit ħo kgT, the following expression,
h
€(w, T) =
c exp(Bho)-1
reduces to the classical form given by:
e(w, T)= Ac ³(kgT)w²:
² = 0 [₁²³ (²+²)].
Ac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc44715e-e997-4081-aac6-58f75fd336c7%2Fb5ad1cd4-5985-457b-a6f1-aaec06c9c8c4%2Fpw6pwa4_processed.png&w=3840&q=75)
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- We are going to use Heisenberg's uncertainty principle to estimate the ground- state energy of hydrogen. In our model, the electron is confined in a one- dimensional well with a length about the size of hydrogen, so that Ax = 0.0529 nm. Estimate Ap, and then assume that the ground-state energy is roughly Ap2/2me. (Give your answer in Joules or electron-volts.)Write down the expression for the quantized energy levels, En, of a particle of mass m inan infinite square well of width L.b) State the frequency of a photon that would be emitted from a particle transitioning from then = 2 energy level to the n = 1 energy level of this infinite square well.The wavelength λmax at which the Planck distribution is a maximum can be found by solving dρ(λ,T)/dT = 0. Differentiate ρ(λ,T) with respect to T and show that the condition for the maximum can be expressed as xex − 5(ex − 1) = 0, where x = hc/λkT. There are no analytical solutions to this equation, but a numerical approach gives x = 4.965 as a solution. Use this result to confirm Wien’s law, that λmaxT is a constant, deduce an expression for the constant, and compare it to the value quoted in the text.
- (2.13) Selection rules in hydrogen Hydrogen atoms are excited (by a pulse of laser light that drives a multi-photon process) to a spe- cific configuration and the subsequent spontaneous emission is resolved using a spectrograph. Infra- red and visible spectral lines are detected only at the wavelengths 4.05 um, 1.87 µm and 0.656 µm. Explain these observations and give the values of n and l for the configurations involved in these transitions.Complete the derivation of E = Taking the derivatives we find (Use the following as necessary: k₁, K₂ K3, and 4.) +- ( ²) (²) v² = SO - #2² - = 2m so the Schrödinger equation becomes (Use the following as necessary: K₁, K₂, K3, ħ, m and p.) 亢 2mm(K² +K ² + K² v k₁ = E = = EU The quantum numbers n, are related to k, by (Use the following as necessary: n, and L₁.) лħ n₂ π²h² 2m √2m h²²/0₁ 2m X + + by substituting the wave function (x, y, z) = A sin(kx) sin(k₂y) sin(kz) into - 13³3). X What is the origin of the three quantum numbers? O the Schrödinger equation O the Pauli exclusion principle O the uncertainty principle Ⓒthe three boundary conditions 2² 7²4 = E4. 2mI need the answer as soon as possible